Even in the 1700s, there'd been some specific solutions found—like Euler's collinear configuration, and Lagrange's equilateral triangle.
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They also found that the laser was more efficient and had better range when the laser was collinear with the debris and had the same inclination, or tilt.
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There are two main forms of PDS: Collinear and Transverse. Collinear PDS was introduced in a 1980 paper by A.C. Boccara, D. Fournier, et al. In collinear, two beams pass through and intersect in a medium. The pump beam heats the material and the probe beam is deflected.
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A complete quadrangle consists of four points, no three of which are collinear. In the Fano plane, the three points not on a complete quadrangle are the diagonal points of that quadrangle and are collinear. This contradicts the Fano axiom, often used as an axiom for the Euclidean plane, which states that the three diagonal points of a complete quadrangle are never collinear.
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The theorem is clearly true for three non-collinear points. We proceed by induction. Assume n > 3 and the theorem is true for n − 1\. Let P be a set of n points not all collinear.
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An angle consists of a point O (the vertex) and two non- collinear rays out from O (the sides). A triangle is given by three non- collinear points (called vertices) and their three segments AB, BC, and CA. If three points A, B, and C are non-collinear, then a plane ABC is the set of all points collinear with pairs of points on one or two of the sides of triangle ABC. If four points A, B, C, and D are non-coplanar, then a space (3-space) ABCD is the set of all points collinear with pairs of points selected from any of the four faces (planar regions) of the tetrahedron ABCD.
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Yambo can treat molecules and periodic systems (both metallic an insulating) in three dimensions (crystalline solids) two dimensions (surfaces) and one dimension (e.g., nanotubes, nanowires, polymer chains). It can also handle collinear (i.e., spin-polarized wave functions) and non-collinear (spinors) magnetic systems.
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Because most nonlinear crystals are birefringent, beams that are collinear inside a crystal may not be collinear outside of it. The phase fronts (wave vector) do not point in the same direction as the energy flow (Poynting vector) because of walk-off. The phase matching angle makes possible any gain at all (0th order). In a collinear setup, the freedom to choose the center wavelength allows a constant gain up to first order in wavelength.
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The isolation between the collinear ports is however limited by the performance of the matching structure.
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Two distinct points always determine a (straight) line. Three distinct points are either collinear or determine a unique plane. On the other hand, four distinct points can either be collinear, coplanar, or determine the entire space. Two distinct lines can either intersect, be parallel or be skew.
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In our particular case, linear equations between homogeneous point coordinates, Möbius called a permutation [Verwandtschaft] of both point spaces in particular a collineation. This signification would be changed later by Chasles to homography. Möbius’ expression is immediately comprehended when we follow Möbius in calling points collinear when they lie on the same line. Möbius' designation can be expressed by saying, collinear points are mapped by a permutation to collinear points, or in plain speech, straight lines stay straight.
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Thus, the problem of detecting collinear points can be converted to the problem of finding concurrent curves.
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Collinear dipole array on repeater for radio station JOHG-FM on Mt. Shibisan, Kagoshima, Japan In telecommunications, a collinear antenna array (sometimes colinear antenna array) is an array of dipole or quarter-wave antennas mounted in such a manner that the corresponding elements of each antenna are parallel and collinear; that is, they are located along a common axis. Collinear arrays are high gain omnidirectional antennas. Both dipoles and quarter-wavelength monopoles have an omnidirectional radiation pattern in free space when oriented vertically; they radiate equal radio power in all azimuthal directions perpendicular to the antenna, with the signal strength dropping to zero on the antenna axis. The purpose of stacking multiple antennas in a vertical collinear array is to increase the power radiated in horizontal directions and reduce the power radiated into the sky or down toward the earth, where it is wasted.
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A permutation of the seven points that carries collinear points (points on the same line) to collinear points is called a collineation or symmetry of the plane. The collineations of a geometry form a group under composition, and for the Fano plane this group (PΓL(3,2) = PGL(3,2)) has 168 elements.
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An antenna mast with four collinear directional arrays. In telecommunications, a collinear (or co-linear) antenna array is an array of dipole antennas mounted in such a manner that the corresponding elements of each antenna are parallel and aligned, that is they are located along a common line or axis.
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If the E-plane port is also matched, then half power will leave by the E-plane port. In this circumstance, there is no power 'left over' either to be reflected out of the first collinear port or to be transmitted to the other collinear port. Despite apparently being in direct communication with each other, the two collinear ports are 'magically' isolated. The isolation between the E-plane and H-plane ports is wide-band and is as perfect as is the symmetry of the device.
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A related concept in algebraic geometry is general position, whose precise meaning depends on the context. For example, in the Euclidean plane, three points in general position are not collinear. This is because the property of not being collinear is a generic property of the configuration space of three points in R2.
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For even higher gain, multiple Yagis or helicals can be mounted together to make array antennas. Vertical collinear arrays of dipoles can be used to make high gain omnidirectional antennas, in which more of the antenna's power is radiated in horizontal directions. Television and FM broadcasting stations use collinear arrays of specialized dipole antennas such as batwing antennas.
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A permutation of the Fano plane's seven points that carries collinear points (points on the same line) to collinear points is called a collineation of the plane. The full collineation group is of order 168 and is isomorphic to the group PSL(2,7) ≈ PSL(3,2), which in this special case is also isomorphic to the general linear group .
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Simply, a collineation is a one-to-one map from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. One may formalize this using various ways of presenting a projective space. Also, the case of the projective line is special, and hence generally treated differently.
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In 1767, Leonhard Euler found three families of periodic solutions in which the three masses are collinear at each instant. See Euler's three-body problem. In 1772, Lagrange found a family of solutions in which the three masses form an equilateral triangle at each instant. Together with Euler's collinear solutions, these solutions form the central configurations for the three-body problem.
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Euler showed in 1765 that in any triangle, the orthocenter, circumcenter and centroid are collinear. Reprinted in Opera Omnia, ser. I, vol. XXVI, pp.
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If four collinear points are represented in homogeneous coordinates by vectors a, b, c, d such that and , then their cross-ratio is k.
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The collinear J antenna improves the Super-J by separating the two radiating half-wave sections to optimize gain using a phasing coil. The resulting gain is closer to the optimum 3 dB over a conventional J-pole or halfwave antenna. The approximate gain in the H-plane of the Collinear J antenna is from 4.6 to 5.2 dBi (2.4 dBd to 3.1 dBd).
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Oxygen difluoride, an example of a molecule with the bent coordination geometry. In chemistry, molecules with a non-collinear arrangement of two adjacent bonds have bent molecular geometry. Certain atoms, such as oxygen, will almost always set their two (or more) covalent bonds in non-collinear directions due to their electron configuration. Water (H2O) is an example of a bent molecule, as well as its analogues.
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"Higher gain" in this case means that the antenna radiates less energy at higher and lower elevation angles and more in the horizontal directions. High-gain omnidirectional antennas are generally realized using collinear dipole arrays. These consist of multiple half-wave dipoles mounted collinearly (in a line), fed in phase. The coaxial collinear (COCO) antenna uses transposed coaxial sections to produce in-phase half-wavelength radiators.
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If we are to show that , , are collinear for concyclic , then notice that and are similar, and that and will correspond to the isogonal conjugate if we overlap the similar triangles. This means that , hence making collinear. A short proof can be constructed using cross-ratio preservation. Projecting tetrad from onto line , we obtain tetrad , and projecting tetrad from onto line , we obtain tetrad .
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For instance, in spherical geometry, where lines are represented in the standard model by great circles of a sphere, sets of collinear points lie on the same great circle. Such points do not lie on a "straight line" in the Euclidean sense, and are not thought of as being in a row. A mapping of a geometry to itself which sends lines to lines is called a collineation; it preserves the collinearity property. The linear maps (or linear functions) of vector spaces, viewed as geometric maps, map lines to lines; that is, they map collinear point sets to collinear point sets and so, are collineations.
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As there can be no communication with the E-plane port, and again considering the symmetry of the structure, then the power in this signal must be divided equally between the two collinear ports. Similarly for the E-plane port, if the matching structure eliminates any reflection from this port, then the power entering it must be divided equally between the two collinear ports. Now by reciprocity, the coupling between any pair of ports is the same in either direction (the scattering matrix is symmetric). So if the H-plane port is matched, then half the power entering either one of the collinear ports will leave by the H-plane port.
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In a convex quadrilateral, the quasiorthocenter H, the "area centroid" G, and the quasicircumcenter O are collinear in this order on the Euler line, and HG = 2GO..
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A fundamental application is that, in the plane, five points determine a conic, as long as the points are in general linear position (no three are collinear).
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Points which are collinear with respect to the homothetic center but are not homologous are said to be antihomologous, e.g., points Q and P′ in Figure 4.
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When produced, the seed parton is expected to undergo a parton shower, which may include a series of nearly-collinear splittings before the hadronization starts. Furthermore, the jet algorithm must be robust when it comes to fluctuations in the detector response. Theoretically, If a jet algorithm is not infrared and collinear safe, it can not be guaranteed that a finite cross-section can be obtained at any order of perturbation theory.
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The two known examples of point sets with fewer than n/2 ordinary lines. While the Sylvester–Gallai theorem states that an arrangement of points, not all collinear, must determine an ordinary line, it does not say how many must be determined. Let t_2(n) be the minimum number of ordinary lines determined over every set of n non-collinear points. Melchior's proof showed that t_2(n)\ge 3.
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The theorem was further generalized by Dao Thanh Oai. The generalization as follows: First generalization: Let ABC be a triangle, P be a point on the plane, let three parallel segments AA', BB', CC' such that its midpoints and P are collinear. Then PA', PB', PC' meet BC, CA, AB respectively at three collinear points. Dao's second generalization Second generalization: Let a conic S and a point P on the plane.
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The principle behind the operation of acousto-optic tunable filters is based on the wavelength of the diffracted light being dependent on the acoustic frequency. By tuning the frequency of the acoustic wave, the desired wavelength of the optical wave can be diffracted acousto-optically. There are two types of the acousto-optic filters, the collinear and non-collinear filters. The type of filter depends on geometry of acousto-optic interaction.
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A theorem concerning Fano subplanes due to is: :If every quadrangle in a finite projective plane has collinear diagonal points, then the plane is desarguesian (of even order).
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The proof is invalid if C, c, X happen to be collinear. In that case an alternative proof can be provided, for example, using a different projective reference.
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If all five points are collinear, then the remaining line is free, which leaves 2 parameters free. Given four points in general linear position (no three collinear; in particular, no two coincident), there are exactly three pairs of lines (degenerate conics) passing through them, which will in general be intersecting, unless the points form a trapezoid (one pair is parallel) or a parallelogram (two pairs are parallel). Given three points, if they are non- collinear, there are three pairs of parallel lines passing through them – choose two to define one line, and the third for the parallel line to pass through, by the parallel postulate. Given two distinct points, there is a unique double line through them.
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A general conic is defined by five points: given five points in general position, there is a unique conic passing through them. If three of these points lie on a line, then the conic is reducible, and may or may not be unique. If no four points are collinear, then five points define a unique conic (degenerate if three points are collinear, but the other two points determine the unique other line). If four points are collinear, however, then there is not a unique conic passing through them – one line passing through the four points, and the remaining line passes through the other point, but the angle is undefined, leaving 1 parameter free.
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One can distinguish FT pulse shapers by their optical design: i.e., collinear shapers and transverse shapers, and by their programmability, i.e., static (or manually adjustable) shapers and programmable shapers.
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The three collinear Lagrange points (L1, L2, L3) were discovered by Leonhard Euler a few years before Joseph-Louis Lagrange discovered the remaining two. (16MB) In 1772, Lagrange published an "Essay on the three-body problem". In the first chapter he considered the general three-body problem. From that, in the second chapter, he demonstrated two special constant-pattern solutions, the collinear and the equilateral, for any three masses, with circular orbits.
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A jet algorithm is infrared safe if it yields the same set of jets after modifying an event to add a soft radiation. Similarly, a jet algorithm is collinear safe if the final set of jets is not changed after introducing a collinear splitting of one of the inputs. There are several reasons why a jet algorithm must fulfill these two requirements. Experimentally, jets are useful if they carry information about the seed parton.
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More importantly, the Nagel point N, the "area centroid" G, and the incenter I are collinear in this order, and NG = 2GI. This line is called the Nagel line of a tangential quadrilateral.. In a tangential quadrilateral ABCD with incenter I and where the diagonals intersect at P, let HX, HY, HZ, HW be the orthocenters of triangles AIB, BIC, CID, DIA. Then the points P, HX, HY, HZ, HW are collinear.
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An affine transformation preserves: # collinearity between points: three or more points which lie on the same line (called collinear points) continue to be collinear after the transformation. # parallelism: two or more lines which are parallel, continue to be parallel after the transformation. # convexity of sets: a convex set continues to be convex after the transformation. Moreover, the extreme points of the original set are mapped to the extreme points of the transformed set.
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If in the statement of Miquel's theorem the points A´, B´ and C´ form a triangle (that is, are not collinear) then the theorem was named the Pivot theorem in . (In the diagram these points are labeled P, Q and R.) If A´, B´ and C´ are collinear then the Miquel point is on the circumcircle of ∆ABC and conversely, if the Miquel point is on this circumcircle, then A´, B´ and C´ are on a line.
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One also says "two generic lines intersect in a point", which is formalized by the notion of a generic point. Similarly, three generic points in the plane are not collinear; if three points are collinear (even stronger, if two coincide), this is a degenerate case. This notion is important in mathematics and its applications, because degenerate cases may require an exceptional treatment; for example, when stating general theorems or giving precise statements thereof, and when writing computer programs (see generic complexity).
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In quantum field theory, soft-collinear effective theory (or SCET) is a theoretical framework for doing calculations that involve interacting particles carrying widely different energies. The motivation for developing SCET was to control the infrared divergences that occur in quantum chromodynamics (QCD) calculations that involve particles that are soft—carrying much lower energy or momentum than other particles in the process—or collinear—traveling in the same direction as another particle in the process. SCET is an effective theory for highly energetic quarks interacting with collinear and/or soft gluons. It has been used for calculations of the decays of B mesons (quark-antiquark bound states involving a bottom quark) and the properties of jets (sprays of hadrons that emerge from particle collisions when a quark or gluon is produced).
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The compass-only construction of the intersection points of a line and a circle breaks into two cases depending upon whether the center of the circle is or is not collinear with the line.
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Collinear transfer zones include areas which the major fault boundaries are in line with one another. In many cases this geometry relies on the faults to splay at their terminations and interfinger with one another.
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The Sylvester–Gallai theorem states that there is a line containing exactly two points of P. Such two point lines are called ordinary lines. Let a and b be the two points of P on an ordinary line. If the removal of point a produces a set of collinear points then P generates a near pencil of n lines (the n - 1 ordinary lines through a plus the one line containing the other n - 1 points). Otherwise, the removal of a produces a set, P' , of n − 1 points that are not all collinear.
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In projective geometry, a collineation is a one-to-one and onto map (a bijection) from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. A collineation is thus an isomorphism between projective spaces, or an automorphism from a projective space to itself. Some authors restrict the definition of collineation to the case where it is an automorphism.For instance, , and The set of all collineations of a space to itself form a group, called the collineation group.
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These points cannot be covered by only, which gives us . Since degenerate conics are a union of at most two lines, there are always four out of seven points on a degenerate conic that are collinear. Consequently: :If no seven points out of lie on a non-degenerate conic, and no four points out of lie on a line, then the vector space of cubic homogeneous polynomials that vanish on (the affine cones of) has dimension two. On the other hand, assume are collinear and no seven points out of are co-conic.
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A famous result, due to Andrew M. Gleason states that if every complete quadrangle in a finite projective plane extends to a Fano subplane (that is, has collinear diagonal points) then the plane is Desarguesian. Gleason called any projective plane satisfying this condition a Fano plane thus creating some confusion with modern terminology. To compound the confusion, Fano's axiom states that the diagonal points of a complete quadrangle are never collinear, a condition that holds in the Euclidean and real projective planes. Thus, what Gleason called Fano planes do not satisfy Fano's axiom.
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At any fixed point \vec x, the field will also vary sinusoidally with time; it will be a scalar multiple of the amplitude A, between +A and -A When the amplitude A is a vector orthogonal to \vec n, the wave is said to be transverse. Such waves may exhibit polarization, if A can be oriented along two non-collinear directions. When A is a vector collinear with \vec n, the wave is said to be longitudinal. These two possibilities are exemplifiec by the S (shear) waves and P (pressure) waves studied in seismology.
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Eugene Wigner (1902–1995) In theoretical physics, the composition of two non- collinear Lorentz boosts results in a Lorentz transformation that is not a pure boost but is the composition of a boost and a rotation. This rotation is called Thomas rotation, Thomas–Wigner rotation or Wigner rotation. The rotation was discovered by Llewellyn Thomas in 1926, and derived by Wigner in 1939. If a sequence of non-collinear boosts returns an object to its initial velocity, then the sequence of Wigner rotations can combine to produce a net rotation called the Thomas precession.
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An oval in a finite projective plane of order is a ()-arc, in other words, a set of points, no three collinear. Ovals in the Desarguesian (pappian) projective plane for odd are just the nonsingular conics. However, ovals in for even have not yet been classified. In an arbitrary finite projective plane of odd order , no sets with more points than , no three of which are collinear, exist, as first pointed out by Bose in a 1947 paper on applications of this sort of mathematics to the statistical design of experiments.
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Monge's theorem. The intersection of the red lines, that of the blue lines, and that of the green lines are collinear, all falling on the black line. In geometry, Monge's theorem, named after Gaspard Monge, states that for any three circles in a plane, none of which is completely inside one of the others, the intersection points of each of the three pairs of external tangent lines are collinear. For any two circles in a plane, an external tangent is a line that is tangent to both circles but does not pass between them.
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The Droz-Farny line theorem says that the midpoints of the three segments A_1A_2, B_1B_2, and C_1C_2 are collinear. The theorem was stated by Arnold Droz-Farny in 1899, but it is not clear whether he had a proof.
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Equivalently for three points in a plane, the points are collinear if and only if the slope between one pair of points equals the slope between any other pair of points (in which case the slope between the remaining pair of points will equal the other slopes). By extension, k points in a plane are collinear if and only if any (k–1) pairs of points have the same pairwise slopes. In Euclidean geometry, the Euclidean distance d(a,b) between two points a and b may be used to express the collinearity between three points by:Alessandro Padoa, Un nouveau système de définitions pour la géométrie euclidienne, International Congress of Mathematicians, 1900Bertrand Russell, The Principles of Mathematics, p. 410 :The points a, b and c are collinear if and only if d(x,a) = d(c,a) and d(x,b) = d(c,b) implies x=c.
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Although there can be no infinite non-collinear set of points with integer distances, there are infinite non-collinear sets of points whose distances are rational numbers. The (still unsolved) Erdős–Ulam problem asks whether there can exist a dense set of points in the plane at rational distances from each other. For any finite set S of points at rational distances from each other, it is possible to find a similar set of points at integer distances from each other, by expanding S by a factor of the least common denominator of the distances in S. Therefore, there exist arbitrarily large finite sets of non-collinear points with integer distances from each other. However, including more points into S may cause the expansion factor to increase, so this construction does not allow infinite sets of points at rational distances to be transformed into infinite sets of points at integer distances.
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Thus, it follows by mathematical induction. The example of a near-pencil, a set of n-1 collinear points together with one additional point that is not on the same line as the other points, shows that this bound is tight.
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The principle behind an electromagnetic lock is the use of electromagnetism to lock a door when energized. The holding force should be collinear with the load, and the lock and armature plate should be face-to- face to achieve optimal operation.
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A (non-degenerate) conic is completely determined by five points in general position (no three collinear) in a plane and the system of conics which pass through a fixed set of four points (again in a plane and no three collinear) is called a pencil of conics.. The four common points are called the base points of the pencil. Through any point other than a base point, there passes a single conic of the pencil. This concept generalizes a pencil of circles.Berger, M., Geometry Revealed: A Jacob's Ladder to Modern Higher Geometry (Berlin/Heidelberg: Springer, 2010), p. 127.
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The Super-J variation of the J-pole antenna adds another collinear half-wave radiator above the conventional J and connects the two with a phase stub to ensure both vertical half-wave sections radiate in current phase. The phasing stub between the two half-wave sections is often of the Franklin style. The Super-J antenna compresses the vertical beamwidth and has more gain than the conventional J-pole design. Both radiating sections have insufficient separation to realize the maximum benefits of collinear arrays, resulting in slightly less than the optimal 3 dB over a conventional J-pole or halfwave antenna.
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Generalized quadrangle with three points per line; a polar space of rank 2 A polar space of rank two is a generalized quadrangle; in this case, in the latter definition, the set of points of a line ℓ collinear with a point p is the whole ℓ only if p ∈ ℓ. One recovers the former definition from the latter under the assumptions that lines have more than 2 points, points lie on more than 2 lines, and there exist a line ℓ and a point p not on ℓ so that p is collinear to all points of ℓ.
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The interior angle bisectors of a triangle are concurrent in a point called the incenter of the triangle, as seen in the diagram at right. The bisectors of two exterior angles and the bisector of the other interior angle are concurrent. Three intersection points, each of an external angle bisector with the opposite extended side, are collinear (fall on the same line as each other). Three intersection points, two of them between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended, are collinear.
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Every realization of this configuration in the real projective plane is equivalent, under a projective transformation, to a realization constructed in this way from a regular pentagon. Therefore, in every realization, there are four points having the same cross-ratio as the cross- ratio of the four collinear points in the realization derived from the regular pentagon. But, these four points have 1+\varphi as their cross-ratio, where \varphi is the golden ratio, an irrational number. Every four collinear points with rational coordinates have a rational cross ratio, so the Perles configuration cannot be realized by rational points.
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The intrinsic quark structure of the target photon beam is revealed by observing characteristic patterns of the scattered electrons in the final state. Figure 1. Electron–photon scattering generic Feynman diagram. The incoming target photon splits into a nearly collinear quark–antiquark pair.
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A single discrete Thomas rotation (as opposed to the series of infinitesimal rotations that add up to the Thomas precession) is present in situations anytime there are three or more inertial frames in non-collinear motion, as can be seen using Lorentz transformations.
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Some other rubber ducky antennas use a spring of non-conducting material for support and comprise a collinear array antenna. Such antennas are still called rubber ducky antennas even though they function quite differently (and often better) than the original spring antenna.
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In case of a double knot, the length of the knot span becomes zero and the peak reaches one exactly. The basis function is no longer differentiable at that point. The curve will have a sharp corner if the neighbour control points are not collinear.
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Collinear transverse acoustic waves or perpendicular longitudinal waves can change the polarization. The acoustic waves induce a birefringent phase-shift, much like in a Pockels cell. The acousto-optic tunable filter, especially the dazzler, which can generate variable pulse shapes, is based on this principle.
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Presentation of collinear stimuli flanking a target can enhance responses to the target in cortex, an effect known as flanker or collinear facilitation, which has been shown to be weaker in those with schizophrenia than in unaffected adults or those with bipolar disorder. Publications from multiple research groups indicate that those with schizophrenia perform more poorly than healthy adults when asked to identify contours composed of separated line segments embedded in backgrounds made up of randomly oriented segments. This includes evidence from an fMRI experiment indicating abnormally reduced activation in visual areas V2-4. Another group used EEG to examine illusory contour processing deficits in schizophrenia.
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Given three non-collinear points, there are three lines connecting them, but with four points, no three collinear, there are six connecting lines and three additional "diagonal points" determined by their intersections. The science of projective geometry captures this surplus determined by four points through a quaternary relation and the projectivities which preserve the complete quadrangle configuration. An harmonic quadruple of points on a line occurs when there is a complete quadrangle two of whose diagonal points are in the first and third position of the quadruple, and the other two positions are points on the lines joining two quadrangle points through the third diagonal point.Halsted, pp.
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The inequality can also be used to prove Beck's theorem, that if a finite point set does not have a linear number of collinear points, then it determines a quadratic number of distinct lines.. Similarly, Tamal Dey used it to prove upper bounds on geometric k-sets.
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The magic tee is a combination of E and H plane tees. Arm 3 forms an H-plane tee with arms 1 and 2. Arm 4 forms an E-plane tee with arms 1 and 2. Arms 1 and 2 are sometimes called the side or collinear arms.
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The Nagel point is the isotomic conjugate of the Gergonne point. The Nagel point, the centroid, and the incenter are collinear on a line called the Nagel line. The incenter is the Nagel point of the medial triangle; equivalently, the Nagel point is the incenter of the anticomplementary triangle.
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Some authors consider collinear points (sets of points all belonging to a single line) to be a special case of concyclic points, with the line being viewed as a circle of infinite radius. This point of view is helpful, for instance, when studying inversion through a circle and Möbius transformations, as these transformations preserve the concyclicity of points only in this extended sense.. In the complex plane (formed by viewing the real and imaginary parts of a complex number as the x and y Cartesian coordinates of the plane), concyclicity has a particularly simple formulation: four points in the complex plane are either concyclic or collinear if and only if their cross-ratio is a real number..
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In any projective plane a set of four points, no three of which are collinear, and the six lines joining pairs of these points is a configuration known as a complete quadrangle. The lines are called sides and pairs of sides that do not meet at one of the four points are called opposite sides. The points at which opposite sides meet are called diagonal points and there are three of them. If this configuration lies in a projective plane and the three diagonal points are collinear, then the seven points and seven lines of the expanded configuration form a subplane of the projective plane that is isomorphic to the Fano plane and is called a Fano subplane.
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Doing so with horizontal dipole antennas retains those dipoles' directionality and null in the direction of their elements. However if each dipole is vertically oriented, in a so-called collinear antenna array (see graphic), that null direction becomes vertical and the array acquires an omnidirectional radiation pattern (in the horizontal plane) as is typically desired. Vertical collinear arrays are used in the VHF and UHF frequency bands at which wavelengths the size of the elements are small enough to practically stack several on a mast. They are a higher-gain alternative to quarter-wave ground plane antennas used in fixed base stations for mobile two-way radios, such as police, fire, and taxi dispatchers.
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A (non- degenerate) conic is completely determined by five points in general position (no three collinear) in a plane and the system of conics which pass through a fixed set of four points (again in a plane and no three collinear) is called a pencil of conics.. The four common points are called the base points of the pencil. Through any point other than a base point, there passes a single conic of the pencil. This concept generalizes a pencil of circles. In a projective plane defined over an algebraically closed field any two conics meet in four points (counted with multiplicity) and so, determine the pencil of conics based on these four points.
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In ultrasonic flow meter measurement, ToF is used to measure speed of signal propagation upstream and downstream of flow of a media, in order to estimate total flow velocity. This measurement is made in a collinear direction with the flow. In planar Doppler velocimetry (optical flow meter measurement), ToF measurements are made perpendicular to the flow by timing when individual particles cross two or more locations along the flow (collinear measurements would require generally high flow velocities and extremely narrow-band optical filters). In optical interferometry, the pathlength difference between sample and reference arms can be measured by ToF methods, such as frequency modulation followed by phase shift measurement or cross correlation of signals.
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The NP1 gene is in an alternate reading frame to VP1 and overlaps the start of VP1 by 13 nucleotides. Similarly, VP3 is collinear to VP1 and VP2 and results from initiation of translation at a downstream ATG and co-terminates. VP2 is translated from a non-canonical start codon GUG.
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They can be complex numbers, as in a complex exponential plane wave. When the values of F are vectors, the wave is said to be a longitudinal wave if the vectors are always collinear with the vector \vec n, and a transverse wave if they are always orthogonal (perpendicular) to it.
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Therefore, angles OBA and OBC are equal. Finally, because they form a complete circle, we have :∠OBA + ∠ABD + ∠DBC + ∠CBO = 360° but, due to the congruences, angle OBA = angle OBC and angle DBA = angle DBC, thus :2 × ∠OBA + 2 × ∠DBA = 360° :∠OBA + ∠DBA = 180° therefore points O, B, and D are collinear.
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In 1982 the first CARS microscope was demonstrated. In 1999, CARS microscopy using a collinear geometry and high numerical aperture objective were developed in Xiaoliang Sunney Xie's lab at Harvard University. This advancement made the technique more compatible with modern laser scanning microscopes. Since then, CRS's popularity in biomedical research started to grow.
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Notation for Kelly's proof This proof is by Leroy Milton Kelly. call it "simply the best" of the many proofs of this theorem. Suppose that a finite set S of points is not all collinear. Define a connecting line to be a line that contains at least two points in the collection.
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The Collinear Fast-Beam Laser Spectroscopy (CFBS) experiment at TRIUMF is designed to exploit the high beam-intensity and radioisotope-production capability of TRIUMF's ISAC facility, as well as modern ion-trap beam-cooling techniques, in order to measure the hyperfine energy levels and isotope shifts of short-lived isotopes using laser spectroscopy.
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In the case of three non-collinear points in the plane, the triangle with these points as its vertices has a unique Steiner inellipse that is tangent to the triangle's sides at their midpoints. The major axis of this ellipse falls on the orthogonal regression line for the three vertices.Minda and Phelps (2008), Corollary 2.4.
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Skew infinite polygons (apeirogons) have vertices which are not all collinear. A zig-zag skew polygon or antiprismatic polygonRegular complex polytopes, p. 6 has vertices which alternate on two parallel planes, and thus must be even-sided. Regular skew polygons in 3 dimensions (and regular skew apeirogons in two dimensions) are always zig-zag.
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In microholography, focused beams of light are used to record submicrometre-sized holograms in a photorefractive material, usually by the use of collinear beams. The writing process may use the same kinds of media that are used in other types of holographic data storage, and may use two–photon processes to form the holograms.
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Two numbers m and n are not coprime--that is, they share a common factor other than 1--if and only if for a rectangle plotted on a square lattice with vertices at (0, 0), (m, 0), (m, n), and (0, n), at least one interior point is collinear with (0, 0) and (m, n).
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Triangle = Tri (three) + Angle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane (i.e.
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On the other hand, convergence (even to a local extremum) is not guaranteed when using this method in isolation. For example, if the three points are collinear, the resulting parabola is degenerate and thus does not provide a new candidate point. Furthermore, if function derivatives are available, Newton's method is applicable and exhibits quadratic convergence.
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A gear coupling A gear coupling is a mechanical device for transmitting torque between two shafts that are not collinear. It consists of a flexible joint fixed to each shaft. The two joints are connected by a third shaft, called the spindle. Each joint consists of a 1:1 gear ratio internal/external gear pair.
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Major fault boundaries within rift systems tend to expand with time based on the assumption that the extensional stress and strain is consistent in a given rift system. Transfer zones have been identified according to the fault propagation evolution; they include approaching, overlapping, collateral, and collinear. This classification assists in visualization and deformational history of the transfer zone.
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The graph compares the E-plane gain of the above three variations to the conventional J antenna. The conventional J antenna and SlimJIM variation are nearly identical in gain and pattern. The Super-J reveals the benefit of properly phasing and orienting a second radiator above the first. The Collinear J shows slightly higher performance over the Super-J.
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Finding the smallest set of triangles covering a given polygon is NP-hard. It is also hard to approximate - every polynomial-time algorithm might find a covering with size (1+1/19151) times the minimal covering. If the polygon is in general position (i.e. no two edges are collinear), then every triangle can cover at most 3 polygon edges.
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The Water Jar asterism was seen to the ancient Chinese as the tomb, Fenmu. Nearby, the emperors' mausoleum Xiuliang stood, demarcated by Kappa Aquarii and three other collinear stars. Ku ("crying") and Qi ("weeping"), each composed of two stars, were located in the same region. Three of the Chinese lunar mansions shared their name with constellations.
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If one pencil is of elliptic type, the other is of hyperbolic type and vice versa. The radical axis of any pencil of circles, interpreted as an infinite-radius circle, belongs to the pencil. Any three circles belong to a common pencil whenever all three pairs share the same radical axis and their centers are collinear.
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For three circles denoted by C1, C2, and C3, there are three pairs of circles (C1C2, C2C3, and C1C3). Since each pair of circles has two homothetic centers, there are six homothetic centers altogether. Gaspard Monge showed in the early 19th century that these six points lie on four lines, each line having three collinear points.
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There also exist eight other (103103) configurations (that is, sets of points and lines in the Euclidean plane with three lines per point and three points per line) that are not incidence-isomorphic to the Desargues configuration, one of which is shown at right. In all of these configurations, each point has three other points that are not collinear with it. But in the Desargues configuration, these three points are always collinear with each other (if the chosen point is the center of perspectivity, then the three points form the axis of perspectivity) while in the other configuration shown in the illustration these three points form a triangle of three lines. As with the Desargues configuration, the other depicted configuration can be viewed as a pair of mutually inscribed pentagons.
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Then no five points of and no three points of are collinear. Since will always contain the whole line through on account of Bézout's theorem, the vector space of cubic homogeneous polynomials that vanish on (the affine cones of) is isomorphic to the vector space of quadratic homogeneous polynomials that vanish (the affine cones of) , which has dimension two. Although the sets of conditions for both dimension two results are different, they are both strictly weaker than full general positions: three points are allowed to be collinear, and six points are allowed to lie on a conic (in general two points determine a line and five points determine a conic). For the Cayley-Bacharach theorem, it is necessary to have a family of cubics passing through the nine points, rather than a single one.
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The following proofSee Michèle Audin, Géométrie, éditions BELIN, Paris 1998: indication for exercise 1.37, p. 273 uses only notions of affine geometry, notably homothecies. Whether or not D, E, and F are collinear, there are three homothecies with centers D, E, F that respectively send B to C, C to A, and A to B. The composition of the three then is an element of the group of homothecy- translations that fixes B, so it is a homothecy with center B, possibly with ratio 1 (in which case it is the identity). This composition fixes the line DE if and only if F is collinear with D and E (since the first two homothecies certainly fix DE, and the third does so only if F lies on DE).
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A degenerate case of Pascal's theorem (four points) is interesting; given points on a conic , the intersection of alternate sides, , , together with the intersection of tangents at opposite vertices and are collinear in four points; the tangents being degenerate 'sides', taken at two possible positions on the 'hexagon' and the corresponding Pascal line sharing either degenerate intersection. This can be proven independently using a property of pole-polar. If the conic is a circle, then another degenerate case says that for a triangle, the three points that appear as the intersection of a side line with the corresponding side line of the Gergonne triangle, are collinear. Six is the minimum number of points on a conic about which special statements can be made, as five points determine a conic.
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In this configuration, the individual radiators within the array are often constructed of coaxial feedlines with the center conductor of one element being connected electrically to the shield of the one above, and so on in alternating phase for as many elements are specified by gain or overall length requirements. The final or 'top' element in the stack is a quarter-wave radiator connected directly to the center conductor of the element below it. This style of collinear antenna is usually housed in a fiberglass radome, to provide both support and environmental protection to the relatively fragile coaxial elements. A third type of collinear array, rarely seen outside of amateur radio VHF/UHF applications, uses half-wavelength monopole elements with phasing coils between each consecutive pair of elements to achieve the necessary phase shift.
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Four sided polygons (generally referred to as quads) and triangles are the most common shapes used in polygonal modeling. A group of polygons, connected to each other by shared vertices, is generally referred to as an element. Each of the polygons making up an element is called a face. In Euclidean geometry, any three non-collinear points determine a plane.
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However, in a pappian projective plane a conic is a circle only if it passes through two specific points on the line at infinity, so a circle is determined by five non-collinear points, three in the affine plane and these two special points. Similar considerations explain the smaller than expected number of points needed to define pencils of circles.
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In a triangle, three intersection points, each of an external angle bisector with the opposite extended side, are collinear.Johnson, Roger A., Advanced Euclidean Geometry, Dover Publ., 2007 (orig. 1929). In a triangle, three intersection points, two of them between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended, are collinear.
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Construct three lines da, db, dc through P such that they meet the conic at A, A'; B, B' ; C, C' respectively. Let D be a point on the polar of point P with respect to (S) or D lies on the conic (S). Let DA' ∩ BC =A0; DB' ∩ AC = B0; DC' ∩ AB= C0. Then A0, B0, C0 are collinear.
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Axiom 1 requires a unique line for each pair of distinct points, and a unique point of intersection of non-parallel lines. Axiom 2 depends on a line and a point; it requires a unique parallel to the line and through the point. Axiom 3 requires three non-collinear points. Axiom 4a requires a translation to move any point to any other.
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Quaternary fission (two-alpha accompanied fission) was experimentally discovered by Goennenwein et al. Pyatkov and Kamanin et al. in JINR Dubna are pursuing experiments on collinear ternary fission. In 2005, when Alexandru Proca's death was commemorated, Poenaru used the opportunity to disseminate information about his relativistic equations of the massive vector boson field, as well as his life in Romania and in France.
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In Euclidean and projective geometry, just as two (distinct) points determine a line (a degree-1 plane curve), five points determine a conic (a degree-2 plane curve). There are additional subtleties for conics that do not exist for lines, and thus the statement and its proof for conics are both more technical than for lines. Formally, given any five points in the plane in general linear position, meaning no three collinear, there is a unique conic passing through them, which will be non-degenerate; this is true over both the Euclidean plane and any pappian projective plane. Indeed, given any five points there is a conic passing through them, but if three of the points are collinear the conic will be degenerate (reducible, because it contains a line), and may not be unique; see further discussion.
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Paul Erdős (in ) observed that, when n is a prime number, the set of n grid points (i, i2 mod n), for 0 ≤ i < n, contains no three collinear points. When n is not prime, one can perform this construction for a p × p grid contained in the n × n grid, where p is the largest prime that is at most n. As a consequence, for any ε and any sufficiently large n, one can place :(1 - \epsilon)n points in the n × n grid with no three points collinear. Erdős' bound has been improved subsequently: show that, when n/2 is prime, one can obtain a solution with 3(n - 2)/2 points by placing points on the hyperbola xy ≡ k (mod n/2), where k may be chosen arbitrarily as long as it is nonzero mod n/2.
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Transfer zones can be farther identified by its maturity or (fault propagation evolution); whether the major fault relationship is approaching, overlapping, collateral or collinear. Since transfer zones are normally found in extensional settings many studies have been done within the East African rift system and the Gulf of Suez rift system. Transfer zones have also played a role in hydrocarbon exploration and extraction within the Albertine graben.
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Points of the line X = Y have coordinates which may be written as (1,1,c). Three points, one from each of these lines, are collinear if and only if a = b + c. By selecting all the points on these lines where a, b and c are the field elements with absolute trace 0, the condition in the definition of a projective triad is satisfied.
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Composite Bezier curves can be smoothed to any desired degree of smoothness using Stärk's construction. C2 continuous composite cubic Bezier curves are actually cubic B-splines, and vice versa. Individual curves are by definition C1 and C2 continuous. The geometric condition for C1 continuity when transiting across an endpoint joining two curves is that the associated control points are mutually opposed and collinear with the endpoint.
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An alternative way of stating the theorem is that a non- collinear set of points in the plane with integer distances can only be extended by adding finitely many additional points, before no more points can be added. A set of points with both integer coordinates and integer distances, to which no more can be added while preserving both properties, forms an Erdős–Diophantine graph.
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All collisions conserve momentum. What distinguishes different types of collisions is whether they also conserve kinetic energy. The line of impact is the line that is collinear to the common normal of the surfaces that are closest or in contact during impact. This is the line along which internal force of collision acts during impact, and Newton's coefficient of restitution is defined only along this line.
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Non- collinear sets of points in the three-dimensional grid were considered by . They proved that the maximum number of points in the n × n × n grid with no three points collinear is \Theta(n^2). Similarly to Erdős's 2D construction, this can be accomplished by using points (x, y, x2 \+ y2) mod p, where p is a prime congruent to 3 mod 4. Another analogue in higher dimensions is to find sets of points that do not all lie in the same plane (or hyperplane). For the no-four-coplanar problem in three dimensions, it was reported by Ed Pegg, Oleg567 et al, that 8 such points can be selected in a 3x3x3 grid (exactly one solution up to rotation/reflection), 10 such points can be found for 4x4x4 (232 different solutions), and 13 such points can be found for 5x5x5 (38 different solutions).
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Adding more mirrors does not add more possibilities (in the plane), because they can always be rearranged to cause cancellation. :Proof. An isometry is completely determined by its effect on three independent (not collinear) points. So suppose p1, p2, p3 map to q1, q2, q3; we can generate a sequence of mirrors to achieve this as follows. If p1 and q1 are distinct, choose their perpendicular bisector as mirror.
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Although its empirical formula, AgO, suggests that silver is in the +2 oxidation state in this compound, AgO is in fact diamagnetic. X-ray diffraction studies show that the silver atoms adopt two different coordination environments, one having two collinear oxide neighbours and the other four coplanar oxide neighbours.Wells A.F. (1984) Structural Inorganic Chemistry 5th edition Oxford Science Publications AgO is therefore formulated as AgIAgIIIO2 p. 1181. or Ag2O·Ag2O3.
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If the incircle is tangent to the bases at P and Q, then P, I and Q are collinear, where I is the incenter.J. Wilson, Problem Set 2.2, The University of Georgia, 2010, . The angles AID and BIC in a tangential trapezoid ABCD, with bases AB and DC, are right angles. The incenter lies on the median (also called the midsegment; that is, the segment connecting the midpoints of the legs).
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If the result is 0, the points are collinear; if it is positive, the three points constitute a "left turn" or counter-clockwise orientation, otherwise a "right turn" or clockwise orientation (for counter-clockwise numbered points). This process will eventually return to the point at which it started, at which point the algorithm is completed and the stack now contains the points on the convex hull in counterclockwise order.
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A polar point group has no unique origin because each of those unmoved points can be chosen as one. One or more unique polar axes could be made through two such collinear unmoved points. Polar crystallographic point groups include 1, 2, 3, 4, 6, m, mm2, 3m, 4mm, and 6mm. A chiral (often also called enantiomorphic) point group is one containing only proper (often called "pure") rotation symmetry.
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An object resting on a surface and the corresponding free body diagram showing the forces acting on the object. The normal force N is equal, opposite, and collinear to the gravitational force mg so the net force and moment is zero. Consequently, the object is in a state of static mechanical equilibrium. In classical mechanics, a particle is in mechanical equilibrium if the net force on that particle is zero.
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They radiate vertically polarized radio waves. Theoretically, when stacking idealized lossless antennas in such a fashion, doubling their number will produce double the gain, with an increase of 3.01 dB. In practice, the gain realized will be below this due to imperfect radiation spread and losses. Collinear arrays are frequently constructed as a stack of dipoles, but can also be constructed as a stack of phased quarter- wave antennas.
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Portable radios usually use whips or rubber ducky antennas, while base stations usually use larger fiberglass whips or collinear arrays of vertical dipoles. For directional antennas, the Yagi antenna is the most widely used as a high gain or "beam" antenna. For television reception, the Yagi is used, as well as the log-periodic antenna due to its wider bandwidth. Helical and turnstile antennas are used for satellite communication since they employ circular polarization.
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Diacu also obtained some important results on a conjecture due to Donald G. Saari,F. Diacu, E. Pérez-Chavela and M. Santoprete, Saari's conjecture for the collinear n-body problem, Transactions of the American Mathematical Society 357 (2005), no. 10, 4215–4223. F. Diacu, T. Fujiwara, E. Pérez-Chavela, and M. Santoprete, Saari's homographic conjecture of the three-body problem, Transactions of the American Mathematical Society 360 (2008), no. 12, 6447–6473.
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In a finite projective plane (not necessarily Desarguesian) a set of points such that no three points of are collinear (on a line) is called a . If the plane has order then , however the maximum value of can only be achieved if is even. In a plane of order , a -arc is called an oval and, if is even, a -arc is called a hyperoval. Every conic in the Desarguesian projective plane PG(2,), i.e.
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M21 is normal in PGL(3,4), of index 3. PGL(3,4) has an outer automorphism induced by transposing conjugate elements in F4 (the field automorphism). PGL(3,4) can therefore be extended to the group PΓL(3,4) of projective semilinear transformations, which is a split extension of M21 by the symmetric group S3. PΓL(3,4) has an embedding as a maximal subgroup of M24. A hyperoval has no 3 points that are collinear.
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Let and be any two of the four points given by the incenter and the three excenters of a triangle . Then and are collinear with one of the three triangle vertices. The circle with as diameter passes through the other two vertices and is centered on the circumcircle of . When one of or is the incenter, this is the trillium theorem, with line as the (internal) angle bisector of one of the triangle's angles.
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The ethynyl radical is observed in the microwave portion of the spectrum via pure rotational transitions. In its ground electronic and vibrational state, the nuclei are collinear, and the molecule has a permanent dipole moment estimated to be μ = 0.8 D = . The ground vibrational and electronic (vibronic) state exhibits a simple rigid rotor-type rotational spectrum. However, each rotational state exhibits fine and hyperfine structure, due to the spin-orbit and electron-nucleus interactions, respectively.
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Furthermore, by Qvist's theorem, through any point not on an oval there pass either zero or two tangent lines of that oval. A hyperoval (the 4 red points) in the 7 point Fano plane. When q is even, the situation is completely different. In this case, sets of points, no three of which collinear, may exist in a finite projective plane of order and they are called hyperovals; these are maximal arcs of degree 2.
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The segment AB is the set of points P such that [APB]. The interval AB is the segment AB and its end points A and B. The ray A/B (read as "the ray from A away from B") is the set of points P such that [PAB]. The line AB is the interval AB and the two rays A/B and B/A. Points on the line AB are said to be collinear.
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Depending on the number of quarks which are treated as light, one uses either SU(2) ChiPT or SU(3) ChiPT . Other effective theories are heavy quark effective theory (which expands around heavy quark mass near infinity), and soft-collinear effective theory (which expands around large ratios of energy scales). In addition to effective theories, models like the Nambu–Jona-Lasinio model and the chiral model are often used when discussing general features.
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In 1857, in the second Beiträge, von Staudt contributed a route to number through geometry called the algebra of throws (). It is based on projective range and the relation of projective harmonic conjugates. Through operations of addition of points and multiplication of points, one obtains an "algebra of points", as in chapter 6 of Veblen & Young's textbook on projective geometry. The usual presentation relies on cross ratio (CA,BD) of four collinear points.
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Higher gain omnidirectional UHF antennas can be made of collinear arrays of dipoles and are used for mobile base stations and cellular base station antennas. The short wavelengths also allow high gain antennas to be conveniently small. High gain antennas for point-to-point communication links and UHF television reception are usually Yagi, log periodic, corner reflectors, or reflective array antennas. At the top end of the band slot antennas and parabolic dishes become practical.
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The cross-ratio can therefore be regarded as measuring the quadruple's deviation from this ratio; hence the name anharmonic ratio. The cross-ratio is preserved by linear fractional transformations. It is essentially the only projective invariant of a quadruple of collinear points; this underlies its importance for projective geometry. The cross-ratio had been defined in deep antiquity, possibly already by Euclid, and was considered by Pappus, who noted its key invariance property.
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This therefore means that , where one of the points in the two tetrads overlap, hence meaning that other lines connecting the other three pairs must coincide to preserve cross ratio. Therefore, are collinear. Another proof for Pascal's theorem for a circle uses Menelaus' theorem repeatedly. Dandelin, the geometer who discovered the celebrated Dandelin spheres, came up with a beautiful proof using "3D lifting" technique that is analogous to the 3D proof of Desargues' theorem.
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So this last set of three lines is concurrent if all the other eight sets are because multiplication is commutative, so pq = qp. Equivalently, X, Y, Z are collinear. The proof above also shows that for Pappus's theorem to hold for a projective space over a division ring it is both sufficient and necessary that the division ring is a (commutative) field. German mathematician Gerhard Hessenberg proved that Pappus's theorem implies Desargues's theorem.
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This style tends to be less efficient due to coil losses, but has the advantage that it can be constructed with the elements supporting themselves, doing away with the need for a protective radome. Collinear arrays are often used as the antennas for base stations for land mobile radio systems that communicate with mobile two-way radios in vehicles, such as police, fire, ambulance, and taxi dispatchers. They are also sometimes used for broadcasting.
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A type of continuously variable air core inductor is the variometer. This consists of two coils with the same number of turns connected in series, one inside the other. The inner coil is mounted on a shaft so its axis can be turned with respect to the outer coil. When the two coils' axes are collinear, with the magnetic fields pointing in the same direction, the fields add and the inductance is maximum.
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For the convex polygon, a linear time algorithm for the minimum- area enclosing rectangle is known. It is based on the observation that a side of a minimum-area enclosing box must be collinear with a side of the convex polygon.. It is possible to enumerate boxes of this kind in linear time with the approach called rotating calipers by Godfried Toussaint in 1983.. The same approach is applicable for finding the minimum-perimeter enclosing rectangle.
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Parametric amplification overlaps a weak probe beam with a higher energy pump beam in a non-linear crystal such that the weak beam gets amplified and the remaining energy goes out as a new beam called the idler. This approach has the capability of generating output pulses that are shorter than the input ones. Different schemes of this approach have been implemented. Examples are optical parametric oscillator (OPO), optical parametric amplifier (OPA), non-collinear parametric amplifier (NOPA).
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SCET has also been used to calculate electroweak interactions in Higgs boson production. The new feature of SCET is its ability to handle more than one soft energy scale. For example, processes involving quarks carrying a high energy Q interacting with gluons have two soft scales: the transverse momentum pT of the collinear particles, plus the even softer scale pT2/Q. SCET provides a power-counting formalism for doing perturbation theory in the small parameter ΛQCD/Q.
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In the 1970s he introduced a coordinate transformation (now known as the McGehee transformation) which he used to regularize singularities arising in the Newtonian three-body problem. In 1975 he, with John N. Mather, proved that for the Newtonian collinear four-body problem there exist solutions which become unbounded in a finite time interval. In 1978 he was an Invited Speaker on the subject of Singularities in classical celestial mechanics at the International Congress of Mathematicians in Helsinki.
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The multiple-axis systems generally use three individual antennas aimed in specific directions to steer the acoustic beam. Using three independent (i.e. non-collinear) axes is enough to retrieve the three components of the wind speed, although using more axes would add redundancy and increase robustness to noise when estimating the wind speed, using a least-squares approach. One antenna is generally aimed vertically, and the other two are tilted slightly from the vertical at an orthogonal angle.
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In a uniform matroid U{}^r_n, the circuits are the sets of exactly r+1 elements. Therefore, a uniform matroid is Eulerian if and only if r+1 is a divisor of n. For instance, the n-point lines U{}^2_n are Eulerian if and only if n is divisible by three. The Fano plane has two kinds of circuits: sets of three collinear points, and sets of four points that do not contain any line.
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If collinear points A, B, C, and D are such that D is the harmonic conjugate of C with respect to A and B, then the distances from any one of these points to the three remaining points form harmonic progression.Chapters on the modern geometry of the point, line, and circle, Vol. II by Richard Townsend (1865) p. 24Modern geometry of the point, straight line, and circle: an elementary treatise by John Alexander Third (1898) p.
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The Darboux cubic may be defined from the de Longchamps point, as the locus of points X such that X, the isogonal conjugate of X, and the de Longchamps point are collinear. It is the only cubic curve invariant of a triangle that is both isogonally self-conjugate and centrally symmetric; its center of symmetry is the circumcenter of the triangle.. The de Longchamps point itself lies on this curve, as does its reflection the orthocenter.
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In geometry, collinearity of a set of points is the property of their lying on a single line.The concept applies in any geometry , but is often only defined within the discussion of a specific geometry , A set of points with this property is said to be collinear (sometimes spelled as colinearColinear (Merriam-Webster dictionary)). In greater generality, the term has been used for aligned objects, that is, things being "in a line" or "in a row".
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A five-pointed star A five-pointed star (☆), geometrically a regular concave decagon, is a common ideogram in modern culture. Comparatively rare in classical heraldry, it was notably introduced for the flag of the United States in the Flag Act of 1777 and since has become widely used in flags. It has also become a symbol of fame or "stardom" in Western culture, among other uses. If the collinear edges are joined together a pentagram is produced.
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Non-birefringent methods, to rotate the linear polarization of light beams, include the use of prismatic polarization rotators which use total internal reflection in a prism set designed for efficient collinear transmission. A polariser changing the orientation of linearly polarised light. In this picture, θ1 – θ0 = θi. Media that reduce the amplitude of certain polarization modes are called dichroic, with devices that block nearly all of the radiation in one mode known as polarizing filters or simply "polarisers".
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The antenna radiates horizontally polarized radiation in the horizontal plane. Each group of four elements at a single level is referred to as a bay. The radiation pattern is close to omnidirectional but has four small lobes (maxima) in the directions of the four elements. To reduce power radiated in the unwanted axial directions, in broadcast applications multiple bays fed in phase are stacked vertically with a spacing of approximately one wavelength, to create a collinear array.
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In both collinear and transverse PDS, the surface is heated using a periodically modulated light source, such as an optical beam passing through a mechanical chopper or regulated with a function generator. A lock-in amplifier is then used to measure deflections found at the modulation frequency. Another scheme uses a pulsed laser as the excitation source. In that case, a boxcar average can be used to measure the temporal deflection of the probe beam to the excitation radiation.
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For instance, the Sylvester–Gallai theorem, stating that any non-collinear set of points in the plane has an ordinary line containing exactly two points, transforms under projective duality to the statement that any arrangement of lines with more than one vertex has an ordinary point, a vertex where only two lines cross. The earliest known proof of the Sylvester–Gallai theorem, by , uses the Euler characteristic to show that such a vertex must always exist.
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A unisecant in this example need not be a tangent line to the circle. This terminology is often used in incidence geometry and discrete geometry. For instance, the Sylvester–Gallai theorem of incidence geometry states that if points of Euclidean geometry are not collinear then there must exist a 2-secant of them. And the original orchard-planting problem of discrete geometry asks for a bound on the number of 3-secants of a finite set of points.
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1, No. 1, pp. 18–25. The laws of rational trigonometry, being algebraic, introduce subtleties into the solutions of problems, such as non-additivity of quadrances of collinear points (via the triple quad formula) or spreads of concurrent lines (via the triple spread formula) to give rational-valued outputs. By contrast, in the classical subject linearity is incorporated into distance and angular measurements to simplify these operations, albeit by 'transcendental' techniques employing real numbers entailing approximate valued output.
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In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. However, a set of four or more distinct points will, in general, not lie in a single plane. Two lines in three-dimensional space are coplanar if there is a plane that includes them both.
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If M1 and M2 are the midpoints of the diagonals AC and BD respectively in a tangential quadrilateral ABCD with incenter I, and if the pairs of opposite sides meet at J and K with M3 being the midpoint of JK, then the points M3, M1, I, and M2 are collinear. The line containing them is the Newton line of the quadrilateral. If the extensions of opposite sides in a tangential quadrilateral intersect at J and K, and the extensions of opposite sides in its contact quadrilateral intersect at L and M, then the four points J, L, K and M are collinear.. If the incircle is tangent to the sides AB, BC, CD, DA at T1, T2, T3, T4 respectively, and if N1, N2, N3, N4 are the isotomic conjugates of these points with respect to the corresponding sides (that is, AT1 = BN1 and so on), then the Nagel point of the tangential quadrilateral is defined as the intersection of the lines N1N3 and N2N4. Both of these lines divide the perimeter of the quadrilateral into two equal parts.
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Fernandes and Oliveira suggested an improved voting scheme for the Hough transform that allows a software implementation to achieve real-time performance even on relatively large images (e.g., 1280×960). The Kernel-based Hough transform uses the same (r,\theta) parameterization proposed by Duda and Hart but operates on clusters of approximately collinear pixels. For each cluster, votes are cast using an oriented elliptical-Gaussian kernel that models the uncertainty associated with the best-fitting line with respect to the corresponding cluster.
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The cross-ratio of four collinear points is an invariant under the homography that is fundamental for the study of the homographies of the lines. Three distinct points , and on a projective line over a field form a projective frame of this line. There is therefore a unique homography of this line onto that maps to , to 0, and to 1. Given a fourth point on the same line, the cross-ratio of the four points , , and , denoted , is the element of .
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Anhydrous nickel(II) acetylacetonate exists as molecules of Ni3(acac)6. The three nickel atoms are approximately collinear and each pair of them is bridged by two μ2 oxygen atoms. Each nickel atom has tetragonally distorted octahedral geometry, caused by the difference in the length of the Ni-O bonds between the bridging and non-bridging oxygens. Ni3(acac)6 molecules are almost centrosymmetric, despite the non-centrosymmetric point group of the cis-Ni(acac)2 "monomers," which is uncommon.
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Thus, in Euclidean geometry three non-collinear points determine a circle (as the circumcircle of the triangle they define), but four points in general do not (they do so only for cyclic quadrilaterals), so the notion of "general position with respect to circles", namely "no four points lie on a circle" makes sense. In projective geometry, by contrast, circles are not distinct from conics, and five points determine a conic, so there is no projective notion of "general position with respect to circles".
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D is the harmonic conjugate of C w.r.t. A and B. A, D, B, C form a harmonic range. KLMN is a complete quadrangle generating it. In projective geometry, the harmonic conjugate point of an ordered triple of points on the real projective line is defined by the following construction: :Given three collinear points A, B, C, let L be a point not lying on their join and let any line through C meet LA, LB at M, N respectively.
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This is important because it is unlikely that all the centroids of symbols in a text line are actually collinear. # For each pair of text lines, one can compute a minimum distance between their corresponding line segments. If this distance is within some tolerance of the between-line spacing calculated in step 7, then the two text lines are grouped into the same text block. # Finally, one can calculate a bounding box for each text block, and the document layout analysis is complete.
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Desargues's theorem is true for any projective space of dimension at least 3, and more generally for any projective space that can be embedded in a space of dimension at least 3\. Desargues's theorem can be stated as follows: :If lines and are concurrent (meet at a point), then :the points and are collinear. The points and are coplanar (lie in the same plane) because of the assumed concurrency of and . Therefore, the lines and belong to the same plane and must intersect.
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Gabriel Andrew Dirac (13 March 1925 – 20 July 1984) was a Hungarian/British mathematician who mainly worked in graph theory. He served as Erasmus Smith's Professor of Mathematics at Trinity College Dublin 1964-1966. In 1952, he gave a sufficient condition for a graph to contain a Hamiltonian circuit. The previous year, he conjectured that n points in the plane, not all collinear, must span at least [n/2] two-point lines, where [x] is the largest integer not exceeding x.
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In various plane geometries the notion of interchanging the roles of "points" and "lines" while preserving the relationship between them is called plane duality. Given a set of collinear points, by plane duality we obtain a set of lines all of which meet at a common point. The property that this set of lines has (meeting at a common point) is called concurrency, and the lines are said to be concurrent lines. Thus, concurrency is the plane dual notion to collinearity.
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This model, first proposed in 1951, considers bonding of three collinear atoms. For example, bonding in is described by a set of three molecular orbitals (MOs) derived from p-orbitals on each atom. Bonding results from the combination of a filled p-orbital from Xe with one half-filled p-orbital from each F atom, resulting in a filled bonding orbital, a filled non-bonding orbital, and an empty antibonding orbital. The highest occupied molecular orbital is localized on the two terminal atoms.
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If there is an antiferromagnetic interaction between spins, then it is not possible to arrange the spins so that all interactions between spins are antiparallel. There are six nearest-neighbor interactions, four of which are antiparallel and thus favourable, but two of which (between 1 and 2, and between 3 and 4) are unfavourable. It is impossible to have all interactions favourable, and the system is frustrated. Geometrical frustration is also possible if the spins are arranged in a non-collinear way.
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The inverse cross ratio is invariant under projective transformations and thus makes sense for points in the projective line. However, the formula above only makes sense for points in the affine line. In the slightly more general set-up below, the cross ratio makes sense for any four collinear points in projective space One just identifies the line containing the points with the projective line by a suitable projective transformation and then uses the formula above. The result is independent of any choices made in the identification.
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In 1900 Grossmann graduated from the Federal Polytechnic School (ETH) and became an assistant to the geometer Wilhelm Fiedler. He continued to do research on non- Euclidean geometry and taught in high schools for the next seven years. In 1902, he earned his doctorate from the University of Zurich with the thesis Ueber die metrischen Eigenschaften kollinearer Gebilde (translated On the Metrical Properties of Collinear Structures) with Fiedler as advisor. In 1907, he was appointed full professor of descriptive geometry at the Federal Polytechnic School.
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The Holographic Versatile Disc (HVD) is an optical disc technology developed between April 2004 and mid-2008 that can store up to several terabytes of data on an optical disc 10 cm or 12 cm in diameter. The reduced radius reduces cost and materials used. It employs a technique known as collinear holography, whereby a green and red laser beam are collimated in a single beam. The green laser reads data encoded as laser interference fringes from a holographic layer near the top of the disc.
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A question raised by J.J. Sylvester in 1893 and finally settled by Tibor Gallai concerned incidences of a finite set of points in the Euclidean plane. Theorem (Sylvester-Gallai): A finite set of points in the Euclidean plane is either collinear or there exists a line incident with exactly two of the points. A line containing exactly two of the points is called an ordinary line in this context. Sylvester was probably led to the question while pondering about the embeddability of the Hesse configuration.
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A Fano subplane is a subplane isomorphic to PG(2,2), the unique projective plane of order 2. If you consider a quadrangle (a set of 4 points no three collinear) in this plane, the points determine six of the lines of the plane. The remaining three points (called the diagonal points of the quadrangle) are the points where the lines that do not intersect at a point of the quadrangle meet. The seventh line consists of all the diagonal points (usually drawn as a circle or semicircle).
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But in the Euclidean plane, every finite set of points is either collinear, or includes a pair of points whose line does not contain any other points of the set; this is the Sylvester–Gallai theorem. Because the Hesse configuration disobeys the Sylvester–Gallai theorem, it has no Euclidean realization. This example also shows that the Sylvester–Gallai theorem cannot be generalized to the complex projective plane. However, in complex spaces, the Hesse configuration and all Sylvester–Gallai configurations must lie within a two-dimensional flat subspace..
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The non-geocentric model of the Universe was proposed by the Pythagorean philosopher Philolaus (d. 390 BC), who taught that at the center of the Universe was a "central fire", around which the Earth, Sun, Moon and planets revolved in uniform circular motion. This system postulated the existence of a counter-earth collinear with the Earth and central fire, with the same period of revolution around the central fire as the Earth. The Sun revolved around the central fire once a year, and the stars were stationary.
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The code below uses a function ccw: ccw > 0 if three points make a counter-clockwise turn, clockwise if ccw < 0, and collinear if ccw = 0. (In real applications, if the coordinates are arbitrary real numbers, the function requires exact comparison of floating- point numbers, and one has to beware of numeric singularities for "nearly" collinear points.) Then let the result be stored in the `stack`. let points be the list of points let stack = empty_stack() find the lowest y-coordinate and leftmost point, called P0 sort points by polar angle with P0, if several points have the same polar angle then only keep the farthest for point in points: # pop the last point from the stack if we turn clockwise to reach this point while count stack > 1 and ccw(next_to_top(stack), top(stack), point) <= 0: pop stack push point to stack end Now the stack contains the convex hull, where the points are oriented counter-clockwise and P0 is the first point. Here, `next_to_top()` is a function for returning the item one entry below the top of stack, without changing the stack, and similarly, `top()` for returning the topmost element.
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Ratios are not equal in this sense; but they may be the same. :KJ/JL = (KJ/AG)(AG/JL) = (JD/GD)(BG/JB). The last compound ratio (namely JD : GD & BG : JB) is what is known today as the cross ratio of the collinear points J, G, D, and B in that order; it is denoted today by (J, G; D, B). So we have shown that this is independent of the choice of the particular straight line JD that crosses the three straight lines that concur at A. In particular :(J, G; D, B) = (J, Z; H, E). It does not matter on which side of A the straight line JE falls. In particular, the situation may be as in the next diagram, which is the diagram for Lemma X. :Pappus- collection-7-136 Just as before, we have (J, G; D, B) = (J, Z; H, E). Pappus does not explicitly prove this; but Lemma X is a converse, namely that if these two cross ratios are the same, and the straight lines BE and DH cross at A, then the points G, A, and Z must be collinear.
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A line segment is a part of a line that is bounded by two distinct end points and contains every point on the line between its end points. Depending on how the line segment is defined, either of the two end points may or may not be part of the line segment. Two or more line segments may have some of the same relationships as lines, such as being parallel, intersecting, or skew, but unlike lines they may be none of these, if they are coplanar and either do not intersect or are collinear.
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Construction of the circumcircle (red) and the circumcenter Q (red dot) The circumcenter of a triangle can be constructed by drawing any two of the three perpendicular bisectors. For three non- collinear points, these two lines cannot be parallel, and the circumcenter is the point where they cross. Any point on the bisector is equidistant from the two points that it bisects, from which it follows that this point, on both bisectors, is equidistant from all three triangle vertices. The circumradius is the distance from it to any of the three vertices.
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No continuous arc of a conic can be constructed with straightedge and compass. However, there are several straightedge-and-compass constructions for any number of individual points on an arc. One of them is based on the converse of Pascal's theorem, namely, if the points of intersection of opposite sides of a hexagon are collinear, then the six vertices lie on a conic. Specifically, given five points, and a line passing through , say , \a point that lies on this line and is on the conic determined by the five points can be constructed.
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For the sake of simplicity, the description below assumes that the points are in general position, i.e., no three points are collinear. The algorithm may be easily modified to deal with collinearity, including the choice whether it should report only extreme points (vertices of the convex hull) or all points that lie on the convex hull. Also, the complete implementation must deal with degenerate cases when the convex hull has only 1 or 2 vertices, as well as with the issues of limited arithmetic precision, both of computer computations and input data.
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The special theory of relativity, formulated in 1905 by Albert Einstein, implies that addition of velocities does not behave in accordance with simple vector addition. In relativistic physics, a velocity-addition formula is a three-dimensional equation that relates the velocities of objects in different reference frames. Such formulas apply to successive Lorentz transformations, so they also relate different frames. Accompanying velocity addition is a kinematic effect known as Thomas precession, whereby successive non-collinear Lorentz boosts become equivalent to the composition of a rotation of the coordinate system and a boost.
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In a projective plane a complete k-arc is a set of k points, no three collinear, which can not be extended to a larger arc (thus, every point not on the arc is on a secant line of the arc-a line meeting the arc in two points.) Theorem: Let K be a complete k-arc in Π = PG(2,q) with k < q + 2. The dual in Π of the set of secant lines of K is a blocking set, B, of size k(k - 1)/2.
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A cubic polynomial has three zeroes in the complex number plane, which in general form a triangle, and the Gauss–Lucas theorem states that the roots of its derivative lie within this triangle. Marden's theorem states their location within this triangle more precisely: :Suppose the zeroes , , and of a third-degree polynomial are non-collinear. There is a unique ellipse inscribed in the triangle with vertices , , and tangent to the sides at their midpoints: the Steiner inellipse. The foci of that ellipse are the zeroes of the derivative .
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In any geometry, the set of points on a line are said to be collinear. In Euclidean geometry this relation is intuitively visualized by points lying in a row on a "straight line". However, in most geometries (including Euclidean) a line is typically a primitive (undefined) object type, so such visualizations will not necessarily be appropriate. A model for the geometry offers an interpretation of how the points, lines and other object types relate to one another and a notion such as collinearity must be interpreted within the context of that model.
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A set of 20 points in a 10 × 10 grid, with no three points in a line. In mathematics, in the area of discrete geometry, the no-three-in-line problem asks for the maximum number of points that can be placed in the n × n grid so that no three points are collinear. This number is at most 2n, since if 2n + 1 points are placed in the grid, then by the pigeonhole principle some row and some column will contain three points. The problem was introduced by Henry Dudeney in 1917.
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The deltoid curve is another type of curvilinear triangle, but one in which the curves replacing each side of an equilateral triangle are concave rather than convex. It is not composed of circular arcs, but may be formed by rolling one circle within another of three times the radius. Other planar shapes with three curved sides include the arbelos, which is formed from three semicircles with collinear endpoints,. and the Bézier triangle.. The Reuleaux triangle may also be interpreted as the conformal image of a spherical triangle with 120° angles.
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Minkowski in his earlier works in 1907 and 1908 followed Poincaré in representing space and time together in complex form (x,y,z,ict) emphasizing the formal similarity with Euclidean space. He noted that space-time is in a certain sense a four- dimensional non-Euclidean manifold.Goettingen lecture 1907, see comments in Walter 1999 Sommerfeld (1910) used Minkowski's complex representation to combine non-collinear velocities by spherical geometry and so derive Einstein's addition formula. Subsequent writers,Walter (1999b) principally Varićak, dispensed with the imaginary time coordinate, and wrote in explicitly non-Euclidean (i.e.
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Variometer used in 1920s radio receiver A variometer is a type of continuously variable air-core RF inductor with two windings. One common form consisted of a coil wound on a short hollow cylindrical form, with a second smaller coil inside, mounted on a shaft so its magnetic axis can be rotated with respect to the outer coil. The two coils are connected in series. When the two coils are collinear, with their magnetic fields pointed in the same direction, the two magnetic fields add, and the inductance is maximum.
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If the inner coil is rotated so its axis is at an angle to the outer coil, the magnetic fields do not add and the inductance is less. If the inner coil is rotated so it is collinear with the outer coil but their magnetic fields point in opposite directions, the fields cancel each other out and the inductance is very small or zero. The advantage of the variometer is that inductance can be adjusted continuously, over a wide range. Variometers were widely used in 1920s radio receivers.
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These statements follow from the Central Angle Theorem and the fact that three non-collinear points give a unique circle. It can also be shown that, for fixations along a given Vieth-Müller circle, all the corresponding horopter circles intersect at the point of symmetric convergence. This result implies that each member of the infinite family of horopters is also composed of a circle in the fixation plane and a perpendicular straight line passing through the point of symmetric convergence (located on the circle) so long as the eyes are in primary or secondary position.
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Like many organic farmers, Coleman advocates the prevention-not-treatment approach to weed control. He therefore favors fast, light, frequent cultivation with purpose-built hoe types, skimming weed seedlings off the soil surface with an action that is more like shaving than chopping (hoes "like razors rather than axes"). To that end, he developed the collinear hoe (or collineal hoe). Coleman is a leader in developing and sharing the concept that in season extension a distinction can be made between extending the growing season and extending the harvest season.
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The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the third side. That sum can equal the length of the third side only in the case of a degenerate triangle, one with collinear vertices. It is not possible for that sum to be less than the length of the third side. A triangle with three given positive side lengths exists if and only if those side lengths satisfy the triangle inequality.
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This can be understood as follows: if L and L′ are two lines not passing through Q then the perspective transformation from L to L′ with the center Q is a projective transformation that takes the quadruple {Pi} of points on L into the quadruple {Pi′} of points on L′. Therefore, the invariance of the cross-ratio under projective automorphisms of the line implies (in fact, is equivalent to) the independence of the cross-ratio of the four collinear points {Pi} on the lines {Li} from the choice of the line that contains them.
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400px We assume that P is not collinear with any two vertices of ABC. Let A', B' and C' be the points in which the lines AP, BP, CP meet sidelines BC, CA and AB (extended if necessary). Reflecting A', B', C' in the midpoints of sides BC, CA, AB will give points A", B" and C" respectively. The isotomic lines AA", BB" and CC" joining these new points to the vertices meet at a point (which can be proved using Ceva's theorem), the isotomic conjugate of P.
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The etymology of the term is related to the notion of planetary phases, since the brightness of an object and its appearance as a "phase" is the function of the phase angle. The phase angle varies from 0° to 180°. The value of 0° corresponds to the position where the illuminator, the observer, and the object are collinear, with the illuminator and the observer on the same side of the object. The value of 180° is the position where the object is between the illuminator and the observer, known as inferior conjunction.
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This latter provision would later be regretted by everyone involved. The roads of Cincinnati and Covington were laid collinear with each other, in the hope that a bridge would be built sometime in the future. When the Ohio legislature decided to choose its own location for the bridge, it failed to pick such an obvious spot, hoping to defend Cincinnati's preeminence over Newport and Covington, the rival Kentucky cities. The bridge was actually located almost entirely in Kentucky because the state boundary follows the north bank of the river.
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Experts also recommend Dongguan-Shenzhen inter-city lines and their routes overlap No. 1 line and for further research to make a unified coordination and optimization. It pointed out that the proposed R1 line with Guangzhou-Dongguan-Shenzhen inter-city line collinear operations should avoid duplication and wasteful investment. However the proposal was rejected, R1 line and Guangzhou-Dongguan- Shenzhen inter-city line to be built as separate lines. July 2009, "Dongguan city rapid rail transit construction plan (2009-2015)" was put to the NDRC and the Ministry of Construction for approval.
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D.C. Clary and J.N.L. Connor, 'Isotope and Potential energy surface effects in Vibrational Bonding.' Journal of Physical Chemistry. 1984 One year after the theoretical discovery of vibrational bonds, J. Manz and his team confirmed the calculations that were previously made, and elaborated on them by showing that the vibrational bonds were most likely to occur during symmetric reactions, but stated that vibrational bonds may also be possible with asymmetric reactions.J. Manz, E. Pollak, J. Romelt 'A classical analysis of quantum resonances in isotopic collinear H + H2 Reactions' Chemical Physics letters, 1982.
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Radiative transfer refers to energy transfer through an atmosphere or other medium by means of electromagnetic waves or (equivalently) photons. The simplest form of radiative transfer involves a collinear beam of radiation traveling through a sample to a detector. That flux can be reduced by absorption, scattering or reflection, resulting in energy transmission over a path of less than 100%. The concept of radiative transfer extends beyond simple laboratory phenomena to include thermal emission of radiation by the medium - which can result in more photons arriving at the end of a path than entering it.
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Only sound with parallel collinear phase velocity vectors interfere to produce this nonlinear effect. Even-numbered iterations will produce their modulation products, baseband and high frequency, as reflected emissions from the target. Odd- numbered iterations will produce their modulation products as reflected emissions off the emitter. This effect still holds when the emitter and the reflector are not parallel, though due to diffraction effects the baseband products of each iteration will originate from a different location each time, with the originating location corresponding to the path of the reflected high frequency self-modulation products.
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In geometry, a unital is a set of n3 \+ 1 points arranged into subsets of size n + 1 so that every pair of distinct points of the set are contained in exactly one subset. n ≥ 3 is required by some authors to avoid small exceptional cases. This is equivalent to saying that a unital is a 2-(n3 \+ 1, n + 1, 1) block design. Some unitals may be embedded in a projective plane of order n2 (the subsets of the design become sets of collinear points in the projective plane).
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When the inner coil is turned so its axis is at an angle with the outer, the mutual inductance between them is smaller so the total inductance is less. When the inner coil is turned 180° so the coils are collinear with their magnetic fields opposing, the two fields cancel each other and the inductance is very small. This type has the advantage that it is continuously variable over a wide range. It is used in antenna tuners and matching circuits to match low frequency transmitters to their antennas.
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In order for the signals to add together, they need to arrive in-phase. Consider two dipole antennas placed in a line end-to-end, or collinear. If the resulting array is pointed directly at the source signal, both dipoles will see the same instantaneous signal, and thus their reception will be in-phase. However, if one were to rotate the antenna so it was at an angle to the signal, the extra path from the signal to the more distant dipole means it receives the signal slightly out of phase.
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As the reflection of the orthocenter around the circumcenter, the de Longchamps point belongs to the line through both of these points, which is the Euler line of the given triangle. Thus, it is collinear with all the other triangle centers on the Euler line, which along with the orthocenter and circumcenter include the centroid and the center of the nine-point circle... See in particular Section 5, "Six notable points on the Euler line", pp. 380–383. The de Longchamp point is also collinear, along a different line, with the incenter and the Gergonne point of its triangle.. The three circles centered at A, B, and C, with radii s-a, s-b, and s-c respectively (where s is the semiperimeter) are mutually tangent, and there are two more circles tangent to all three of them, the inner and outer Soddy circles; the centers of these two circles also lie on the same line with the de Longchamp point and the incenter. The de Longchamp point is the point of concurrence of this line with the Euler line, and with three other lines defined in a similar way as the line through the incenter but using instead the three excenters of the triangle.
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The first consequence of such a requirement is that budget sets do not fill the available space and are typically smaller than hyperplanes. Because the dimension of vectors orthogonal to the budget set is larger than one there is no reason for the price systems supporting an equilibrium to be unique up to scaling, likewise the first order conditions no longer implies that gradient of agents are collinear at equilibrium. Both happen to fail to hold generically: the first theorem of welfare economics is hence the first victim of incompleteness. Pareto-optimality of equilibria generally does not hold.
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A design with the parameters of the extension of an affine plane, i.e., a 3-(n2 + 1, n + 1, 1) design, is called a finite inversive plane, or Möbius plane, of order n. It is possible to give a geometric description of some inversive planes, indeed, of all known inversive planes. An ovoid in PG(3,q) is a set of q2 + 1 points, no three collinear. It can be shown that every plane (which is a hyperplane since the geometric dimension is 3) of PG(3,q) meets an ovoid O in either 1 or q + 1 points.
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Recognition of conformational epitopes by B cells. Segments widely separated in the primary structure have come in contact in the three-dimensional tertiary structure forming part of the same epitope In figure at left, the various segments that form the epitope have been shown to be continuously collinear, meaning that they have been shown as sequential; however, for the situation being discussed here (i.e., the antigen recognition by the B cell), this explanation is too simplistic. Such epitopes are known as sequential or linear epitopes, as all the amino acids on them are in the same sequence (line).
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The university is located on a plain known as Lundtoftesletten in the northeastern end of the city of Lyngby. The area was previously home to the airfield Lundtofte Flyveplads. The campus is roughly divided in half by the road Anker Engelunds Vej going in the east-west direction, and, perpendicular to that, by two lengthy, collinear roads located on either side of a parking lot. The campus is thus divided into four parts, referred to as quadrants, numbered 1 through 4 in correspondence with the conventional numbering of quadrants in the Cartesian coordinate system with north upwards.
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Ultrafast processes operating at picosecond, femtosecond, and even attosecond scale are both driven by, and studied using, optical methods that are at the front line of modern science. The physics underpinning the observations at these short time scales is governed by non- equilibrium dynamics, and usually makes use of resonant processes. One demonstration of ultrafast processes is the switching from collinear antiferromagnetic state to spiral antiferromagnetic state in CuO under excitation by 40 fs 800 nm laser pulse. A second example shows the possibility for the direct control of spin waves with THz radiation on antiferromagnetic NiO.
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If such a subset existed, it would form a universal point set that could be used to draw all planar graphs with rational edge lengths (and therefore, after scaling them appropriately, with integer edge lengths). However, Ulam conjectured that dense rational-distance sets do not exist.. According to the Erdős–Anning theorem, infinite non- collinear point sets with all distances being integers cannot exist. This does not rule out the existence of sets with all distances rational, but it does imply that in any such set the denominators of the rational distances must grow arbitrarily large.
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Two solutions whose sides pass through A, B, C In geometry, the Cramer–Castillon problem is a problem stated by the Swiss mathematician Gabriel Cramer solved by the Italian mathematician, resident in Berlin, Jean de Castillon in 1776., page 1. The problem consists of (see the image): Given a circle Z and three points A, B, C in the same plane and not on Z, to construct every possible triangle inscribed in Z whose sides (or their elongations) pass through A, B, C respectively. Centuries before, Pappus of Alexandria had solved a special case: when the three points are collinear.
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There are many ways to derive the Lorentz transformations utilizing a variety of physical principles, ranging from Maxwell's equations to Einstein's postulates of special relativity, and mathematical tools, spanning from elementary algebra and hyperbolic functions, to linear algebra and group theory. This article provides a few of the easier ones to follow in the context of special relativity, for the simplest case of a Lorentz boost in standard configuration, i.e. two inertial frames moving relative to each other at constant (uniform) relative velocity less than the speed of light, and using Cartesian coordinates so that the x and x′ axes are collinear.
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First, it must be proven that points O, B, D are collinear. This may be easily seen by observing that the linkage is mirror-symmetric about line OD, so point B must fall on that line. More formally, triangles BAD and BCD are congruent because side BD is congruent to itself, side BA is congruent to side BC, and side AD is congruent to side CD. Therefore, angles ABD and CBD are equal. Next, triangles OBA and OBC are congruent, since sides OA and OC are congruent, side OB is congruent to itself, and sides BA and BC are congruent.
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Similarly, three points determine a 2-dimensional linear system (net), two points determine a 3-dimensional linear system (web), one point determines a 4-dimensional linear system, and zero points place no constraints on the 5-dimensional linear system of all conics. The Apollonian circles are two 1-parameter families determined by 2 points. As is well known, three non-collinear points determine a circle in Euclidean geometry and two distinct points determine a pencil of circles such as the Apollonian circles. These results seem to run counter the general result since circles are special cases of conics.
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A related result is the de Bruijn–Erdős theorem. Nicolaas Govert de Bruijn and Paul Erdős proved the result in the more general setting of projective planes, but it still holds in the Euclidean plane. The theorem is:Weisstein, Eric W., "de Bruijn–Erdős Theorem" from MathWorld ::In a projective plane, every non-collinear set of points determines at least distinct lines. As the authors pointed out, since their proof was combinatorial, the result holds in a larger setting, in fact in any incidence geometry in which there is a unique line through every pair of distinct points.
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Pappus's hexagon theorem states that, if a hexagon is drawn in such a way that vertices and lie on a line and vertices and lie on a second line, then each two opposite sides of the hexagon lie on two lines that meet in a point and the three points constructed in this way are collinear. A plane in which Pappus's theorem is universally true is called Pappian. According to , Hessenberg's original proof is not complete; he disregarded the possibility that some additional incidences could occur in the Desargues configuration. A complete proof is provided by .
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Quasi-phase-matching is a technique in nonlinear optics which allows a positive net flow of energy from the pump frequency to the signal and idler frequencies by creating a periodic structure in the nonlinear medium. Momentum is conserved, as is necessary for phase-matching, through an additional momentum contribution corresponding to the wavevector of the periodic structure. Consequently, in principle any three-wave mixing process that satisfies energy conservation can be phase-matched. For example, all the optical frequencies involved can be collinear, can have the same polarization, and travel through the medium in arbitrary directions.
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Figure 1: Diagram illustrating σ molecular orbitals of the triiodide anion. The σ molecular orbitals (MOs) of triiodide can be constructed by considering the in-phase and out-of-phase combinations of the central atom's p orbital (collinear with the bond axis) with the p orbitals of the peripheral atoms. This exercise generates the diagram at right (Figure 1). Three molecular orbitals result from the combination of the three relevant atomic orbitals, with the four electrons occupying the two MOs lowest in energy – a bonding MO delocalized across all three centers, and a non-bonding MO localized on the peripheral centers.
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This arrangement has only m ordinary lines, the lines that connect a vertex v with the point at infinity collinear with the two neighbors of v. As with any finite configuration in the real projective plane, this construction can be perturbed so that all points are finite, without changing the number of ordinary lines. For odd n, only two examples are known that match Dirac's lower bound conjecture, that is, with t_2(n)=(n-1)/2 One example, by , consists of the vertices, edge midpoints, and centroid of an equilateral triangle; these seven points determine only three ordinary lines.
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As Paul Erdős observed, the Sylvester–Gallai theorem immediately implies that any set of n points that are not collinear determines at least n different lines. This result is known as the De Bruijn–Erdős theorem. As a base case, the result is clearly true for n=3. For any larger value of n, the result can be reduced from n points to n-1 points, by deleting an ordinary line and one of the two points on it (taking care not to delete a point for which the remaining subset would lie on a single line).
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Trilinear equation: [cyclic sum bc(a4 − b2c2)x(y2 \+ z2] = 0 Barycentric equation: [cyclic sum (a4 − b2c2)x(c2y2 \+ b2z2] = 0 Let A′B′C′ be the 1st Brocard triangle. For arbitrary point X, let XA, XB, XC be the intersections of the lines XA′, XB′, XC′ with the sidelines BC, CA, AB, respectively. The 1st Brocard cubic is the locus of X for which the points XA, XB, XC are collinear. The 1st Brocard cubic passes through the centroid, symmedian point, Steiner point, other triangle centers, and the vertices of the 1st and 3rd Brocard triangles.
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If X_{1} is highly correlated with another independent variable, X_{2}, in the given data set, then we have a set of observations for which X_{1} and X_{2} have a particular linear stochastic relationship. We don't have a set of observations for which all changes in X_{1} are independent of changes in X_{2}, so we have an imprecise estimate of the effect of independent changes in X_{1}. In some sense, the collinear variables contain the same information about the dependent variable. If nominally "different" measures actually quantify the same phenomenon then they are redundant.
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The simplest affine plane contains only four points; it is called the affine plane of order 2. (The order of an affine plane is the number of points on any line, see below.) Since no three are collinear, any pair of points determines a unique line, and so this plane contains six lines. It corresponds to a tetrahedron where non- intersecting edges are considered "parallel", or a square where not only opposite sides, but also diagonals are considered "parallel". More generally, a finite affine plane of order n has n2 points and lines; each line contains n points, and each point is on lines.
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In the study of collineations, the case of projective lines is special due to the small dimension. When the line is viewed as a projective space in isolation, any permutation of the points of a projective line is a collineation,, , since every set of points are collinear. However, if the projective line is embedded in a higher-dimensional projective space, the geometric structure of that space can be used to impose a geometric structure on the line. Thus, in synthetic geometry, the homographies and the collineations of the projective line that are considered are those obtained by restrictions to the line of collineations and homographies of spaces of higher dimension.
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A conic in a projective plane that contains the two absolute points is called a circle. Since five points determine a conic, a circle (which may be degenerate) is determined by three points. To obtain the extended Euclidean plane, the absolute line is chosen to be the line at infinity of the Euclidean plane and the absolute points are two special points on that line called the circular points at infinity. Lines containing two points with real coordinates do not pass through the circular points at infinity, so in the Euclidean plane a circle, under this definition, is determined by three points that are not collinear.
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Three non-collinear points in space suffice to determine an orbital plane. A common example would be the positions of the centers of a massive body (host) and of an orbiting celestial body at two different times/points of its orbit. The orbital plane is defined in relation to a reference plane by two parameters: inclination (i) and longitude of the ascending node (Ω). By definition, the reference plane for the Solar System is usually considered to be Earth's orbital plane, which defines the ecliptic, the circular path on the celestial sphere that the Sun appears to follow over the course of a year.
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In the Euclidean plane, the real projective plane, higher-dimensional Euclidean spaces or real projective spaces, or spaces with coordinates in an ordered field, the Sylvester–Gallai theorem shows that the only possible Sylvester–Gallai configurations are one-dimensional: they consist of three or more collinear points. was inspired by this fact and by the example of the Hesse configuration to ask whether, in spaces with complex-number coordinates, every Sylvester–Gallai configuration is at most two-dimensional. repeated the question. answered Serre's question affirmatively; simplified Kelly's proof, and proved analogously that in spaces with quaternion coordinates, all Sylvester–Gallai configurations must lie within a three-dimensional subspace.
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In geometry, the Segment Addition Postulate states that given 2 points A and C, a third point B lies on the line segment AC if and only if the distances between the points satisfy the equation AB + BC = AC. This is related to the triangle inequality, which states that AB + BC \geq AC with equality if and only if A, B, and C are collinear (on the same line). This in turn is equivalent to the proposition that the shortest distance between two points lies on a straight line. The segment addition postulate is often useful in proving results on the congruence of segments.
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The collinearity equations are a set of two equations, used in photogrammetry and remote sensing to relate coordinates in an image (sensor) plane (in two dimensions) to object coordinates (in three dimensions). In the photography setting, the equations are derived by considering the central projection of a point of the object through the optical centre of the camera to the image in the image (sensor) plane. The three points, object point, image point and optical centre, are always collinear. Another way to say this is that the line segments joining the object points with their image points are all concurrent at the optical centre.
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Fisica relativista, Kapelusz Editorial, Buenos Aires, Argentina (1955). They were again rediscovered in 1955 by Henri Amar, who subsequently wrote in 1957 in American Journal of Physics: "I regret my unfamiliarity with South American literature and wish to acknowledge the priority of Professor Loedel's work", along with a note by Loedel Palumbo citing his publications on the geometrical representation of Lorentz transformations. Those diagrams are therefore called "Loedel diagrams", and have been cited by some textbook authors on the subject.E. Beneditto, M. Capriolo, A. Feoli, D. Tucci (2013) Some remarks about underused Loedel diagrams European Journal of Physics 34(1) Suppose there are two collinear velocities v and w.
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Nevertheless, the projective viewpoint allows certain configurations to be described more easily. In particular, it allows the use of projective duality, in which the roles of points and lines in statements of projective geometry can be exchanged for each other. Under projective duality, the existence of an ordinary line for a set of non- collinear points in RP2 is equivalent to the existence of an ordinary point in a nontrivial arrangement of finitely many lines. An arrangement is said to be trivial when all its lines pass through a common point, and nontrivial otherwise; an ordinary point is a point that belongs to exactly two lines.
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The easy example is looking at a cube in the direction where the face normal is collinear with the view vector. The first type of silhouette edge is sometimes troublesome to handle because it does not necessarily correspond to a physical edge in the CAD model. The reason that this can be an issue is that a programmer might corrupt the original model by introducing the new silhouette edge into the problem. Also, given that the edge strongly depends upon the orientation of the model and view vector, this can introduce numerical instabilities into the algorithm (such as when a trick like dilution of precision is considered).
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Axiom 4b requires a dilation at P to move Q to R when the three points are collinear. Artin writes the line through P and Q as P + Q. To define a dilation he writes, "Let two distinct points P and Q and their images P′ and Q′ be given." To suggest the role of incidence in geometry, a dilation is specified by this property: "If l′ is the line parallel to P + Q which passes through P′, then Q′ lies on l′." Of course, if P′ ≠ Q′, then this condition implies P + Q is parallel to P′ + Q′, so that the dilation is an affine transformation.
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If is a polarity of a finite projective plane (which need not be desarguesian), , of order then the number of its absolute points (or absolute lines), is given by: : , where is a non- negative integer. Since is an integer, if is not a square, and in this case, is called an orthogonal polarity. R. Baer has shown that if is odd, the absolute points of an orthogonal polarity form an oval (that is, points, no three collinear), while if is even, the absolute points lie on a non-absolute line. In summary, von Staudt conics are not ovals in finite projective planes (desarguesian or not) of even order.
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This configuration is named after Pappus of Alexandria. Pappus's hexagon theorem states that every two triples of collinear points ABC and abc (none of which lie on the intersection of the two lines) can be completed to form a Pappus configuration, by adding the six lines Ab, aB, Ac, aC, Bc, and bC, and their three intersection points , , and . These three points are the intersection points of the "opposite" sides of the hexagon AbCaBc. According to Pappus' theorem, the resulting system of nine points and eight lines always has a ninth line containing the three intersection points X, Y, and Z, called the Pappus line.
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In 1864 Boulton looked at the problems of combustion at constant pressure, in connection with the operation of an industrial gas turbine. His British patent, No. 1636 of 1864, contains points of interest. He realized that the high velocity of the gas jet exiting his combustion chamber nozzle offered a practical difficulty, and proposed to remedy this by the use of successive induced jets of increasing volume and consequently lower velocity. This was shown in his drawing, with gases being delivered through collinear nozzles of increasing diameter, with the outer nozzles operating at increased gas volumes with reduced velocities, similar to the exhaust of a high-bypass turbofan jet engine.
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Rear view of presidential motorcade. ECM suburban is on the left USSS Electronic Countermeasures Suburban is the United States Secret Service Electronic Counter Measures Chevrolet Suburban, an element of a United States presidential motorcade or vice-presidential motorcade. This vehicle is usually in front of the Presidential Sparecoach and usually trails the vice- presidential limo behind the Vice Presidential Follow-Up Vehicle and is used to counter guided attacks, such as IEDs, rocket-propelled grenades, and anti- tank guided missiles. Its most identifying mark are two collinear antennas mounted on the roof that are solely used in barrage jamming applications, the primary method used to counter IED threats.
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For a general input polarization, the net effect of the rhomb is identical to that of a birefringent (doubly-refractive) quarter-wave plate, except that a simple birefringent plate gives the desired 90° separation at a single frequency, and not (even approximately) at widely different frequencies, whereas the phase separation given by the rhomb depends on its refractive index, which varies only slightly over a wide frequency range (see Dispersion). Two Fresnel rhombs can be used in tandem (usually cemented to avoid reflections at their interface) to achieve the function of a half-wave plate. The tandem arrangement, unlike a single Fresnel rhomb, has the additional feature that the emerging beam can be collinear with the original incident beam.
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In mathematics, a ternary relation or triadic relation is a finitary relation in which the number of places in the relation is three. Ternary relations may also be referred to as 3-adic, 3-ary, 3-dimensional, or 3-place. Just as a binary relation is formally defined as a set of pairs, i.e. a subset of the Cartesian product of some sets A and B, so a ternary relation is a set of triples, forming a subset of the Cartesian product of three sets A, B and C. An example of a ternary relation in elementary geometry can be given on triples of points, where a triple is in the relation if the three points are collinear.
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An arbelos is formed from three collinear points A, B, and C, by the three semicircles with diameters AB, AC, and BC. Let the two smaller circles have radii r1 and r2, from which it follows that the larger semicircle has radius r = r1+r2. Let the points D and E be the center and midpoint, respectively, of the semicircle with the radius r1. Let H be the midpoint of line AC. Then two of the four quadruplet circles are tangent to line HE at the point E, and are also tangent to the outer semicircle. The other two quadruplet circles are formed in a symmetric way from the semicircle with radius r2.
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Since the radical axis is a line this means that the three homothetic centers are collinear Let C1 and C2 be a conjugate pair of circles tangent to all three given circles (Figure 11). By conjugate we imply that both tangent circles belong to the same family with respect to any one of the given pairs of circles. As we've already seen, the radical axis of any two tangent circles from the same family passes through the homothetic center of the two given circles. Since the tangent circles are common for all three pairs of given circles then their homothetic centers all belong to the radical axis of C1 and C2 e.g.
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Two central configurations are considered to be equivalent if they are similar, that is, they can be transformed into each other by some combination of rotation, translation, and scaling. With this definition of equivalence, there is only one configuration of one or two points, and it is always central. In the case of three bodies, there are three one-dimensional central configurations, found by Leonhard Euler. The finiteness of the set of three- point central configurations was shown by Joseph-Louis Lagrange in his solution to the three-body problem; Lagrange showed that there is only one non-collinear central configuration, in which the three points form the vertices of an equilateral triangle.
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One way of constructing the Perles configuration is to start with a regular pentagon and its five diagonals, which form the sides of a smaller regular pentagon within the initial one. The nine points of the configuration consist of four out of the five vertices of each pentagon and the shared center of the two pentagons; the two missing pentagon vertices are chosen to be collinear with the center. The nine lines of the configuration consist of the five lines that are diagonals of the outer pentagon and sides of the inner pentagon, and the four lines that pass through the center and through corresponding pairs of vertices from the two pentagons.
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In a plane cubic model three points sum to zero in the group if and only if they are collinear. For an elliptic curve defined over the complex numbers the group is isomorphic to the additive group of the complex plane modulo the period lattice of the corresponding elliptic functions. The intersection of two quadric surfaces is, in general, a nonsingular curve of genus one and degree four, and thus an elliptic curve, if it has a rational point. In special cases, the intersection either may be a rational singular quartic or is decomposed in curves of smaller degrees which are not always distinct (either a cubic curve and a line, or two conics, or a conic and two lines, or four lines).
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The twin circles (red) of an arbelos (grey) Animation of twin circles for various positions of point B on AC segment In geometry, the twin circles are two special circles associated with an arbelos. An arbelos is determined by three collinear points , , and , and is the curvilinear triangular region between the three semicircles that have , , and as their diameters. If the arbelos is partitioned into two smaller regions by a line segment through the middle point of , , and , perpendicular to line , then each of the two twin circles lies within one of these two regions, tangent to its two semicircular sides and to the splitting segment. These circles first appeared in the Book of Lemmas, which showed (Proposition V) that the two circles are congruent.
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The more interesting case is when the material's electron spontaneously break above-mentioned symmetry. For ferromagnetism in the ground state, there is a common spin quantization axis and a global excess of electrons of a given spin quantum number, there are more electrons pointing in one direction than in the other, giving a macroscopic magnetization (typically, the majority electrons are chosen to point up). In the most simple (collinear) cases of antiferromagnetism, there is still a common quantization axis, but the electronic spins are pointing alternatingly up and down, leading again to cancellation of the macroscopic magnetization. However, specifically in the case of frustration of the interactions, the resulting structures can become much more complicated, with inherently three-dimensional orientations of the local spins.
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Elliptic case Hyperbolic case In geometry, the ', named for 18th century British mathematicians William Braikenridge and Colin Maclaurin, is the converse to Pascal's theorem. It states that if the three intersection points of the three pairs of lines through opposite sides of a hexagon lie on a line L, then the six vertices of the hexagon lie on a conic C; the conic may be degenerate, as in Pappus's theorem. The Braikenridge–Maclaurin theorem may be applied in the Braikenridge–Maclaurin construction, which is a synthetic construction of the conic defined by five points, by varying the sixth point. Namely, Pascal's theorem states that given six points on a conic (the vertices of a hexagon), the lines defined by opposite sides intersect in three collinear points.
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A reflective array antenna for radar consisting of numerous dipoles fed in-phase (thus realizing a broadside array) in front of a large reflector (horizontal wires) to make it uni-directional. On the other hand, for a rotating antenna (or one used only towards a particular direction) one may desire increased gain and directivity in a particular horizontal direction. If the broadside array discussed above (whether collinear or not) is turned horizontal, then the one obtains a greater gain in the horizontal direction perpendicular to the antennas, at the expense of most other directions. Unfortunately that also means that the direction opposite the desired direction also has a high gain, whereas high gain is usually desired in one single direction.
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The power which is wasted in the reverse direction, however, can be redirected, for instance by using a large planar reflector, as is accomplished in the reflective array antenna, increasing the gain in the desired direction by another 3 dB An alternative realization of a uni-directional antenna is the end-fire array. In this case the dipoles are again side by side (but not collinear), but fed in progressing phases, arranged so that their waves add coherently in one direction but cancel in the opposite direction. So now, rather than being perpendicular to the array direction as in a broadside array, the directivity is in the array direction (i.e. the direction of the line connecting their feedpoints) but with one of the opposite directions suppressed.
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It corresponds to the direction (upward or downward) of the cross product of the two coplanar vectors defined by the two pairs of points (p_1, p_2) and (p_1, p_3). The sign of the acute angle is the sign of the expression : P = (x_2-x_1)(y_3-y_1)-(y_2-y_1)(x_3-x_1), which is the signed length of the cross product of the two vectors. In the "right-handed" coordinate system, if the result is 0, the points are collinear; if it is positive, the three points constitute a positive angle of rotation around p_1 from p_2 to p_3, otherwise a negative angle. From another point of view, the sign of P tells whether p_3 lies to the left or to the right of line p_1, p_2.
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A generalization of the Droz-Farny line theorem was proved in 1930 by René Goormaghtigh. As above, let T be a triangle with vertices A, B, and C. Let P be any point distinct from A, B, and C, and L be any line through P. Let A_1, B_1, and C_1 be points on the side lines BC, CA, and AB, respectively, such that the lines PA_1, PB_1, and PC_1 are the images of the lines PA, PB, and PC, respectively, by reflection against the line L. Goormaghtigh's theorem then says that the points A_1, B_1, and C_1 are collinear. The Droz-Farny line theorem is a special case of this result, when P is the orthocenter of triangle T.
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If, by means of a suitable internal structure, the E-plane (difference) and H-plane (sum) ports are simultaneously matched, then by symmetry, reciprocity and conservation of energy it may be shown that the two collinear ports are also matched, and are 'magically' isolated from each other. The E-field of the dominant mode in each port is perpendicular to the broad wall of the waveguide. The signals in the E-plane and H-plane ports therefore have orthogonal polarizations, and so (considering the symmetry of the structure) there can be no communication between these two ports. For a signal entering the H-plane port, a well-designed matching structure will prevent any of the power in the signal being reflected back out of the same port.
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As early as 1983, at the "superluminal workshop" held at Jodrell Bank Observatory, referring to the seven then-known superluminal jets, > Schilizzi ... presented maps of arc-second resolution [showing the large- > scale outer jets] ... which ... have revealed outer double structure in all > but one (3C 273) of the known superluminal sources. An embarrassment is that > the average projected size [on the sky] of the outer structure is no smaller > than that of the normal radio-source population. In other words, the jets are evidently not, on average, close to our line-of- sight. (Their apparent length would appear much shorter if they were.) In 1993, Thomson et al. suggested that the (outer) jet of the quasar 3C 273 is nearly collinear to our line-of-sight.
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These are Lemmas XII, XIII, XV, and XVII in the part of Book VII consisting of lemmas to the first of the three books of Euclid's Porisms. The lemmas are proved in terms of what today is known as the cross ratio of four collinear points. Three earlier lemmas are used. The first of these, Lemma III, has the diagram below (which uses Pappus's lettering, with G for Γ, D for Δ, J for Θ, and L for Λ). :Pappus-collection-7-129 Here three concurrent straight lines, AB, AG, and AD, are crossed by two lines, JB and JE, which concur at J. Also KL is drawn parallel to AZ. Then :KJ : JL :: (KJ : AG & AG : JL) :: (JD : GD & BG : JB).
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A plane wave can be studied by ignoring the directions perpendicular to the direction vector \vec n; that is, by considering the function G(z,t) = F(z \vec n, t) as a wave in a one-dimensional medium. Any local operator, linear or not, applied to a plane wave yields a plane wave. Any linear combination of plane waves with the same normal vector \vec n is also a plane wave. For a scalar plane wave in two or three dimensions, the gradient of the field is always collinear with the direction \vec n; specifically, abla F(\vec x,t) = \vec n\partial_1 G(\vec x \cdot \vec n, t), where \partial_1 G is the partial derivative of G with respect to the first argument.
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Cu2SO4 crystallizes in the orthorhombic space group Fddd with cubic closest pack cell unit properties a = 474.8, b = 1396, and c = 1086 pm, Z = 8, Dx = 4.12 g cm−3. The structure is formed by the tendency of Cu(I) to form two collinear sp bonds to oxygen atoms which result in short O-Cu-O groups with the O-Cu bond length being approximately 196vpm. Due to the bonding properties of the Cu(I) metal the structure is built of layers of composition Cu2SO4. Due to the loss of only one electron and the bonding properties of copper(I) the structure is built up from four oxygen atoms of each sulfate group bonding to four other sulfate groups of the same layer via symmetrical O-Cu-O bridging.
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Subclasses of the planar graphs may, in general, have smaller universal sets (sets of points that allow straight-line drawings of all n-vertex graphs in the subclass) than the full class of planar graphs, and in many cases universal point sets of exactly n points are possible. For instance, it is not hard to see that every set of n points in convex position (forming the vertices of a convex polygon) is universal for the n-vertex outerplanar graphs, and in particular for trees. Less obviously, every set of n points in general position (no three collinear) remains universal for outerplanar graphs.. Planar graphs that can be partitioned into nested cycles, 2-outerplanar graphs and planar graphs of bounded pathwidth, have universal point sets of nearly-linear size.; .
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There are still ongoing discussions about the correct form of equations for the Thomas rotation in different reference systems with contradicting results. Goldstein: :The spatial rotation resulting from the successive application of two non-collinear Lorentz transformations have been declared every bit as paradoxical as the more frequently discussed apparent violations of common sense, such as the twin paradox. Einstein's principle of velocity reciprocity (EPVR) reads :We postulate that the relation between the coordinates of the two systems is linear. Then the inverse transformation is also linear and the complete non- preference of the one or the other system demands that the transformation shall be identical with the original one, except for a change of to With less careful interpretation, the EPVR is seemingly violated in some models.
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However, many other types of hyperovals of PG(2, q) can be found if q > 8\. Hyperovals of PG(2, q) for q even have only been classified for q < 64 to date. In PG(2,2h), h > 0, a hyperoval contains at least four points no three of which are collinear. Thus, by the Fundamental Theorem of Projective Geometry we can always assume that the points with projective coordinates (1,0,0), (0,1,0), (0,0,1) and (1,1,1) are contained in any hyperoval. The remaining points of the hyperoval (when h > 1) will have the form (t, f(t),1) where t ranges through the values of the finite field GF(2h) and f is a function on that field which represents a permutation and can be uniquely expressed as a polynomial of degree at most 2h \- 2, i.e.
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There are thousands of different constructions that find a special point associated with (and often inside) a triangle, satisfying some unique property: see the article Encyclopedia of Triangle Centers for a catalogue of them. Often they are constructed by finding three lines associated in a symmetrical way with the three sides (or vertices) and then proving that the three lines meet in a single point: an important tool for proving the existence of these is Ceva's theorem, which gives a criterion for determining when three such lines are concurrent. Similarly, lines associated with a triangle are often constructed by proving that three symmetrically constructed points are collinear: here Menelaus' theorem gives a useful general criterion. In this section just a few of the most commonly encountered constructions are explained.
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Rational trigonometry follows an approach built on the methods of linear algebra to the topics of elementary (high school level) geometry. Distance is replaced with its squared value (quadrance) and 'angle' is replaced with the squared value of the usual sine ratio (spread) associated to either angle between two lines. (The complement of Spread, known as cross, also corresponds to a scaled form of the inner product between line segments taken as vectors). The three main laws in trigonometry – Pythagoras's theorem, the sine law and the cosine law – are given in rational (square-equivalent) form, and are augmented by two further laws – the triple quad formula (relating the quadrances of three collinear points) and the triple spread formula (relating the spreads of three concurrent lines) –, giving the five main laws of the subject.
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An arbelos is a shape bounded by three mutually-tangent semicircular arcs with collinear endpoints, with the two smaller arcs nested inside the larger one; let the endpoints of these three arcs be (in order along the line containing them) A, B, and C. Let K1 and K2 be two more arcs, centered at A and C, respectively, with radii AB and CB, so that these two arcs are tangent at B; let K3 be the largest of the three arcs of the arbelos. A circle, with the center A1, is then created tangent to the arcs K1,K2, and K3. This circle is congruent with Archimedes' twin circles, making it an Archimedean circle; it is one of the Schoch circles. The Schoch line is perpendicular to the line AC and passes through the point A1.
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Diagram showing the process of electron–positron pair production. In reality the produced pair are nearly collinear. For photons with high photon energy (MeV scale and higher), pair production is the dominant mode of photon interaction with matter. These interactions were first observed in Patrick Blackett's counter- controlled cloud chamber, leading to the 1948 Nobel Prize in Physics. If the photon is near an atomic nucleus, the energy of a photon can be converted into an electron–positron pair: : -> + The photon's energy is converted to particle mass in accordance with Einstein’s equation, ; where is energy, is mass and is the speed of light. The photon must have higher energy than the sum of the rest mass energies of an electron and positron (2 ⋅ 511 keV = 1.022 MeV, resulting in a photon-wavelength of 1.2132 picometer) for the production to occur.
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It is named after the Greek astronomer and mathematician Ptolemy. The four points can be ordered in any of three distinct ways (counting reversals as not distinct) to form three different quadrilaterals, for each of which the sum of the products of opposite sides is at least as large as the product of the diagonals. Thus the three product terms in the inequality can be additively permuted to put any one of them on the right side of the inequality, so the three products of opposite sides or of diagonals of any one of the quadrilaterals must obey the triangle inequality.. As a special case, Ptolemy's theorem states that the inequality becomes an equality when the four points lie in cyclic order on a circle. The other case of equality occurs when the four points are collinear in order.
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In extreme cases, where the interference is either deliberate or all attempts to get rid of the offending device have proved futile, it may be possible to look at changing the parameters of the network. Changing collinear antennas for high gain directional dishes normally works very well, since the narrow beam from a high gain dish will not physically "see" the interference. Often sector antennae have sharp "nulls" in their vertical pattern, so changing the tilt angle of sector antennas with a spectrum analyzer connected to monitor the strength of the interference can place the offending device within the null of the sector. High gain antennas on the transmitter end can "overpower" the interference, although their use may cause the effective radiated power (ERP) of the signal to become too high, and so their use may not be legal.
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Under a molecular orbital theory framework, the oxygen-oxygen bond in triplet dioxygen is better described as one full σ bond plus two π half-bonds, each half-bond accounted for by two-center three- electron (2c-3e) bonding, to give a net bond order of two (1+2×), while also accounting for the spin state (S = 1). In the case of triplet dioxygen, each 2c-3e bond consists of two electrons in a πu bonding orbital and one electron in a πg antibonding orbital to give a net bond order contribution of . The usual rules for constructing Lewis structures must be modified to accommodate molecules like triplet dioxygen or nitric oxide that contain 2c-3e bonds. There is no consensus in this regard; Pauling has suggested the use of three closely spaced collinear dots to represent the three-electron bond (see illustration).
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Collinear folded dipole array Many types of array antennas are constructed using multiple dipoles, usually half-wave dipoles. The purpose of using multiple dipoles is to increase the directional gain of the antenna over the gain of a single dipole; the radiation of the separate dipoles interferes to enhance power radiated in desired directions. In arrays with multiple dipole driven elements, the feedline is split using an electrical network in order to provide power to the elements, with careful attention paid to the relative phase delays due to transmission between the common point and each element. In order to increase antenna gain in horizontal directions (at the expense of radiation towards the sky or towards the ground) one can stack antennas in the vertical direction in a broadside array where the antennas are fed in phase.
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The magnetic properties of the β-CrPO4 are a result of the cation-cation distances along the octahedral chains which give rise to strong direct-exchange interactions and even metal- metal bonding. Neutron diffraction studies reveal that the spiral moments in β-CrPO4 are collinear and anti-ferromagnetically coupled along the chains in the 001 planes, at low temperature (5K, µeff = 2.55µB). Observations from a diffraction study has shown that at low temperature(2K), the α-CrPO4 octahedra CrO6 units build up an infinite, three-dimensional network expected to provide strong Cr-O-Cr magnetic superexchange linkages with exchange pathway through the phosphate group. These linkages give the structure its anti-ferromagnetic characteristic (Ɵ = -35.1 K, µeff = 3.50µB) which results in the anti-parallel magnetic spins in the plane that is perpendicular to the chains of the octahedral CrO6.
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In statistics and machine learning, lasso (least absolute shrinkage and selection operator; also Lasso or LASSO) is a regression analysis method that performs both variable selection and regularization in order to enhance the prediction accuracy and interpretability of the statistical model it produces. It was originally introduced in geophysics literature in 1986, and later independently rediscovered and popularized in 1996 by Robert Tibshirani, who coined the term and provided further insights into the observed performance. Lasso was originally formulated for linear regression models and this simple case reveals a substantial amount about the behavior of the estimator, including its relationship to ridge regression and best subset selection and the connections between lasso coefficient estimates and so-called soft thresholding. It also reveals that (like standard linear regression) the coefficient estimates do not need to be unique if covariates are collinear.
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The happy ending problem: every set of five points in general position contains the vertices of a convex quadrilateral The "happy ending problem" (so named by Paul Erdős because it led to the marriage of George Szekeres and Esther KleinA world of teaching and numbers - times two, Michael Cowling, The Sydney Morning Herald, 2005-11-07, cited 2014-09-04) is the following statement: :Theorem: any set of five points in the plane in general positionIn this context, general position means that no two points coincide and no three points are collinear. has a subset of four points that form the vertices of a convex quadrilateral. This was one of the original results that led to the development of Ramsey theory. The happy ending theorem can be proven by a simple case analysis: if four or more points are vertices of the convex hull, any four such points can be chosen.
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All NEMA 6 devices are three-wire grounding devices (hot-hot-ground) used for 208 and 240 V circuits and rated for 250 V maximum, with the 6-15, 6-20 and 6-30 being grounding versions of the 2-15, 2-20 and 2-30, respectively. The 6-15 resembles the 5-15, but with collinear horizontal pins, spaced center-to- center. The 20 A plug has a blade rotated 90°, and the 6-20R receptacle may have a T-shaped hole, to accept both 6-15P and 6-20P plugs (similar to the 5-20R receptacle accepting 5-15P and 5-20P plugs). 6-15R and 6-20R receptacles are usually manufactured on the same assembly line as "Industrial" or "Commercial" grade 5-15R and 5-20R receptacles, with all 4 receptacles sharing the same "triple wipe" T contacts behind the varying faceplates.
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The antiparallelogram is an important feature in the design of Hart's inversor, a linkage that (like the Peaucellier–Lipkin linkage) can convert rotary motion to straight-line motion.. An antiparallelogram-shaped linkage can also be used to connect the two axles of a four-wheeled vehicle, decreasing the turning radius of the vehicle relative to a suspension that only allows one axle to turn.. A pair of nested antiparallelograms was used in a linkage defined by Alfred Kempe as part of his universality theorem stating that any algebraic curve may be traced out by the joints of a suitably defined linkage. Kempe called the nested-antiparallelogram linkage a "multiplicator", as it could be used to multiply an angle by an integer. Antiparallelogram braced to stop it turning into a normal parallelogram. The points PQRS are the midpoints of the sides and are collinear, and X can be any distance away on the perpendicular bisector of SQ. With this linkage .
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Edelsbrunner's most heavily cited research contributionAccording to Google scholar, August 2008. is his work with Ernst Mücke on alpha shapes, a technique for defining a sequence of multiscale approximations to the shape of a three-dimensional point cloud. In this technique, one varies a parameter alpha ranging from 0 to the diameter of the point cloud; for each value of the parameter, the shape is approximated as the union of line segments, triangles, and tetrahedra defined by 2, 3, or 4 of the points respectively such that there exists a sphere of radius at most alpha containing only the defining points. Another heavily cited paper, also with Mücke, concerns “simulation of simplicity.” This is a technique for automatically converting algorithms that work only when their inputs are in general position (for instance, algorithms that may misbehave when some three input points are collinear) into algorithms that work robustly, correctly, and efficiently in the face of special-position inputs.
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History of multiferroics: number of papers per year on magnetoelectrics or the magnetoelectric effect (in blue), and on multiferroics (in red) A Web of Science search for the term multiferroic yields the year 2000 paper "Why are there so few magnetic ferroelectrics?" from N. A. Spaldin (then Hill) as the earliest result. This work explained the origin of the contraindication between magnetism and ferroelectricity and proposed practical routes to circumvent it, and is widely credited with starting the modern explosion of interest in multiferroic materials . The availability of practical routes to creating multiferroic materials from 2000 stimulated intense activity. Particularly key early works were the discovery of large ferroelectric polarization in epitaxially grown thin films of magnetic BiFeO3, the observation that the non-collinear magnetic ordering in orthorhombic TbMnO3 and TbMn2O5 causes ferroelectricity, and the identification of unusual improper ferroelectricity that is compatible with the coexistence of magnetism in hexagonal manganite YMnO3.
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If three spheres are given, with their centers non-collinear, then their six centers of similitude form the six points of a complete quadrilateral, the four lines of which are called the axes of similitude. And if four spheres are given, with their centers non- coplanar, then they determine 12 centers of similitude and 16 axes of similitude, which together form an instance of the Reye configuration . The Reye configuration can also be realized by points and lines in the Euclidean plane, by drawing the three-dimensional configuration in three-point perspective. An 83122 configuration of eight points in the real projective plane and 12 lines connecting them, with the connection pattern of a cube, can be extended to form the Reye configuration if and only if the eight points are a perspective projection of a parallelepiped The 24 permutations of the points (\pm 1, \pm 1, 0, 0) form the vertices of a 24-cell centered at the origin of four-dimensional Euclidean space.
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An arc diagram As well as for straight-line graph drawing, universal point sets have been studied for other drawing styles; in many of these cases, universal point sets with exactly n points exist, based on a topological book embedding in which the vertices are placed along a line in the plane and the edges are drawn as curves that cross this line at most once. For instance, every set of n collinear points is universal for an arc diagram in which each edge is represented as either a single semicircle or a smooth curve formed from two semicircles.. By using a similar layout, every convex curve in the plane can be shown to contain an n-point subset that is universal for polyline drawing with at most one bend per edge.. This set contains only the vertices of the drawing, not the bends; larger sets are known that can be used for polyline drawing with all vertices and all bends placed within the set..
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Lone pair–lone pair (lp–lp) repulsions are considered stronger than lone pair–bonding pair (lp–bp) repulsions, which in turn are considered stronger than bonding pair–bonding pair (bp–bp) repulsions, distinctions that then guide decisions about overall geometry when 2 or more non-equivalent positions are possible. For instance, when 5 valence electron pairs surround a central atom, they adopt a trigonal bipyramidal molecular geometry with two collinear axial positions and three equatorial positions. An electron pair in an axial position has three close equatorial neighbors only 90° away and a fourth much farther at 180°, while an equatorial electron pair has only two adjacent pairs at 90° and two at 120°. The repulsion from the close neighbors at 90° is more important, so that the axial positions experience more repulsion than the equatorial positions; hence, when there are lone pairs, they tend to occupy equatorial positions as shown in the diagrams of the next section for steric number five.
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Saffron has advised businesses and institutions all over the world on brand and strategy. Their clients come from a wide variety of sectors, from energy to telecommunications, retail to museums, Silicon Valley giants to European start-ups. Some notable clients include YouTube, Siemens, Engie, Goldman Sachs, Santander, T-Mobile, Vueling, BBVA, Bankinter, Sodexo, Flying Tiger Copenhagen, Pacha, A1 Telekom Austria, C&A;, Fujitsu, Volotea, V Festival, Doha Film Institute, King's College London, The Institute of Cancer Research, Madrid Open and Turkcell, Vueling, Coca Cola, C&A;, Fujitsu, Raiffeisen Bank International, LVMH, Iberia, Lloyd's of London, Swiss Re. Saffron has worked for a number of cities and countries on place brand strategy and identities including Vienna, the world’s most liveable city, and London, Northern Ireland, Turkey, Trinidad & Tobago and Poland.[13] The company also works to define Employee Value Proposition for clients, notably Sodexo, one of the world’s largest employers. Recent clients of Saffron include London’s Victoria & Albert Museum, Despegar – Latin America’s largest online travel agency, HK Express airline, Collinear, Gulf Air and Repsol.
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For any d-dimensional polytope, one can specify its collection of facet directions and measures by a finite set of d-dimensional nonzero vectors, one per facet, pointing perpendicularly outward from the facet, with length equal to the (d-1)-dimensional measure of its facet. As Hermann Minkowski proved, a finite set of nonzero vectors describes a polytope in this way if and only if it spans the whole d-dimensional space, no two are collinear with the same sign, and the sum of the set is the zero vector. The polytope described by this set has a unique shape, in the sense that any two polytopes described by the same set of vectors are translates of each other. The Blaschke sum X\\# Y of two polytopes X and Y is defined by combining the vectors describing their facet directions and measures, in the obvious way: form the union of the two sets of vectors, except that when both sets contain vectors that are parallel and have the same sign, replace each such pair of parallel vectors by its sum.
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The first theorem considers any four circles passing through a common point M and otherwise in general position, meaning that there are six additional points where exactly two of the circles cross and that no three of these crossing points are collinear. Every set of three of these four circles has among them three crossing points, and (by the assumption of non-collinearity) there exists a circle passing through these three crossing points. The conclusion is that, like the first set of four circles, the second set of four circles defined in this way all pass through a single point P (in general not the same point as M). The second theorem considers five circles in general position passing through a single point M. Each subset of four circles defines a new point P according to the first theorem. Then these five points all lie on a single circle C. The third theorem considers six circles in general position that pass through a single point M. Each subset of five circles defines a new circle by the second theorem.
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In connection with Tverberg's theorem, conjectured that, for every set of r(d + 1) points in d-dimensional Euclidean space, colored with d + 1 colors in such a way that there are r points of each color, there is a way to partition the points into rainbow simplices (sets of d + 1 points with one point of each color) in such a way that the convex hulls of these sets have a nonempty intersection.. For instance, the two-dimensional case (proven by Bárány and Larman) with r = 3 states that, for every set of nine points in the plane, colored with three colors and three points of each color, it is possible to partition the points into three intersecting rainbow triangles, a statement similar to Rota's basis conjecture which states that it is possible to partition the points into three non-degenerate rainbow triangles. The conjecture of Bárány and Larman allows a collinear triple of points to be considered as a rainbow triangle, whereas Rota's basis conjecture disallows this; on the other hand, Rota's basis conjecture does not require the triangles to have a common intersection. Substantial progress on the conjecture of Bárány and Larman was made by ..
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