99 Sentences With "conic section" | Random Sentence Generator

Nicoteles of Cyrene () (c. 250 BCE) was a Greek mathematician from Cyrene. He is mentioned in the preface to Book IV of the Conics of Apollonius, as criticising Conon concerning the maximum number of points with which a conic section can meet another conic section. Apollonius states that Nicoteles claimed that the case in which a conic section meets opposite sections could be solved, but had not demonstrated how.

It touches the sides at its midpoints. There is no other (non-degenerate) conic section with the same properties, because a conic section is determined by 5 points/tangents. b) By a simple calculation. c) The circumcircle is mapped by a scaling, with factor 1/2 and the centroid as center, onto the incircle.

If these points are real, the curve is a hyperbola; if they are imaginary conjugates, it is an ellipse; if there is only one double point, it is a parabola. If the points at infinity are the cyclic points and , the conic section is a circle. If the coefficients of a conic section are real, the points at infinity are either real or complex conjugate.

To distinguish the degenerate cases from the non-degenerate cases (including the empty set with the latter) using matrix notation, let be the determinant of the 3×3 matrix of the conic section—that is, ; and let be the discriminant. Then the conic section is non-degenerate if and only if . If we have a point when , two parallel lines (possibly coinciding) when , or two intersecting lines when .

The directrix of a conic section can be found using Dandelin's construction. Each Dandelin sphere intersects the cone at a circle; let both of these circles define their own planes. The intersections of these two parallel planes with the conic section's plane will be two parallel lines; these lines are the directrices of the conic section. However, a parabola has only one Dandelin sphere, and thus has only one directrix.

A conic section can be created using five points near the end of the known data. If the conic section created is an ellipse or circle, when extrapolated it will loop back and rejoin itself. An extrapolated parabola or hyperbola will not rejoin itself, but may curve back relative to the X-axis. This type of extrapolation could be done with a conic sections template (on paper) or with a computer.

Hyperbolas may be seen in many sundials. On any given day, the sun revolves in a circle on the celestial sphere, and its rays striking the point on a sundial traces out a cone of light. The intersection of this cone with the horizontal plane of the ground forms a conic section. At most populated latitudes and at most times of the year, this conic section is a hyperbola.

If there is an intersection point of multiplicity at least 3, the two curves are said to be osculating. If there is only one intersection point, which has multiplicity 4, the two curves are said to be superosculating.. Furthermore, each straight line intersects each conic section twice. If the intersection point is double, the line is a tangent line. Intersecting with the line at infinity, each conic section has two points at infinity.

The second theorem is that for any conic section, the distance from a fixed point (the focus) is proportional to the distance from a fixed line (the directrix), the constant of proportionality being called the eccentricity. A conic section has one Dandelin sphere for each focus. An ellipse has two Dandelin spheres touching the same nappe of the cone, while hyperbola has two Dandelin spheres touching opposite nappes. A parabola has just one Dandelin sphere.

Dandelin spheres are touching the pale yellow plane that intersects the cone.In geometry, the Dandelin spheres are one or two spheres that are tangent both to a plane and to a cone that intersects the plane. The intersection of the cone and the plane is a conic section, and the point at which either sphere touches the plane is a focus of the conic section, so the Dandelin spheres are also sometimes called focal spheres.Taylor, Charles.

In these methods, it is considered that the body is always moving in a conic section, however the conic section is constantly changing due to the perturbations. If all perturbations were to cease at any particular instant, the body would continue in this (now unchanging) conic section indefinitely; this conic is known as the osculating orbit and its orbital elements at any particular time are what are sought by the methods of general perturbations. General perturbations takes advantage of the fact that in many problems of celestial mechanics, the two-body orbit changes rather slowly due to the perturbations; the two-body orbit is a good first approximation. General perturbations is applicable only if the perturbing forces are about one order of magnitude smaller, or less, than the gravitational force of the primary body.

Together they worked on free fall, catenary, conic section, and fluid statics. Both believed that it was necessary to create a method that thoroughly linked mathematics and physics.Durandin, Guy. 1970. Les Principes de la Philosophie.

1\. Definition of the Steiner generation of a conic section The Steiner conic or more precisely Steiner's generation of a conic, named after the Swiss mathematician Jakob Steiner, is an alternative method to define a non- degenerate projective conic section in a projective plane over a field. The usual definition of a conic uses a quadratic form (see Quadric (projective geometry)). Another alternative definition of a conic uses a hyperbolic polarity. It is due to K. G. C. von Staudt and sometimes called a von Staudt conic.

It is zero if the conic section degenerates into two lines, a double line or a single point. The second discriminant, which is the only one that is considered in many elementary textbooks, is the discriminant of the homogeneous part of degree two of the equation. It is equal to :b^2 - ac, and determines the shape of the conic section. If this discriminant is negative, the curve either has no real points, or is an ellipse or a circle, or, if degenerated, is reduced to a single point.

Apollonius, Conics, Book I, Definition 4. Refer also to If the figure of the conic section is cut by a grid of parallel lines, the diameter bisects all the line segments included between the branches of the figure.

The calculator also has a special section for advanced conic section graphing. Dynamic graphing provides all the functionality of regular graphing, but allows the binding of a variable in the graph equation to time over a value range.

In mathematics, a quadratic set is a set of points in a projective space that bears the same essential incidence properties as a quadric (conic section in a projective plane, sphere or cone or hyperboloid in a projective space).

Brianchon's theorem In geometry, Brianchon's theorem is a theorem stating that when a hexagon is circumscribed around a conic section, its principal diagonals (those connecting opposite vertices) meet in a single point. It is named after Charles Julien Brianchon (1783–1864).

This conic section is the intersection of the cone of light rays with the flat surface. This cone and its conic section change with the seasons, as the Sun's declination changes; hence, sundials that follow the motion of such light- spots or shadow-tips often have different hour-lines for different times of the year. This is seen in shepherd's dials, sundial rings, and vertical gnomons such as obelisks. Alternatively, sundials may change the angle or position (or both) of the gnomon relative to the hour lines, as in the analemmatic dial or the Lambert dial.

Hence for any line g the image \pi(g)=\pi_a\pi_b(g) can be constructed and therefore the images of an arbitrary set of points. The lines u and v contain only the conic points U and V resp.. Hence u and v are tangent lines of the generated conic section. A proof that this method generates a conic section follows from switching to the affine restriction with line w as the line at infinity, point O as the origin of a coordinate system with points U,V as points at infinity of the x- and y-axis resp. and point E=(1,1).

A toric section is an intersection of a plane with a torus, just as a conic section is the intersection of a plane with a cone. Special cases have been known since antiquity, and the general case was studied by Jean Gaston Darboux..

In geometry, two diameters of a conic section are said to be conjugate if each chord parallel to one diameter is bisected by the other diameter. For example, two diameters of a circle are conjugate if and only if they are perpendicular.

The proof makes use of the property that for every conic section we can find a one-sheet hyperboloid which passes through the conic. There also exists a simple proof for Pascal's theorem for a circle using the law of sines and similarity.

This discussion started from the definition of a parabola as a conic section, but it has now led to a description as a graph of a quadratic function. This shows that these two descriptions are equivalent. They both define curves of exactly the same shape.

Under standard assumptions, a body moving under the influence of a force, directed to a central body, with a magnitude inversely proportional to the square of the distance (such as gravity), has an orbit that is a conic section (i.e. circular orbit, elliptic orbit, parabolic trajectory, hyperbolic trajectory, or radial trajectory) with the central body located at one of the two foci, or the focus (Kepler's first law). If the conic section intersects the central body, then the actual trajectory can only be the part above the surface, but for that part the orbit equation and many related formulas still apply, as long as it is a freefall (situation of weightlessness).

Pascal's theorem (also known as the "Hexagrammum Mysticum Theorem") states that if an arbitrary hexagon is inscribed in any conic section, and pairs of opposite sides are extended until they meet, the three intersection points will lie on a straight line, the "Pascal line" of that configuration.

In any case, the intersection curve of a plane and a quadric (sphere, cylinder, cone,...) is a conic section. For details, see.CDKG: Computerunterstützte Darstellende und Konstruktive Geometrie (TU Darmstadt) (PDF; 3,4 MB), p. 87–124 An important application of plane sections of quadrics is contour lines of quadrics.

This was given by Isaac Newton through his Inverse Square Law. (Proposition 4), and relationships between centripetal forces varying as the inverse-square of the distance to the center and orbits of conic-section form (Propositions 5–10). Propositions 11–31 establish properties of motion in paths of eccentric conic-section form including ellipses, and their relation with inverse-square central forces directed to a focus, and include Newton's theorem about ovals (lemma 28). Propositions 43–45 are demonstration that in an eccentric orbit under centripetal force where the apse may move, a steady non-moving orientation of the line of apses is an indicator of an inverse- square law of force.

Relativity: An Introduction to the Special Theory, pp. 5–10. World Scientific. . Although this may be the first suggestion that a conic section could play a role in astronomy, al-Zarqālī did not apply the ellipse to astronomical theory and neither he nor his Iberian or Maghrebi contemporaries used an elliptical deferent in their astronomical calculations.

The symmetry of point and line is expressed as projective duality. Starting with perspectivities, the transformations of projective geometry are formed by composition, producing projectivities. Steiner identified sets preserved by projectivities such as a projective range and pencils. He is particularly remembered for his approach to a conic section by way of projectivity called the Steiner conic.

Conics, also known as conic sections to emphasize their three-dimensional geometry, arise as the intersection of a plane with a cone. Degeneracy occurs when the plane contains the apex of the cone or when the cone degenerates to a cylinder and the plane is parallel to the axis of the cylinder. See Conic section#Degenerate cases for details.

If six unordered points are given on a conic section, they can be connected into a hexagon in 60 different ways, resulting in 60 different instances of Pascal's theorem and 60 different Pascal lines. This configuration of 60 lines is called the Hexagrammum Mysticum. with a reference to Veblen and Young, Projective Geometry, vol. I, p.

The affine part of the generated curve appears to be the hyperbola y=1/x. Remark: #The Steiner generation of a conic section provides simple methods for the construction of ellipses, parabolas and hyperbolas which are commonly called the parallelogram methods. #The figure that appears while constructing a point (figure 3) is the 4-point-degeneration of Pascal's theorem.

Eccentricity, which is a mathematical constant conveyed in the form of a ratio and essentially describes to what degree a conic section deviates from being circular. Another variable that impacts motion silencing, eccentricity determines to what extent motion causes silencing. Choi, Bovik, and Cormack (2016) observed that when eccentricity in peripheral vision increases, motion silencing decreases.

An illustration of various conic constants In geometry, the conic constant (or Schwarzschild constant, after Karl Schwarzschild) is a quantity describing conic sections, and is represented by the letter K. The constant is given by :K = -e^2, where e is the eccentricity of the conic section. The equation for a conic section with apex at the origin and tangent to the y axis is :y^2-2Rx+(K+1)x^2=0 where R is the radius of curvature at x = 0\. This formulation is used in geometric optics to specify oblate elliptical (K > 0), spherical (K = 0), prolate elliptical (0 > K > −1), parabolic (K = −1), and hyperbolic (K < −1) lens and mirror surfaces. When the paraxial approximation is valid, the optical surface can be treated as a spherical surface with the same radius.

In general the determination of an intersection leads to non-linear equations, which can be solved numerically, for example using Newton iteration. Intersection problems between a line and a conic section (circle, ellipse, parabola, etc.) or a quadric (sphere, cylinder, hyperboloid, etc.) lead to quadratic equations that can be easily solved. Intersections between quadrics lead to quartic equations that can be solved algebraically.

Using the Dandelin spheres, it can be proved that any conic section is the locus of points for which the distance from a point (focus) is proportional to the distance from the directrix.Brannan, A. et al. Geometry, page 19 (Cambridge University Press, 1999). Ancient Greek mathematicians such as Pappus of Alexandria were aware of this property, but the Dandelin spheres facilitate the proof.

Sir Isaac Newton showed that a body controlled by the Sun moves in a conic section—that is, an ellipse, a parabola or a hyperbola. Because the latter two are open curves, a comet which pursued such a path would go off into space never to reappear. A derangement of orbit from closed to open curve has doubtless happened often.

Isaac Newton developed a convenient and inexpensive sundial, in which a small mirror is placed on the sill of a south- facing window.Waugh (1973), pp. 116–121. The mirror acts like a nodus, casting a single spot of light on the ceiling. Depending on the geographical latitude and time of year, the light-spot follows a conic section, such as the hyperbolae of the pelikonon.

Cone with cross-sections The diagram represents a cone with its axis . The point A is its apex. An inclined cross-section of the cone, shown in pink, is inclined from the axis by the same angle , as the side of the cone. According to the definition of a parabola as a conic section, the boundary of this pink cross- section EPD is a parabola.

The Steiner inellipse plays a special role: Its area is the greatest of all inellipses. Because a non-degenerate conic section is uniquely determined by five items out of the sets of vertices and tangents, in a triangle whose three sides are given as tangents one can specify only the points of contact on two sides. The third point of contact is then uniquely determined.

Pencils of circles: in the pencil of red circles, the only degenerate conic is the horizontal axis; the pencil of blue circles has three degenerate conics, the vertical axis and two circles of radius zero. The conic section with equation x^2-y^2 = 0 is degenerate as its equation can be written as (x-y)(x+y)= 0, and corresponds to two intersecting lines forming an "X". This degenerate conic occurs as the limit case a=1, b=0 in the pencil of hyperbolas of equations a(x^2-y^2) - b=0. The limiting case a=0, b=1 is an example of a degenerate conic consisting of twice the line at infinity. Similarly, the conic section with equation x^2 + y^2 = 0, which has only one real point, is degenerate, as x^2+y^2 is factorable as (x+iy)(x-iy) over the complex numbers.

Let ABCDEF be a hexagon formed by six tangent lines of a conic section. Then Brianchon's theorem states that the three main diagonals AD, BE, and CF intersect at a single point. In a hexagon that is tangential to a circle and that has consecutive sides a, b, c, d, e, and f,Gutierrez, Antonio, "Hexagon, Inscribed Circle, Tangent, Semiperimeter", , Accessed 2012-04-17. :a+c+e=b+d+f.

A sharply angled plane with an offset conic section removed was chosen as the most efficient. With this configuration, the head first separated (by force) the sediments to be displaced while supporting the sediments of the bore wall. A vortex of water was created by angled water jets in the conic space. This design massively disturbed sediments in one ‘exhaust’ sector of the SPI image, but minimised disturbance in the remainder.

It keeps an amount of functionality of C.a.R. but uses a different graphical interface which purportedly eliminates some intermediate dialogs and provides direct access to numerous effects. Constructions are done using a main palette, which contains some useful construction shortcuts in addition to the standard compass and ruler tools. These include perpendicular bisector, circle through three points, circumcircular arc through three points, and conic section through five points.

A projective cone (or just cone) in projective geometry is the union of all lines that intersect a projective subspace R (the apex of the cone) and an arbitrary subset A (the basis) of some other subspace S, disjoint from R. In the special case that R is a single point, S is a plane, and A is a conic section on S, the projective cone is a conical surface; hence the name.

A rainbow is perceived as a circle in the sky; and its contributing light rays form a cone. In contrast, a dewbow is perceived as the intersection of that cone and the ground. If the ground is flat and horizontal, and the sun is low in the sky, the dew bow is a hyperbola. Theoretically, when the sun is high, the intersection might be another conic section, like a parabola or an ellipse.

The equation of a conic section in the variable trilinear point x : y : z is :rx^2+sy^2+tz^2+2uyz+2vzx+2wxy=0. It has no linear terms and no constant term. The equation of a circle of radius r having center at actual-distance coordinates (a', b', c' ) is :(x-a')^2\sin 2A+(y-b')^2\sin 2B+(z-c')^2\sin 2C=2r^2\sin A\sin B\sin C.

He argued that a mirror shaped like the part of a conic section, would correct the spherical aberration that flawed the accuracy of refracting telescopes. His design, the "Gregorian telescope", however, remained un-built. In 1666, Isaac Newton argued that the faults of the refracting telescope were fundamental because the lens refracted light of different colors differently. He concluded that light could not be refracted through a lens without causing chromatic aberrations.

In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section (though it may be degeneratethe empty set is included as a degenerate conic since it may arise as a solution of this equation), and all conic sections arise in this way. The most general equation is of the form :Ax^2 + Bxy + Cy^2 +Dx + Ey + F = 0, with all coefficients real numbers and not all zero.

When only two gravitational bodies interact, their orbits follow a conic section. The orbit can be open (implying the object never returns) or closed (returning). Which it is depends on the total energy (kinetic + potential energy) of the system. In the case of an open orbit, the speed at any position of the orbit is at least the escape velocity for that position, in the case of a closed orbit, the speed is always less than the escape velocity.

In projective geometry, a von Staudt conic is the point set defined by all the absolute points of a polarity that has absolute points. In the real projective plane a von Staudt conic is a conic section in the usual sense. In more general projective planes this is not always the case. Karl Georg Christian von Staudt introduced this definition in Geometrie der Lage (1847) as part of his attempt to remove all metrical concepts from projective geometry.

Apollonius of Perga made significant advances in the study of conic sections. Apollonius of Perga (c. 262–190 BC) made significant advances to the study of conic sections, showing that one can obtain all three varieties of conic section by varying the angle of the plane that cuts a double-napped cone. He also coined the terminology in use today for conic sections, namely parabola ("place beside" or "comparison"), "ellipse" ("deficiency"), and "hyperbola" ("a throw beyond").

During his time in Kushtia, Hossain, coming from a relatively poor family, supported himself by various scholarships, lodged at nearby homes as house tutors and later settled at a hostel. Among the teachers, he specially later recalled of Jyotindranath Roy and Jatindra Mohan Biswas. During his years at the school, Jyotindranath Roy, an accomplished teacher of Mathematics and Sciences, laid Hossain's foundation in algebra, geometry, conic section and mechanics. Mr. Roy, recognizing Hossain's interest in mathematics, introduced the mechanics course solely for him.

Originally, before brake drums were available, frying pans were used (Pérez I 1988:310, Pérez II 1988:23, etc.) and possibly plow blades as well (Pérez I 1988:106 and 134). The second category is the bocuses (sing. bocú alt. pl. bocúes), also called fondos ("bottoms"). > “The bokú is a single-headed drum, skin nailed to the shell, shell open at > one end, long, shaped like a conic section and made of staves with iron > hoops that circle them and hold them together.

What should be considered as a degenerate case of a conic depends on the definition being used and the geometric setting for the conic section. There are some authors who define a conic as a two-dimensional nondegenerate quadric. With this terminology there are no degenerate conics (only degenerate quadrics), but we shall use the more traditional terminology and avoid that definition. In the Euclidean plane, using the geometric definition, a degenerate case arises when the cutting plane passes through the apex of the cone.

Quetelet, Adolphe (1819) "Dissertatio mathematica inauguralis de quibusdam locis geometricis nec non de curva focali" (Inaugural mathematical dissertation on some geometric loci and also focal curves), doctoral thesis (University of Ghent ("Gand"), Belgium). (in Latin) The Dandelin spheres can be used to give elegant modern proofs of two classical theorems known to Apollonius of Perga. The first theorem is that a closed conic section (i.e. an ellipse) is the locus of points such that the sum of the distances to two fixed points (the foci) is constant.

Heath, Thomas. A History of Greek Mathematics, page 119 (focus-directrix property), page 542 (sum of distances to foci property) (Clarendon Press, 1921). Neither Dandelin nor Quetelet used the Dandelin spheres to prove the focus-directrix property. The first to do so may have been Pierce Morton in 1829,Numericana's Biographies: Morton, Pierce or perhaps Hugh Hamilton who remarked (in 1758) that a sphere touches the cone at a circle which defines a plane whose intersection with the plane of the conic section is a directrix.

The hypothetical motion that the body follows under the gravitational effect of one other body only is typically a conic section, and can be readily modeled with the methods of geometry. This is called a two-body problem, or an unperturbed Keplerian orbit. The differences between the Keplerian orbit and the actual motion of the body are caused by perturbations. These perturbations are caused by forces other than the gravitational effect between the primary and secondary body and must be modeled to create an accurate orbit simulation.

The magnification in an atom is due to the projection of ions radially away from the small, sharp tip. Subsequently, in the far field, the ions will be greatly magnified. This magnification is sufficient to observe field variations due to individual atoms, thus allowing in field ion and field evaporation modes for the imaging of single atoms. The standard projection model for the atom probe is an emitter geometry that is based upon a revolution of a conic section, such as a sphere, hyperboloid or paraboloid.

If the gnomon (the shadow-casting object) is not an edge but a point (e.g., a hole in a plate), the shadow (or spot of light) will trace out a curve during the course of a day. If the shadow is cast on a plane surface, this curve will be a conic section (usually a hyperbola), since the circle of the Sun's motion together with the gnomon point define a cone. At the spring and fall equinoxes, the cone degenerates into a plane and the hyperbola into a line.

The problem of intersection of an ellipse/hyperbola/parabola with another conic section leads to a system of quadratic equations, which can be solved in special cases easily by elimination of one coordinate. Special properties of conic sections may be used to obtain a solution. In general the intersection points can be determined by solving the equation by a Newton iteration. If a) both conics are given implicitly (by an equation) a 2-dimensional Newton iteration b) one implicitly and the other parametrically given a 1-dimensional Newton iteration is necessary.

Tour EDF's most striking characteristic consists in the extrusion of a conic section of the tower on its northern edge. The resulting conic hole extends from the ground floor to the 26th floor and serves as the main entrance to the tower, an entrance built under a wide circular canopy 24 m (79 feet) in diameter. As a consequence, the length of the tower is slightly less at its base than at its top. The cladding of the tower alternates horizontal stripes of plain steel and tinted windows.

Kig can handle any classical object of the dynamic geometry, but also: # The center of curvature and osculating circle of a curve; # The dilation, generic affinity, inversion, projective application, homography and harmonic homology; # The hyperbola with given asymptotes; # The Bézier curves (2nd and 3rd degree); # The polar line of a point and pole of a line with respect to a conic section; # The asymptotes of a hyperbola; # The cubic curve through 9 points; # The cubic curve with a double point through 6 points; # The cubic curve with a cusp through 4 points.

The intersections of the extended opposite sides of inscribed hexagon ABCDEF lie on the blue Pascal line MNP. The hexagon's extended sides are in gray and red. Pascal's theorem states that if six arbitrary points are chosen on a conic section (i.e., ellipse, parabola or hyperbola) and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon (extended if necessary) meet in three points which lie on a straight line, called the Pascal line of the hexagon.

A cylindric section in which the intersecting plane intersects and is perpendicular to all the elements of the cylinder is called a '. If a right section of a cylinder is a circle then the cylinder is a circular cylinder. In more generality, if a right section of a cylinder is a conic section (parabola, ellipse, hyperbola) then the solid cylinder is said to be parabolic, elliptic and hyperbolic respectively. Cylindric sections of a right circular cylinder For a right circular cylinder, there are several ways in which planes can meet a cylinder.

Book III contains 56 propositions. Apollonius claims original discovery for theorems "of use for the construction of solid loci ... the three-line and four-line locus ...." The locus of a conic section is the section. The three-line locus problem (as stated by Taliafero's appendix to Book III) finds "the locus of points whose distances from three given fixed straight lines ... are such that the square of one of the distances is always in a constant ratio to the rectangle contained by the other two distances." This is the proof of the application of areas resulting in the parabola.

Next, in 1849, Kirkman studied the Pascal lines determined by the intersection points of opposite sides of a hexagon inscribed within a conic section. Any six points on a conic may be joined into a hexagon in 60 different ways, forming 60 different Pascal lines. Extending previous work of Steiner, Kirkman showed that these lines intersect in triples to form 60 points (now known as the Kirkman points), so that each line contains three of the points and each point lies on three of the lines. That is, these lines and points form a projective configuration of type 603603.

Circle (e=0), ellipse (e=1/2), parabola (e=1) and hyperbola (e=2) with fixed focus F and directrix (e=∞). Alternatively, one can define a conic section purely in terms of plane geometry: it is the locus of all points whose distance to a fixed point (called the focus) is a constant multiple (called the eccentricity ) of the distance from to a fixed line (called the directrix). For we obtain an ellipse, for a parabola, and for a hyperbola. A circle is a limiting case and is not defined by a focus and directrix in the Euclidean plane.

It is believed that the first definition of a conic section was given by Menaechmus (died 320 BCE) as part of his solution of the Delian problem (Duplicating the cube).According to Plutarch this solution was rejected by Plato on the grounds that it could not be achieved using only straightedge and compass, however this interpretation of Plutarch's statement has come under criticism. His work did not survive, not even the names he used for these curves, and is only known through secondary accounts. The definition used at that time differs from the one commonly used today.

The special perpendicular is employed to compute the volume of a pyramid (p 35), an equation on skew lines that reduces to zero when they are coplanar, a property of a spherical triangle, and the coincidence of the perpendiculars in a tetrahedron. Chapter four describes equations of geometric figures: line, plane, circle, sphere. The definition of a conic section is taken from Kelland and Tait: "the locus of a point which move so that its distance from a fixed point bears a constant ratio to its distance from a fixed straight line." Ellipse, hyperbola, and parabola are then illustrated.

The result on the Legendre symbol amounts to the formula for the number of points on a conic section that is a projective line over the field of p elements. A paper of André Weil from 1949 very much revived the subject. Indeed, through the Hasse–Davenport relation of the late 20th century, the formal properties of powers of Gauss sums had become current once more. As well as pointing out the possibility of writing down local zeta-functions for diagonal hypersurfaces by means of general Jacobi sums, Weil (1952) demonstrated the properties of Jacobi sums as Hecke characters.

The axis of a cone is the straight line (if any), passing through the apex, about which the base (and the whole cone) has a circular symmetry. In common usage in elementary geometry, cones are assumed to be right circular, where circular means that the base is a circle and right means that the axis passes through the centre of the base at right angles to its plane. If the cone is right circular the intersection of a plane with the lateral surface is a conic section. In general, however, the base may be any shapeGrünbaum, Convex Polytopes, second edition, p. 23.

Equivalence of a quadratic Bézier curve and a parabolic segment A quadratic Bézier curve is also a segment of a parabola. As a parabola is a conic section, some sources refer to quadratic Béziers as "conic arcs". With reference to the figure on the right, the important features of the parabola can be derived as follows: # Tangents to the parabola at the end-points of the curve (A and B) intersect at its control point (C). # If D is the midpoint of AB, the tangent to the curve which is perpendicular to CD (dashed cyan line) defines its vertex (V).

142–146 With respect to the beginnings of projective geometry, Kepler introduced the idea of continuous change of a mathematical entity in this work. He argued that if a focus of a conic section were allowed to move along the line joining the foci, the geometric form would morph or degenerate, one into another. In this way, an ellipse becomes a parabola when a focus moves toward infinity, and when two foci of an ellipse merge into one another, a circle is formed. As the foci of a hyperbola merge into one another, the hyperbola becomes a pair of straight lines.

A point (x,y,z) of the contour line of an implicit surface with equation f(x,y,z)=0 and parallel projection with direction \vec v has to fulfill the condition g(x,y,z)= abla f(x,y,z)\cdot \vec v=0, because \vec v has to be a tangent vector, which means any contour point is a point of the intersection curve of the two implicit surfaces : f(x,y,z)=0 ,\ g(x,y,z)=0. For quadrics, g is always a linear function. Hence the contour line of a quadric is always a plane section (i.e. a conic section).

An elliptic, parabolic, and hyperbolic Kepler orbit: Elliptic orbit by eccentricity The orbital eccentricity of an astronomical object is a dimensionless parameter that determines the amount by which its orbit around another body deviates from a perfect circle. A value of 0 is a circular orbit, values between 0 and 1 form an elliptic orbit, 1 is a parabolic escape orbit, and greater than 1 is a hyperbola. The term derives its name from the parameters of conic sections, as every Kepler orbit is a conic section. It is normally used for the isolated two-body problem, but extensions exist for objects following a Klemperer rosette orbit through the galaxy.

In other cases, the hour-lines are not spaced evenly, even though the shadow rotates uniformly. If the gnomon is not aligned with the celestial poles, even its shadow will not rotate uniformly, and the hour lines must be corrected accordingly. The rays of light that graze the tip of a gnomon, or which pass through a small hole, or reflect from a small mirror, trace out a cone aligned with the celestial poles. The corresponding light-spot or shadow-tip, if it falls onto a flat surface, will trace out a conic section, such as a hyperbola, ellipse or (at the North or South Poles) a circle.

The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane parallel to another plane that is tangential to the conical surface. The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola through the middle) is called the "axis of symmetry". The point where the parabola intersects its axis of symmetry is called the "vertex" and is the point where the parabola is most sharply curved.

The two diagonals and the two tangency chords are concurrent.Yiu, Paul, Euclidean Geometry, , 1998, pp. 156–157.Grinberg, Darij, Circumscribed quadrilaterals revisited, 2008 One way to see this is as a limiting case of Brianchon's theorem, which states that a hexagon all of whose sides are tangent to a single conic section has three diagonals that meet at a point. From a tangential quadrilateral, one can form a hexagon with two 180° angles, by placing two new vertices at two opposite points of tangency; all six of the sides of this hexagon lie on lines tangent to the inscribed circle, so its diagonals meet at a point.

The study of these two hyperboloids is, therefore, in this way connected very simply, through biquaternions, with the study of the sphere; ... In this passage is the operator giving the scalar part of a quaternion, and is the "tensor", now called norm, of a quaternion. A modern view of the unification of the sphere and hyperboloid uses the idea of a conic section as a slice of a quadratic form. Instead of a conical surface, one requires conical hypersurfaces in four-dimensional space with points determined by quadratic forms. First consider the conical hypersurface :P = \lbrace p \ : \ w^2 = x^2 + y^2 + z^2 \rbrace and :H_r = \lbrace p \ :\ w = r \rbrace , which is a hyperplane.

In 1999, Charlie Hughes of Peavey Electronics filed for a patent on a hybrid horn he called Quadratic-Throat Waveguide. The horn was basically a simple conic section but its throat was curved in a circular arc to match the desired throat size for proper mating to the speaker driver. Instead of increasing the horn mouth size with a flare to control midrange beaming, a relatively thin layer of foam covering the mouth edge was found to suit the same end. The QT waveguide, when compared to popular CD horns, produced about lower levels of second harmonic distortion across all frequencies, and an average of lower levels of the more annoying third harmonic distortion.

The path followed by any particle in the classical Kepler problem is a conic section. In particular, if the total energy E of the particle is greater than zero (that is, if the particle is unbound), the path of such a particle is a hyperbola. This property is useful in studying atomic and sub-atomic forces by scattering high-energy particles; for example, the Rutherford experiment demonstrated the existence of an atomic nucleus by examining the scattering of alpha particles from gold atoms. If the short-range nuclear interactions are ignored, the atomic nucleus and the alpha particle interact only by a repulsive Coulomb force, which satisfies the inverse square law requirement for a Kepler problem.

The semi-major (a) and semi-minor axis (b) of an ellipse In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the widest points of the perimeter. The semi-major axis is one half of the major axis, and thus runs from the centre, through a focus, and to the perimeter. The semi-minor axis of an ellipse or hyperbola is a line segment that is at right angles with the semi-major axis and has one end at the center of the conic section. For the special case of a circle, the lengths of the semi-axes are both equal to the radius of the circle.

Prior started playing guitar aged 8, drums at 11, vibraphone at 17 and piano at 24. During his teens, he performed in the bands Mandrake, then Legend with brother Rob Prior (guitar/vocals) and neighbor Paul Williams (bass guitar) at The Kirribilli Hotel, the Mosman Spastic Centre and Sydney Boys High School. With the addition of keyboards player Chris Short and the name change to Conic Section, they composed original jazz-rock fusion, performed at the Limerick Castle Hotel and won the Festival Of Pop band competition at Flemington Sydney in 1974 and a 2SM song competition in 1975. The band TAPP, comprising John and Rob Prior, Peter Astley (bass guitar), Geoff Taylor (keyboards) and Peter Noakes (vocals) recorded at Wirra Willa Studios, Glenfield Park.

This geometric object can also be described as the set of all points swept by a line that intercepts the axis and rotates around it; or the union of all lines that intersect the axis at a fixed point p and at a fixed angle \theta. The aperture of the cone is the angle 2 \theta. More generally, when the directrix C is an ellipse, or any conic section, and the apex is an arbitrary point not on the plane of C, one obtains a conical quadric, which is a special case of a quadric surface. A cylindrical surface can be viewed as a limiting case of a conical surface whose apex is moved off to infinity in a particular direction.

In this expanded plane, we define the polar of the point to be the line at infinity (and is the pole of the line at infinity), and the poles of the lines through are the points of infinity where, if a line has slope its pole is the infinite point associated to the parallel class of lines with slope . The pole of the -axis is the point of infinity of the vertical lines and the pole of the -axis is the point of infinity of the horizontal lines. The construction of a correlation based on inversion in a circle given above can be generalized by using inversion in a conic section (in the extended real plane). The correlations constructed in this manner are of order two, that is, polarities.

A collapsible cone horn with removable flared bell. This horn was patented in 1901 for gramophone record playback The megaphone, a simple cone made of paper or other flexible material, is the oldest and simplest acoustic horn, used prior to loudspeakers as a passive acoustic amplifier for mechanical phonographs and for the human voice; it is still used by cheerleaders and lifeguards. Because the conic section shape describes a portion of a perfect sphere of radiated sound, cones have no phase or amplitude distortion of the wavefront. The small megaphones used in phonographs and as loudhailers were not long enough to reproduce the low frequencies in music; they had a high cutoff frequency which attenuated the bottom two octaves of the sound spectrum, giving the megaphone a characteristic tinny sound.

Cones were constructed by rotating a right triangle about one of its legs so the hypotenuse generates the surface of the cone (such a line is called a generatrix). Three types of cones were determined by their vertex angles (measured by twice the angle formed by the hypotenuse and the leg being rotated about in the right triangle). The conic section was then determined by intersecting one of these cones with a plane drawn perpendicular to a generatrix. The type of the conic is determined by the type of cone, that is, by the angle formed at the vertex of the cone: If the angle is acute then the conic is an ellipse; if the angle is right then the conic is a parabola; and if the angle is obtuse then the conic is a hyperbola (but only one branch of the curve).

The image of spherical pentagon PQRST in the gnomonic projection (a projection from the centre of the sphere) onto any plane tangent to the sphere is a rectilinear pentagon. Its five vertices P'Q'R'S'T' unambiguously determine a conic section; in this case — an ellipse. Gauss showed that the altitudes of pentagram P'Q'R'S'T' (lines passing through vertices and perpendicular to opposite sides) cross in one point O', which is the image of the point of tangency of the plane to sphere. Arthur Cayley observed that, if we set the origin of a Cartesian coordinate system in point O', then the coordinates of vertices P'Q'R'S'T': (x_1, y_1),\ldots, (x_5, y_5) satisfy the equalities x_1 x_4 + y_1 y_4 = x_2 x_5 + y_2 y_5 = x_3 x_1 + y_3 y_1 = x_4 x_2 + y_4 y_2 = x_5 x_3 + y_5 y_3 = -\rho^2, where \rho is the length of the radius of the sphere.

The circle is an instance of a conic section and the nine-point circle is an instance of the general nine-point conic that has been constructed with relation to a triangle ABC and a fourth point P, where the particular nine-point circle instance arises when P is the orthocenter of ABC. The vertices of the triangle and P determine a complete quadrilateral and three "diagonal points" where opposite sides of the quadrilateral intersect. There are six "sidelines" in the quadrilateral; the nine-point conic intersects the midpoints of these and also includes the diagonal points. The conic is an ellipse when P is interior to ABC or in a region sharing vertical angles with the triangle, but a nine-point hyperbola occurs when P is in one of the three adjacent regions, and the hyperbola is rectangular when P lies on the circumcircle of ABC.

The perimeter of the base of a cone is called the "directrix", and each of the line segments between the directrix and apex is a "generatrix" or "generating line" of the lateral surface. (For the connection between this sense of the term "directrix" and the directrix of a conic section, see Dandelin spheres.) The "base radius" of a circular cone is the radius of its base; often this is simply called the radius of the cone. The aperture of a right circular cone is the maximum angle between two generatrix lines; if the generatrix makes an angle θ to the axis, the aperture is 2θ. Illustration from Problemata mathematica... published in Acta Eruditorum, 1734 A cone with a region including its apex cut off by a plane is called a "truncated cone"; if the truncation plane is parallel to the cone's base, it is called a frustum.

The converse is the Braikenridge–Maclaurin theorem, named for 18th- century British mathematicians William Braikenridge and Colin Maclaurin , which states that if the three intersection points of the three pairs of lines through opposite sides of a hexagon lie on a line, then the six vertices of the hexagon lie on a conic; 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. The theorem was generalized by August Ferdinand Möbius in 1847, as follows: suppose a polygon with sides is inscribed in a conic section, and opposite pairs of sides are extended until they meet in points. Then if of those points lie on a common line, the last point will be on that line, too.

With the publication of his Principia roughly eighty years later (1687), Isaac Newton provided a physical theory that accounted for all three of Kepler's laws, a theory based on Newton's laws of motion and his law of universal gravitation. In particular, Newton proposed that the gravitational force between any two bodies was a central force F(r) that varied as the inverse square of the distance r between them. Arguing from his laws of motion, Newton showed that the orbit of any particle acted upon by one such force is always a conic section, specifically an ellipse if it does not go to infinity. However, this conclusion holds only when two bodies are present (the two-body problem); the motion of three bodies or more acting under their mutual gravitation (the n-body problem) remained unsolved for centuries after Newton,Whittaker, pp. 339–385.

In analogy with the conic sections, the set of points whose Cartesian coordinates satisfy the general equation of the second degree, namely, :Ax^2 + By^2 + Cz^2 + Fxy + Gyz + Hxz + Jx + Ky + Lz + M = 0, where and are real numbers and not all of and are zero, is called a quadric surface. There are six types of non-degenerate quadric surfaces: # Ellipsoid # Hyperboloid of one sheet # Hyperboloid of two sheets # Elliptic cone # Elliptic paraboloid # Hyperbolic paraboloid The degenerate quadric surfaces are the empty set, a single point, a single line, a single plane, a pair of planes or a quadratic cylinder (a surface consisting of a non-degenerate conic section in a plane and all the lines of through that conic that are normal to ). Elliptic cones are sometimes considered to be degenerate quadric surfaces as well. Both the hyperboloid of one sheet and the hyperbolic paraboloid are ruled surfaces, meaning that they can be made up from a family of straight lines.

Gauss heard about the problem and tackled it. After three months of intense work, he predicted a position for Ceres in December 1801—just about a year after its first sighting—and this turned out to be accurate within a half-degree when it was rediscovered by Franz Xaver von Zach on 31 December at Gotha, and one day later by Heinrich Olbers in Bremen. This confirmation eventually led to the classification of Ceres as minor-planet designation 1 Ceres: the first asteroid (now dwarf planet) ever discovered. Gauss's method involved determining a conic section in space, given one focus (the Sun) and the conic's intersection with three given lines (lines of sight from the Earth, which is itself moving on an ellipse, to the planet) and given the time it takes the planet to traverse the arcs determined by these lines (from which the lengths of the arcs can be calculated by Kepler's Second Law).

Light path in a Gregorian telescope. In 1636 Marin Mersenne proposed a telescope consisting of a paraboloidal primary mirror and a paraboloidal secondary mirror bouncing the image through a hole in the primary, solving the problem of viewing the image. Mirror Mirror: A History of the Human Love Affair With Reflection by Mark Pendergrast Page 88 James Gregory went into further detail in his book Optica Promota (1663), pointing out that a reflecting telescope with a mirror that was shaped like the part of a conic section, would correct spherical aberration as well as the chromatic aberration seen in refractors. The design he came up with bears his name: the "Gregorian telescope"; but according to his own confession, Gregory had no practical skill and he could find no optician capable of realizing his ideas and after some fruitless attempts, was obliged to abandon all hope of bringing his telescope into practical use.

Foucault's knife edge test determines the shape of a mirror by finding the focal lengths of its areas, commonly called zones and measured from the mirror center. In the test, light from a point source is focused onto the center of curvature of the mirror and reflected back to a knife edge. The test enables the tester to quantify the conic section of the mirror, thereby allowing the tester to validate the actual shape of the mirror, which is necessary to obtain optimal performance of the optical system. The Foucault test is in use to this date, most notably by amateur and smaller commercial telescope makers as it is inexpensive and uses simple, easily made equipment. With Charles Wheatstone’s revolving mirror he, in 1862, determined the speed of light to be 298,000 km/s – 10,000 km/s less than that obtained by previous experimenters and only 0.6% in error of the currently accepted value.

He solved some of them and discussed the conditions of solvability. For example, he was able to solve the problem of inscribing an equilateral pentagon into a square, resulting in a fourth degree equation.Jan Hogendijk (1984) "al-Kuhi's construction of an equilateral pentagon in a given square", Zeitschrift für Gesch. Arab.-Islam. Wiss. 1: 100-144; correction and addendum Volume 4, 1986/87, p.267 He also wrote a treatise on the "perfect compass", a compass with one leg of variable length that allows users to draw any conic section: straight lines, circles, ellipses, parabolas and hyperbolas.Jan Hogendijk (2008) "Two beautiful geometrical theorems by Abu Sahl Kuhi in a 17th century Dutch translation", Ta'rikh-e Elm: Iranian Journal for the History of Science 6: 1-36 It is likely that al-Qūhī invented the device. Alt URL Reviews: Seyyed Hossein Nasr (1998) in Isis 89 (1) pp. 112-113 ; Charles Burnett (1998) in Bulletin of the School of Oriental and African Studies, University of London 61 (2) p.