143 Sentences With "tangency" | Random Sentence Generator

Tangency gained investments from pension funds and family offices in the latest funding round, Hagedorn said.

Tangency Capital has offices in London and Bermuda and was founded by Hagedorn, Michael Jedraszak and Kai Morgenstern.

One essay here is titled "The Point of Tangency: On Digression," and one way to describe this collection is as a series of tangents.

Tangency Capital launched last year to invest directly in non-life reinsurance risks, and is benefiting from the rising rates, Dominik Hagedorn told Reuters by telephone.

Tangency Capital, which has offices in London and Bermuda and will invest directly in non-life reinsurance risks, was founded by Dominik Hagedorn, Michael Jedraszak and Kai Morgenstern.

LONDON (Reuters) - Hedge fund Tangency Capital has raised a further $165 million to invest in the property reinsurance market, bringing its size to $265 million as the hurricane season gets under way, one of its co-founders told Reuters on Wednesday.

All portfolios between the risk-free asset and the tangency portfolio are portfolios composed of risk- free assets and the tangency portfolio, while all portfolios on the linear frontier above and to the right of the tangency portfolio are generated by borrowing at the risk-free rate and investing the proceeds into the tangency portfolio.

Hotine oblique Mercator projection has approximately constant scale along the geodesic of conceptual tangency. Hotine's work was extended by Engels and Grafarend in 1995 to make the geodesic of conceptual tangency have true scale.

The points having the minimum vertical separation are also the tangency points for the maximally separated parallel tangents.

As the target level of output is increased, the relevant isoquant becomes farther and farther out from the origin, and still it is optimal in a cost-minimization sense to operate at the tangency point of the relevant isoquant with an isocost curve. The set of all such tangency points is called the firm's expansion path.

In this formula P is the sub-portfolio of risky assets at the tangency with the Markowitz bullet, F is the risk-free asset, and C is a combination of portfolios P and F. By the diagram, the introduction of the risk-free asset as a possible component of the portfolio has improved the range of risk-expected return combinations available, because everywhere except at the tangency portfolio the half-line gives a higher expected return than the hyperbola does at every possible risk level. The fact that all points on the linear efficient locus can be achieved by a combination of holdings of the risk-free asset and the tangency portfolio is known as the one mutual fund theorem, where the mutual fund referred to is the tangency portfolio.

The extouch triangle of a reference triangle has its vertices at the points of tangency of the reference triangle's excircles with its sides (not extended).

In Möbius geometry, tangency between a line and a circle becomes a special case of tangency between two circles. This equivalence is extended further in Lie sphere geometry. Radius and tangent line are perpendicular at a point of a circle, and hyperbolic-orthogonal at a point of the unit hyperbola. The parametric representation of the unit hyperbola via radius vector is p(a) \ =\ (\cosh a, \sinh a) .

Specifically, the point of tangency between any isoquant and an isocost line gives the lowest-cost combination of inputs that can produce the level of output associated with that isoquant. Equivalently, it gives the maximum level of output that can be produced for a given total cost of inputs. A line joining tangency points of isoquants and isocosts (with input prices held constant) is called the expansion path.Salvatore, Dominick (1989).

The radius of each solution is determined by finding a point of tangency T, which may be done by choosing one of the three intersection points P between the given lines; and drawing a circle centered on the midpoint of C and P of diameter equal to the distance between C and P. The intersections of that circle with the intersecting given lines are the two points of tangency.

In differential geometry the defining characteristic of a tangent space is that it approximates the smooth manifold to first order near the point of tangency. Equivalently, if we zoom in more and more at the point of tangency the manifold appears to become more and more straight, asymptotically tending to approach the tangent space. This turns out to be the correct point of view in geometric measure theory.

Feuerbach's theorem has also been used as a test case for automated theorem proving.. The three points of tangency with the excircles form the Feuerbach triangle of the given triangle.

The income–consumption curve is the set of tangency points of indifference curves with the various budget constraint lines, with prices held constant, as income increases shifting the budget constraint out.

If the two circles α and β cross each other, another two circles γ and δ are each tangent to both α and β, and in addition γ and δ are tangent to each other, then the point of tangency between γ and δ necessarily lies on one of the two circles of antisimilitude. If α and β are disjoint and non-concentric, then the locus of points of tangency of γ and δ again forms two circles, but only one of these is the (unique) circle of antisimilitude. If α and β are tangent or concentric, then the locus of points of tangency degenerates to a single circle, which again is the circle of antisimilitude.Tangencies: Circular Angle Bisectors, The Geometry Junkyard, David Eppstein, 1999.

Tangency points show the lowest cost input combination for producing any given level of output. A curve connecting the tangency points is called the expansion path because it shows how the input usages expand as the chosen level of output expands. In economics, an expansion path (also called a scale lineJain, TR; Khanna OP (2008). Economics. VK Publications, ) is a curve in a graph with quantities of two inputs, typically physical capital and labor, plotted on the axes.

An Apollonius problem is impossible if the given circles are nested, i.e., if one circle is completely enclosed within a particular circle and the remaining circle is completely excluded. This follows because any solution circle would have to cross over the middle circle to move from its tangency to the inner circle to its tangency with the outer circle. This general result has several special cases when the given circles are shrunk to points (zero radius) or expanded to straight lines (infinite radius).

But two of these diagonals are the same as the diagonals of the tangential quadrilateral, and the third diagonal of the hexagon is the line through two opposite points of tangency. Repeating this same argument with the other two points of tangency completes the proof of the result. If the extensions of opposite sides in a tangential quadrilateral intersect at J and K, and the diagonals intersect at P, then JK is perpendicular to the extension of IP where I is the incenter.

4) then they divide the immediate neighbourhood into four regions, one of which (shown as pale green) is preferable for both consumers; therefore a point at which indifference curves cross cannot be an equlibrium, and an equilibrium must be a point of tangency. Secondly, the only price which can hold in the market at the point of tangency is the one given by the gradient of the tangent, since at only this price will the consumers be willing to accept limitingly small exchanges. And thirdly (the most difficult point) all exchanges taking the consumers on the path from ω to equilibrium must take place at the same price. If this is accepted, then that price must be the one operative at the point of tangency, and the result follows.

Example of a substitution effect In the graphical rendition of the theory of consumer choice, as shown in the accompanying graph, the consumer’s choice of the optimal quantities to demand of two goods is the point of tangency between an indifference curve (curved) and the budget constraint (a straight line). The graph shows an initial budget constraint BC1 with resulting choice at tangency point A, and a new budget constraint after a decrease in the absolute price of Y (the good whose quantity is shown horizontally), with resulting choice at tangency point C. In each case the absolute value of the slope of the budget constraint is the ratio of the price of good Y to the price of good X – that is, the relative price of good Y in terms of X.

The above analysis describes optimal behavior of an individual investor. Asset pricing theory builds on this analysis in the following way. Since everyone holds the risky assets in identical proportions to each other—namely in the proportions given by the tangency portfolio—in market equilibrium the risky assets' prices, and therefore their expected returns, will adjust so that the ratios in the tangency portfolio are the same as the ratios in which the risky assets are supplied to the market. Thus relative supplies will equal relative demands.

The center of the Mandart inellipse is the mittenpunkt of the triangle. The three lines connecting the triangle vertices to the opposite points of tangency all meet in a single point, the Nagel point of the triangle.

The anticomplementary circle (red, radius 2r) of ΔABC is tangent to the three Johnson circles, which have centres on the lines (orange) between the common intersection, H, and the points of tangency. These points of tangency form the anticomplementary triangle, ΔPAPBPC, green. # The centers of the Johnson circles lie on a circle of the same radius r as the Johnson circles centered at H. These centers form the Johnson triangle. # The circle centered at H with radius 2r, known as the anticomplementary circle is tangent to each of the Johnson circles.

The process then continues until the market's and household's marginal rates of substitution are equal. Now, if the price of carrots were to change, and the price of all other goods were to remain constant, the gradient of the budget line would also change, leading to a different point of tangency and a different quantity demanded. These price / quantity combinations can then be used to deduce a full demand curve. A line connecting all points of tangency between the indifference curve and the budget constraint is called the expansion path.

In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for tangency. One needs a definition of intersection number in order to state results like Bézout's theorem. The intersection number is obvious in certain cases, such as the intersection of x- and y-axes which should be one. The complexity enters when calculating intersections at points of tangency and intersections along positive dimensional sets.

The tangent lengths and tangency chords The eight tangent lengths (e, f, g, h in the figure to the right) of a tangential quadrilateral are the line segments from a vertex to the points where the incircle is tangent to the sides. From each vertex there are two congruent tangent lengths. The two tangency chords (k and l in the figure) of a tangential quadrilateral are the line segments that connect points on opposite sides where the incircle is tangent to these sides. These are also the diagonals of the contact quadrilateral.

An external tangency is one where the two circles bend away from each other at their point of contact; they lie on opposite sides of the tangent line at that point, and they exclude one another. The distance between their centers equals the sum of their radii. By contrast, an internal tangency is one in which the two circles curve in the same way at their point of contact; the two circles lie on the same side of the tangent line, and one circle encloses the other. In this case, the distance between their centers equals the difference of their radii.

Similarly, the distance d2 between the centers of the solution circle and C2 is either or , again depending on their chosen tangency. Thus, the difference between these distances is always a constant that is independent of rs. This property, of having a fixed difference between the distances to the foci, characterizes hyperbolas, so the possible centers of the solution circle lie on a hyperbola. A second hyperbola can be drawn for the pair of given circles C2 and C3, where the internal or external tangency of the solution and C2 should be chosen consistently with that of the first hyperbola.

For many years the proof of the four-vertex theorem remained difficult, but a simple and conceptual proof was given by , based on the idea of the minimum enclosing circle.. This is a circle that contains the given curve and has the smallest possible radius. If the curve includes an arc of the circle, it has infinitely many vertices. Otherwise, the curve and circle must be tangent at at least two points. At each tangency, the curvature of the curve is greater than that of the circle (else the curve would continue from the tangency outside the circle rather than inside).

A tacnode corresponds to a type A3−-singularity. In fact each type A2n+1−-singularity, where n ≥ 0 is an integer, corresponds to a curve with self intersection. As n increases the order of self intersection increases: transverse crossing, ordinary tangency, etc.

For three circles that are mutually externally tangent, the (unique) circles of antisimilitude for each pair again cross each other at 120° angles in two triple intersection points that are the isodynamic points of the triangle formed by the three points of tangency.

If we reciprocate such a canonical polyhedron about its midsphere, the dual polyhedron will share the same edge-tangency points and so must also be canonical. It is the canonical dual, and the two together form a canonical dual pair., Theorem 3.1, p. 449.

Similarly, tangency to a given line L (tangency is intersection with multiplicity two) is one quadratic condition, so determined a quadric in P5. However the linear system of divisors consisting of all such quadrics is not without a base locus. In fact each such quadric contains the Veronese surface, which parametrizes the conics :(aX + bY + cZ)2 = 0 called 'double lines'. This is because a double line intersects every line in the plane, since lines in the projective plane intersect, with multiplicity two because it is doubled, and thus satisfies the same intersection condition (intersection of multiplicity two) as a nondegenerate conic that is tangent to the line.

The tangent line t and the tangent point T have a conjugate relationship to one another, which has been generalized into the idea of pole points and polar lines. The same reciprocal relation exists between a point P outside the circle and the secant line joining its two points of tangency. If a point P is exterior to a circle with center O, and if the tangent lines from P touch the circle at points T and S, then ∠TPS and ∠TOS are supplementary (sum to 180°). If a chord TM is drawn from the tangency point T of exterior point P and ∠PTM ≤ 90° then ∠PTM = (1/2)∠TOM.

It is assumed in this construction that all circles lie within the triangle, and all points of tangency lie on the sides of the triangle. If the problem is generalized to allow circles that may not be within the triangle, and points of tangency on the lines extending the sides of the triangle, then the sequence of circles eventually reaches a periodic sequence of six circles, but may take arbitrarily many steps to reach this periodicity. The name may also refer to Miquel's six circles theorem, the result that if five circles have four triple points of intersection then the remaining four points of intersection lie on a sixth circle.

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.

Vertical tangent on the function ƒ(x) at x = c. In mathematics, particularly calculus, a vertical tangent is a tangent line that is vertical. Because a vertical line has infinite slope, a function whose graph has a vertical tangent is not differentiable at the point of tangency.

The parameterization above implies that the curve is rational which implies it has genus zero. A line segment can slide with each end on the deltoid and remain tangent to the deltoid. The point of tangency travels around the deltoid twice while each end travels around it once.

A circle packing for a five-vertex planar graph The circle packing theorem (also known as the Koebe–Andreev–Thurston theorem) describes the possible tangency relations between circles in the plane whose interiors are disjoint. A circle packing is a connected collection of circles (in general, on any Riemann surface) whose interiors are disjoint. The intersection graph of a circle packing is the graph having a vertex for each circle, and an edge for every pair of circles that are tangent. If the circle packing is on the plane, or, equivalently, on the sphere, then its intersection graph is called a coin graph; more generally, intersection graphs of interior-disjoint geometric objects are called tangency graphs or contact graphs.

Early examinations of the properties of equilibrium were based on an implicit definition as tangency, and convexity seems to have been implicitly assumed.Oscar Lange, ‘The Foundations of Welfare Economics’ (1942). There was no doubt that equilibrium would be reached: gradient ascent would lead to it. But the results lacked generality.

The tangency of the top surface with the constrained space in the center represents the solution. Another simple problem (see diagram) can be defined by the constraints :x12 − x22 \+ x32 ≤ 2 :x12 \+ x22 \+ x32 ≤ 10 with an objective function to be maximized :f(x) = x1x2 \+ x2x3 where x = (x1, x2, x3).

Risk parity advocates assert that the unlevered risk parity portfolio is quite close to the tangency portfolio, as close as can be measured given uncertainties and noise in the data. Theoretical and empirical arguments are made in support of this contention. One specific set of assumptions that puts the risk parity portfolio on the efficient frontier is that the individual asset classes are uncorrelated and have identical Sharpe ratios. Risk parity critics rarely contest the claim that the risk parity portfolio is near the tangency portfolio but they say that the leveraged investment line is less steep and that the levered risk parity portfolio has slight or no advantage over 60% stocks / 40% bonds, and carries the disadvantage of greater explicit leverage.

In mathematics, transversality is a notion that describes how spaces can intersect; transversality can be seen as the "opposite" of tangency, and plays a role in general position. It formalizes the idea of a generic intersection in differential topology. It is defined by considering the linearizations of the intersecting spaces at the points of intersection.

Consider the intersection of the x-axis with the parabola :y = x^2.\ Then :P = y,\ and :Q = y - x^2,\ so : I_p(P,Q) = I_p(y,y - x^2) = I_p(y,x^2) = I_p(y,x) + I_p(y,x) = 1 + 1 = 2.\, Thus, the intersection degree is two; it is an ordinary tangency.

The nine- point circle passes through these three midpoints; thus, it is the circumcircle of the medial triangle. These two circles meet in a single point, where they are tangent to each other. That point of tangency is the Feuerbach point of the triangle. Associated with the incircle of a triangle are three more circles, the excircles.

In geometry, tangent circles (also known as kissing circles) are circles in a common plane that intersect in a single point. There are two types of tangency: internal and external. Many problems and constructions in geometry are related to tangent circles; such problems often have real-life applications such as trilateration and maximizing the use of materials.

The slope of the secant line equals the average product of labor, where the slope = dQ/dL. The slope of the curve at each intersection marks a point on the average product curve. The slope increases until the line reaches a point of tangency with the total product curve. This point marks the maximum average product of labor.

Hobson, p.309. and may differ from the normal of the secondary wavefront (cf. Fig.2), and therefore from the normal of the primary wavefront at the point of tangency. Hence the ray velocity, in magnitude and direction, is the radial velocity of an infinitesimal secondary wavefront, and is generally a function of location and direction.

Figure 3: The Efficient Portfolio The investor's optimal portfolio is found at the point of tangency of the efficient frontier with the indifference curve. This point marks the highest level of satisfaction the investor can obtain. This is shown in Figure 3. R is the point where the efficient frontier is tangent to indifference curve C3, and is also an efficient portfolio.

Capital market line Capital market line (CML) is the tangent line drawn from the point of the risk-free asset to the feasible region for risky assets. The tangency point M represents the market portfolio, so named since all rational investors (minimum variance criterion) should hold their risky assets in the same proportions as their weights in the market portfolio.

The sketch consists of geometry such as points, lines, arcs, conics (except the hyperbola), and splines. Dimensions are added to the sketch to define the size and location of the geometry. Relations are used to define attributes such as tangency, parallelism, perpendicularity, and concentricity. The parametric nature of SolidWorks means that the dimensions and relations drive the geometry, not the other way around.

Furthermore, it is included in every bitangent plane. The two points of tangency are also double points. Thus the intersection curve, which theory says must be a quartic, contains four double points. But we also know that a quartic with more than three double points must factor (it cannot be irreducible), and by symmetry the factors must be two congruent conics.

The canonical example is :y^2-x^4= 0. A tacnode of an arbitrary curve may then be defined from this example, as a point of self-tangency locally diffeomorphic to the point at the origin of this curve. Another example of a tacnode is given by the links curve shown in the figure, with equation : (x^2+y^2-3x)^2 -4x^2(2-x) = 0.

Canonical dual compound of cuboctahedron (light) and rhombic dodecahedron (dark). Pairs of edges meet on their common midsphere. Any convex polyhedron can be distorted into a canonical form, in which a unit midsphere (or intersphere) exists tangent to every edge, and such that the average position of the points of tangency is the center of the sphere. This form is unique up to congruences.

Notice that Pareto is careful not to say that constant prices are general, merely that they are the commonest and most important case. The task of finding a competitive equilibrium accordingly reduces to the task of finding a point of tangency between two indifference curves for which the tangent passes through a given point. The use of offer curves (described below) provides a systematic procedure for doing this.

These are circles that are each tangent to the three lines through the triangle's sides. Each excircle touches one of these lines from the opposite side of the triangle, and is on the same side as the triangle for the other two lines. Like the incircle, the excircles are all tangent to the nine-point circle. Their points of tangency with the nine-point circle form a triangle, the Feuerbach triangle.

The more usual solution will lie in the non-zero interior at the point of tangency between the objective function and the constraint. For example, in consumer theory the objective function is the indifference-curve map (the utility function) of the consumer. The budget line is the constraint. In the usual case, constrained utility is maximized on the budget constraint with strictly positive quantities consumed of both goods.

To have an envelope, it is necessary that the individual members of the family of curves are differentiable curves as the concept of tangency does otherwise not apply, and there has to be a smooth transition proceeding through the members. But these conditions are not sufficient – a given family may fail to have an envelope. A simple example of this is given by a family of concentric circles of expanding radius.

Animation showing the inversive transformation of an Apollonius problem. The blue and red circles swell to tangency, and are inverted in the grey circle, producing two straight lines. The yellow solutions are found by sliding a circle between them until it touches the transformed green circle from within or without. Many special cases of Apollonius's problem involve finding a circle that is tangent to one or more lines.

Under the same inversion, the corresponding points of tangency of the two solution circles are transformed into one another; for illustration, in Figure 6, the two blue points lying on each green line are transformed into one another. Hence, the lines connecting these conjugate tangent points are invariant under the inversion; therefore, they must pass through the center of inversion, which is the radical center (green lines intersecting at the orange dot in Figure 6).

The Bankoff circle is formed from three semicircles that create an arbelos. A circle C1 is then formed tangent to each of the three semicircles, as an instance of the problem of Apollonius. Another circle C2 is then created, through three points: the two points of tangency of C1 with the smaller two semicircles, and the point where the two smaller semicircles are tangent to each other. C2 is the Bankoff circle.

Under a particular inversion centered on A, the four initial circles of the Pappus chain are transformed into a stack of four equally sized circles, sandwiched between two parallel lines. This accounts for the height formula hn = n dn and the fact that the original points of tangency lie on a common circle. The height hn of the center of the nth circle above the base diameter ACB equals n times dn.Ogilvy, pp. 54-55.

In fact, the curve family of cissoids is named for this example and some authors refer to it simply as the cissoid. It has a single cusp at the pole, and is symmetric about the diameter of the circle which is the line of tangency of the cusp. The line is an asymptote. It is a member of the conchoid of de Sluze family of curves and in form it resembles a tractrix.

In case of symmetry two of the dashed circles may touch in a point on a bisector, making two bitangents coincide there, but still setting up the relevant quadrilaterals for Malfatti's circles. The three bitangents , , and cross the triangle sides at the point of tangency with the third inscribed circle, and may also be found as the reflections of the angle bisectors across the lines connecting pairs of centers of these incircles..

With equilibrium defined as ‘competitive equilibrium’, the first fundamental theorem can be proved even if indifference curves need not be convex: any competitive equilibrium is (globally) Pareto optimal. However the proof is no longer obvious, and the reader is referred to the article on Fundamental theorems of welfare economics. The same result would not have been considered to hold (with non- convex indifference curves) under the tangency definition of equilibrium. The point x of Fig.

Feuerbach's theorem: the nine-point circle is tangent to the incircle and excircles of a triangle. The incircle tangency is the Feuerbach point. In the geometry of triangles, the incircle and nine-point circle of a triangle are internally tangent to each other at the Feuerbach point of the triangle. The Feuerbach point is a triangle center, meaning that its definition does not depend on the placement and scale of the triangle.

The blue region is the feasible region. The tangency of the line with the feasible region represents the solution. The line is the best achievable contour line (locus with a given value of the objective function). A simple problem (shown in the diagram) can be defined by the constraints :x1 ≥ 0 :x2 ≥ 0 :x12 \+ x22 ≥ 1 :x12 \+ x22 ≤ 2 with an objective function to be maximized :f(x) = x1 \+ x2 where x = (x1, x2).

In an isotropic medium, because the propagation speed is independent of direction, the secondary wavefronts that expand from points on a primary wavefront in a given infinitesimal time are spherical, so that their radii are normal to their common tangent surface at the points of tangency. But their radii mark the ray directions, and their common tangent surface is a general wavefront. Thus the rays are normal (orthogonal) to the wavefronts.De Witte, 1959, p.

A tacnode at the origin of the curve defined by (x2+y2 −3x)2−4x2(2−x)=0 In classical algebraic geometry, a tacnode (also called a point of osculation or double cusp). is a kind of singular point of a curve. It is defined as a point where two (or more) osculating circles to the curve at that point are tangent. This means that two branches of the curve have ordinary tangency at the double point.

550px There is a unique circle passing through the top and bottom of the painting and tangent to the eye-level line. By elementary geometry, if the viewer's position were to move along the circle, the angle subtended by the painting would remain constant. All positions on the eye-level line except the point of tangency are outside of the circle, and therefore the angle subtended by the painting from those points is smaller. By Euclid's Elements III.

Fig. 10. An Edgeworth box with multiple equilibria Fig. 11. An Edgeworth box with multiple equilibria (detail) It might be supposed from economic considerations that if a shared tangent exists through a given endowment, and if the indifference curves are not pathological in their shape, then the point of tangency will be unique. This turns out not to be true. Conditions for uniqueness of equilibrium have been the subject of extensive research: see General equilibrium theory. Figs.

Tusi's diagram of the Tusi couple, 13th centuryVatican Library, Vat. ar. 319 fol. 28 verso math19 NS.15 , fourteenth-century copy of a manuscript from Tusi Some modern commentators also call the Tusi couple a "rolling device" and describe it as a small circle rolling inside a large fixed circle. However, Tusi himself described it differently: :If two coplanar circles, the diameter of one of which is equal to half the diameter of the other, are taken to be internally tangent at a point, and if a point is taken on the smaller circle—and let it be at the point of tangency—and if the two circles move with simple motions in opposite direction in such a way that the motion of the smaller [circle] is twice that of the larger so the smaller completes two rotations for each rotation of the larger, then that point will be seen to move on the diameter of the larger circle that initially passes through the point of tangency, oscillating between the endpoints.

Cissoid of Diocles traced by points M with OM = M1M2. 462x462px In geometry, the cissoid of Diocles is a cubic plane curve notable for the property that it can be used to construct two mean proportionals to a given ratio. In particular, it can be used to double a cube. It can be defined as the cissoid of a circle and a line tangent to it with respect to the point on the circle opposite to the point of tangency.

An oval dome is a dome of oval shape in plan, profile, or both. The term comes from the Latin ovum, meaning "egg". The earliest oval domes were used by convenience in corbelled stone huts as rounded but geometrically undefined coverings, and the first examples in Asia Minor date to around 4000 B.C. The geometry was eventually defined using combinations of circular arcs, transitioning at points of tangency. If the Romans created oval domes, it was only in exceptional circumstances.

Seven circles theoremIn geometry, the seven circles theorem is a theorem about a certain arrangement of seven circles in the Euclidean plane. Specifically, given a chain of six circles all tangent to a seventh circle and each tangent to its two neighbors, the three lines drawn between opposite pairs of the points of tangency on the seventh circle all pass through the same point. Though elementary in nature, this theorem was not discovered until 1974 (by Evelyn, Money-Coutts, and Tyrrell).

As described below, Apollonius' problem has ten special cases, depending on the nature of the three given objects, which may be a circle (C), line (L) or point (P). By custom, these ten cases are distinguished by three letter codes such as CCP. Viète solved all ten of these cases using only compass and straightedge constructions, and used the solutions of simpler cases to solve the more complex cases. Figure 4: Tangency between circles is preserved if their radii are changed by equal amounts.

In the second approach, the radii of the given circles are modified appropriately by an amount Δr so that two of them are tangential (touching). Their point of tangency is chosen as the center of inversion in a circle that intersects each of the two touching circles in two places. Upon inversion, the touching circles become two parallel lines: Their only point of intersection is sent to infinity under inversion, so they cannot meet. The same inversion transforms the third circle into another circle.

The general problem of finding a hexlet for three given mutually tangent spheres A, B and C can be reduced to the annular case using inversion. This geometrical operation always transforms spheres into spheres or into planes, which may be regarded as spheres of infinite radius. A sphere is transformed into a plane if and only if the sphere passes through the center of inversion. An advantage of inversion is that it preserves tangency; if two spheres are tangent before the transformation, they remain so after.

36 (alternatively the power-of-a- point theorem), the distance from the wall to the point of tangency is the geometric mean of the heights of the top and bottom of the painting. This means, in turn, that if we reflect the bottom of the picture in the line at eye-level and draw the circle with the segment between the top of the picture and this reflected point as diameter, the circle intersects the line at eye- level in the required position (by Elements II.14).

Arrow and Debreu do not explain why they require global separation, which may have made their proofs easier but can be seen to have unexpected consequences. In Fig. 13 the point x is a point of tangency which is also a point at which indifference curves are locally separated by the dashed price line; but since they are not globally separated the point is not an equilibrium according to Arrow and Debreu’s definition. Fig. 14. A Pareto optimum which is not a ‘competitive equilibrium’In Fig.

The curve has a unique vertex at the point of tangency with its defining circle. That is, this point is the only point where the curvature reaches a local minimum or local maximum. The defining circle of the witch is also its osculating circle at the vertex, the unique circle that "kisses" the curve at that point by sharing the same orientation and curvature. Because this is an osculating circle at the vertex of the curve, it has third-order contact with the curve.

The configuration of a circle tangent to four circles in the plane has special properties, which have been elucidated by Larmor (1891) and Lachlan (1893). Such a configuration is also the basis for Casey's theorem, itself a generalization of Ptolemy's theorem. The extension of Apollonius' problem to three dimensions, namely, the problem of finding a fifth sphere that is tangent to four given spheres, can be solved by analogous methods. For example, the given and solution spheres can be resized so that one given sphere is shrunk to point while maintaining tangency.

The three tangent points are reflections of point H about the vertices of the Johnson triangle. # The points of tangency between the Johnson circles and the anticomplementary circle form another triangle, called the anticomplementary triangle of the reference triangle. It is similar to the Johnson triangle, and is homothetic by a factor 2 centered at H, their common circumcenter. # Johnson's theorem: The 2-wise intersection points of the Johnson circles (vertices of the reference triangle ABC) lie on a circle of the same radius r as the Johnson circles.

In the mathematical area of graph theory, a contact graph or tangency graph is a graph whose vertices are represented by geometric objects (e.g. curves, line segments, or polygons), and whose edges correspond to two objects touching (but not crossing) according to some specified notion. online PDF It is similar to the notion of an intersection graph but differs from it in restricting the ways that the underlying objects are allowed to intersect each other. The circle packing theorem states that every planar graph can be represented as a contact graph of circles.

A Doyle spiral of type (8,16) printed in 1911 in Popular Science as an illustration of phyllotaxis. One of its spiral arms is shaded. Coxeter's loxodromic sequence of tangent circles, a Doyle spiral of type (1,3) In the mathematics of circle packing, a Doyle spiral is a pattern of non-crossing circles in the plane, each tangent to six others. The sequences of circles linked to each other through opposite points of tangency lie on logarithmic spirals (or, in degenerate cases, circles or lines) having, in general, three different shapes of spirals.

A consequence of the hairy ball theorem is that any continuous function that maps an even-dimensional sphere into itself has either a fixed point or a point that maps onto its own antipodal point. This can be seen by transforming the function into a tangential vector field as follows. Let s be the function mapping the sphere to itself, and let v be the tangential vector function to be constructed. For each point p, construct the stereographic projection of s(p) with p as the point of tangency.

However, between each pair of tangencies, the curvature must decrease to less than that of the circle, for instance at a point obtained by translating the circle until it no longer contains any part of the curve between the two points of tangency and considering the last point of contact between the translated circle and the curve. Therefore, there is a local minimum of curvature between each pair of tangencies, giving two of the four vertices. There must be a local maximum of curvature between each pair of local minima, giving the other two vertices..

A polygon is defined to be cyclic if its vertices are all concyclic. For example, all the vertices of a regular polygon of any number of sides are concyclic. A tangential polygon is one having an inscribed circle tangent to each side of the polygon; these tangency points are thus concyclic on the inscribed circle. A convex quadrilateral is orthodiagonal (has perpendicular diagonals) if and only if the midpoints of the sides and the feet of the four altitudes are eight concyclic points, on what is called the eight-point circle.

In automotive design, a class A surface is any of a set of freeform surfaces of high efficiency and quality. Although, strictly, it is nothing more than saying the surfaces have curvature and tangency alignment – to ideal aesthetical reflection quality, many people interpret class A surfaces to have G2 (or even G3) curvature continuity to one another (see free form surface modelling). Bézier surface map definition Class A surfacing is done using computer-aided industrial design applications. Class A surface modellers are also called "digital sculptors" in the industry.

Construction of the envelope of a family of curves. In geometry, an envelope of a planar family of curves is a curve that is tangent to each member of the family at some point, and these points of tangency together form the whole envelope. Classically, a point on the envelope can be thought of as the intersection of two "infinitesimally adjacent" curves, meaning the limit of intersections of nearby curves. This idea can be generalized to an envelope of surfaces in space, and so on to higher dimensions.

As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point. Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized; see Tangent space.

This result can be generalised to cyclic polygons: the circumcentre is equidistant from each of the vertices. Likewise, the incentre of a triangle or any other tangential polygon is equidistant from the points of tangency of the polygon's sides with the circle. Every point on a perpendicular bisector of the side of a triangle or other polygon is equidistant from the two vertices at the ends of that side. Every point on the bisector of an angle of any polygon is equidistant from the two sides that emanate from that angle.

Assume that there are c units of coconut and f units of fish available for consumption in the Crusoe Friday economy. Given this endowment bundle (c,f), the Pareto efficient bundle can be determined at the mutual tangency of Crusoe's and Friday's indifference curves in the Edgeworth box along the Pareto Set (contract curve). These are the bundles at which Crusoe's and Friday's marginal rate of substitution are equal. In a simple exchange economy, the contract curve describes the set of bundles that exhaust the gains from trade.

If even one closed Steiner chain is possible for two given circles (blue), then infinitely many Steiner chains are possible, all related by rotation. Their points of tangency always fall on a circle (orange). If the two given circles are nested, one inside the other, the centers of the Steiner chain circles (black) fall on an ellipse (red); otherwise, they fall on a hyperbola. A Steiner chain between two non-intersecting circles can always be transformed into another Steiner chain of equally sized circles sandwiched between two concentric circles.

The general statement of Apollonius' problem is to construct one or more circles that are tangent to three given objects in a plane, where an object may be a line, a point or a circle of any size. These objects may be arranged in any way and may cross one another; however, they are usually taken to be distinct, meaning that they do not coincide. Solutions to Apollonius' problem are sometimes called Apollonius circles, although the term is also used for other types of circles associated with Apollonius. The property of tangency is defined as follows.

Torricelli extended this work to other curves such as the cycloid, and then the formula was generalized to fractional and negative powers by Wallis in 1656. In a 1659 treatise, Fermat is credited with an ingenious trick for evaluating the integral of any power function directly. Fermat also obtained a technique for finding the centers of gravity of various plane and solid figures, which influenced further work in quadrature. James Gregory, influenced by Fermat's contributions both to tangency and to quadrature, was then able to prove a restricted version of the second fundamental theorem of calculus in the mid-17th century.

Circles of Apollonius The problem of Apollonius is one of the earliest examples of enumerative geometry. This problem asks for the number and construction of circles that are tangent to three given circles, points or lines. In general, the problem for three given circles has eight solutions, which can be seen as 23, each tangency condition imposing a quadratic condition on the space of circles. However, for special arrangements of the given circles, the number of solutions may also be any integer from 0 (no solutions) to six; there is no arrangement for which there are seven solutions to Apollonius' problem.

S1 is small as it passes through the hole between A, B and C, and grows till it becomes a plane tangent to them. The centre of inversion is now also with a point of tangency with the image of S6, so it is also a plane tangent to A, B and C. As S1 proceeds, its concavity is reversed and now it surrounds all the other spheres, tangent to A, B, C, S2 and S6. S2 pushes upwards and grows to become a tangent plane and S6 shrinks. S1 then obtains S6's former position as a tangent plane.

The dimensions in the sketch can be controlled independently, or by relationships to other parameters inside or outside the sketch. In an assembly, the analog to sketch relations are mates. Just as sketch relations define conditions such as tangency, parallelism, and concentricity with respect to sketch geometry, assembly mates define equivalent relations with respect to the individual parts or components, allowing the easy construction of assemblies. SolidWorks also includes additional advanced mating features such as gear and cam follower mates, which allow modeled gear assemblies to accurately reproduce the rotational movement of an actual gear train.

Koebe–Andreev–Thurston theorem: If G is a finite maximal planar graph, then the circle packing whose tangency graph is isomorphic to G is unique, up to Möbius transformations and reflections in lines. Thurston observes that this uniqueness is a consequence of the Mostow rigidity theorem. To see this, let G be represented by a circle packing. Then the plane in which the circles are packed may be viewed as the boundary of a halfspace model for three-dimensional hyperbolic space; with this view, each circle is the boundary of a plane within the hyperbolic space.

Blue curve of Pareto efficient points, at points of tangency of indifference curves in an Edgeworth box. If the initial allocations of the two goods are at a point not on this locus, then the two people can trade to a point on the efficient locus within the lens formed by the indifference curves that they were originally on. The set of all these efficient points that could be traded to is the contract curve. In the graph below, the initial endowments of the two people are at point X, on Kelvin's indifference curve K1 and Jane's indifference curve J1.

The restriction of this projective transformation to the midsphere is a Möbius transformation.. There is a unique way of performing this transformation so that the midsphere is the unit sphere and so that the centroid of the points of tangency is at the center of the sphere; this gives a representation of the given polyhedron that is unique up to congruence, the canonical polyhedron.. Alternatively, a transformed polyhedron that maximizes the minimum distance of a vertex from the midsphere can be found in linear time; the canonical polyhedron chosen in this way has maximal symmetry among all choices of the canonical polyhedron..

Then one can use the Atiyah–Bott fixed-point theorem, of Michael Atiyah and Raoul Bott, to reduce, or localize, the computation of a GW invariant to an integration over the fixed-point locus of the action. Another approach is to employ symplectic surgeries to relate X to one or more other spaces whose GW invariants are more easily computed. Of course, one must first understand how the invariants behave under the surgeries. For such applications one often uses the more elaborate relative GW invariants, which count curves with prescribed tangency conditions along a symplectic submanifold of X of real codimension two.

The same inversion can be used to show that the points where the circles of the Pappus chain are tangent to one another lie on a common circle. As noted above, the inversion centered at point A transforms the arbelos circles CU and CV into two parallel lines, and the circles of the Pappus chain into a stack of equally sized circles sandwiched between the two parallel lines. Hence, the points of tangency between the transformed circles lie on a line midway between the two parallel lines. Undoing the inversion in the circle, this line of tangent points is transformed back into a circle.

Thus, if the inversion transformation is chosen judiciously, the problem can be reduced to a simpler case, such as the annular Soddy's hexlet. Inversion is reversible; repeating an inversion in the same point returns the transformed objects to their original size and position. Inversion in the point of tangency between spheres A and B transforms them into parallel planes, which may be denoted as a and b. Since sphere C is tangent to both A and B and does not pass through the center of inversion, C is transformed into another sphere c that is tangent to both planes; hence, c is sandwiched between the two planes a and b.

In algebraic geometry, the conic sections in the projective plane form a linear system of dimension five, as one sees by counting the constants in the degree two equations. The condition to pass through a given point P imposes a single linear condition, so that conics C through P form a linear system of dimension 4. Other types of condition that are of interest include tangency to a given line L. In the most elementary treatments a linear system appears in the form of equations :\lambda C + \mu C' = 0\ with λ and μ unknown scalars, not both zero. Here C and C′ are given conics.

As the probability density function of the Cauchy distribution, the witch of Agnesi has applications in probability theory. It also gives rise to Runge's phenomenon in the approximation of functions by polynomials, has been used to approximate the energy distribution of spectral lines, and models the shape of hills. The witch is tangent to its defining circle at one of the two defining points, and asymptotic to the tangent line to the circle at the other point. It has a unique vertex (a point of extreme curvature) at the point of tangency with its defining circle, which is also its osculating circle at that point.

From an interior point in a reference triangle, the nearest points on the three sides serve as the vertices of the pedal triangle of that point. If the interior point is the circumcenter of the reference triangle, the vertices of the pedal triangle are the midpoints of the reference triangle's sides, and so the pedal triangle is called the midpoint triangle or medial triangle. The midpoint triangle subdivides the reference triangle into four congruent triangles which are similar to the reference triangle. The Gergonne triangle or intouch triangle of a reference triangle has its vertices at the three points of tangency of the reference triangle's sides with its incircle.

Leibniz, G., "Nova Methodus pro Maximis et Minimis", Acta Eruditorum, Oct. 1684. More precisely, a straight line is said to be a tangent of a curve at a point if the line passes through the point on the curve and has slope , where f is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space. As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point.

There are two broad and conflicting approaches to the problem of generating valid cutter locations, given a CAD model and a tool definition: calculation by offsets, and calculation against triangles. Each is discussed in a later section of this article. The most common example of the general cutter location problem is cutter radius compensation (CRC), in which an endmill (whether square end, ball end, or bull end) must be offset to compensate for its radius. Since the 1950s, CRC calculations finding tangency points on the fly have been done automatically within CNC controls, following the instructions of G-codes such as G40, G41, and G42.

First, a point, line or circle is assumed to be tangent to itself; hence, if a given circle is already tangent to the other two given objects, it is counted as a solution to Apollonius' problem. Two distinct geometrical objects are said to intersect if they have a point in common. By definition, a point is tangent to a circle or a line if it intersects them, that is, if it lies on them; thus, two distinct points cannot be tangent. If the angle between lines or circles at an intersection point is zero, they are said to be tangent; the intersection point is called a tangent point or a point of tangency.

In most situations, it is convenient to set each of these curves tangent to the trajectory at the point of intersection. Curves that obey this condition (and also the further condition that they have the same curvature at the point of tangency as would be produced by the object's gravity towards the central body in the absence of perturbing forces) are called osculating, while the variables parameterising these curves are called osculating elements. In some situations, description of orbital motion can be simplified and approximated by choosing orbital elements that are not osculating. Also, in some situations, the standard (Lagrange-type or Delaunay-type) equations furnish orbital elements that turn out to be non-osculating.

Since a standard torus is the orbit of a point under a two dimensional abelian subgroup of the Möbius group, it follows that the cyclides also are, and this provides a second way to define them. A third property which characterizes Dupin cyclides is that their curvature lines are all circles (possibly through the point at infinity). Equivalently, the curvature spheres, which are the spheres tangent to the surface with radii equal to the reciprocals of the principal curvatures at the point of tangency, are constant along the corresponding curvature lines: they are the tangent spheres containing the corresponding curvature lines as great circles. Equivalently again, both sheets of the focal surface degenerate to conics.

An intersection of these two hyperbolas (if any) gives the center of a solution circle that has the chosen internal and external tangencies to the three given circles. The full set of solutions to Apollonius' problem can be found by considering all possible combinations of internal and external tangency of the solution circle to the three given circles. Isaac Newton (1687) refined van Roomen's solution, so that the solution-circle centers were located at the intersections of a line with a circle. Newton formulates Apollonius' problem as a problem in trilateration: to locate a point Z from three given points A, B and C, such that the differences in distances from Z to the three given points have known values.

A horosphere within the Poincaré disk model tangent to the edges of a hexagonal tiling cell of a hexagonal tiling honeycomb Apollonian sphere packing can be seen as showing horospheres that are tangent to an outer sphere of a Poincaré disk model In hyperbolic geometry, a horosphere (or parasphere) is a specific hypersurface in hyperbolic n-space. It is the boundary of a horoball, the limit of a sequence of increasing balls sharing (on one side) a tangent hyperplane and its point of tangency. For n = 2 a horosphere is called a horocycle. A horosphere can also be described as the limit of the hyperspheres that share a tangent hyperplane at a given point, as their radii go towards infinity.

Malfatti's work was popularized for a wider readership in French by Joseph Diaz Gergonne in the first volume of his Annales (1811), with further discussion in the second and tenth. However, Gergonne only stated the circle-tangency problem, not the area-maximizing one. Malfatti's assumption that the two problems are equivalent is incorrect. , who went back to the original Italian text, observed that for some triangles a larger area can be achieved by a greedy algorithm that inscribes a single circle of maximal radius within the triangle, inscribes a second circle within one of the three remaining corners of the triangle, the one with the smallest angle, and inscribes a third circle within the largest of the five remaining pieces.

Transversality depends on ambient space. The two curves shown are transverse when considered as embedded in the plane, but not if we consider them as embedded in a plane in three-dimensional space Suppose we have transverse maps f_1: L_1 \to M and f_2: L_2 \to M where L_1, L_2 and M are manifolds with dimensions \ell_1, \ell_2 and m respectively. The meaning of transversality differs a lot depending on the relative dimensions of M, L_1 and L_2. The relationship between transversality and tangency is clearest when \ell_1 + \ell_2 = m . We can consider three separate cases: #When \ell_1 + \ell_2 < m , it is impossible for the image of L_1 and L_2's tangent spaces to span M's tangent space at any point.

We have seen that the points of tangency of indifference curves are the Pareto optima, but we also saw previously that the economic equilibria are those points at which indifference curves are tangential to a common price line. It follows that the equilibria are precisely the Pareto optima. This argument applies with one restriction even if the curves are undifferentiable or if the equilibrium is on the boundary. The condition for equilibrium is that no further exchange will take place, and the condition for no further exchange to take place is that there is no direction of motion which benefits one consumer without harming the other; and this is equivalent to the definition of a Pareto optimum.See K. Wicksell, ‘Lectures on Political Economy’ I (1906), Eng. tr.

Assume the existence of an economy with two agents, Octavio and Abby, who consume two goods X and Y of which there are fixed supplies, as illustrated in the above Edgeworth box diagram. Further, assume an initial distribution (endowment) of the goods between Octavio and Abby and let each have normally structured (convex) preferences represented by indifference curves that are convex toward the people's respective origins. If the initial allocation is not at a point of tangency between an indifference curve of Octavio and one of Abby, then that initial allocation must be at a point where an indifference curve of Octavio crosses one of Abby. These two indifference curves form a lens shape, with the initial allocation at one of the two corners of the lens.

A more general version of the theorem, due to , applies to polynomials whose degree may be higher than three, but that have only three roots , , and . For such polynomials, the roots of the derivative may be found at the multiple roots of the given polynomial (the roots whose exponent is greater than one) and at the foci of an ellipse whose points of tangency to the triangle divide its sides in the ratios , , and . Another generalization () is to n-gons: some n-gons have an interior ellipse that is tangent to each side at the side's midpoint. Marden's theorem still applies: the foci of this midpoint-tangent inellipse are zeroes of the derivative of the polynomial whose zeroes are the vertices of the n-gon.

Recently, there has been an effort to experimentally image the 1s and 2p orbitals in a SrTiO3 crystal using scanning transmission electron microscopy with energy dispersive x-ray spectroscopy. Because the imaging was conducted using an electron beam, Coulombic beam- orbital interaction that is often termed as the impact parameter effect is included in the final outcome (see the figure at right). The shapes of p, d and f-orbitals are described verbally here and shown graphically in the Orbitals table below. The three p-orbitals for have the form of two ellipsoids with a point of tangency at the nucleus (the two-lobed shape is sometimes referred to as a "dumbbell"—there are two lobes pointing in opposite directions from each other).

If all extrema of are isolated, then an inflection point is a point on the graph of at which the tangent crosses the curve. A falling point of inflection is an inflection point where the derivative is negative on both sides of the point; in other words, it is an inflection point near which the function is decreasing. A rising point of inflection is a point where the derivative is positive on both sides of the point; in other words, it is an inflection point near which the function is increasing. For an algebraic curve, a non singular point is an inflection point if and only if the intersection number of the tangent line and the curve (at the point of tangency) is greater than 2.

B; with a level of production Q3, input X and input Y can only be combined efficiently in the certain ratio occurring at the kink in the isoquant. The firm will combine the two inputs in the required ratio to maximize profit. Isoquants are typically combined with isocost lines in order to solve a cost-minimization problem for given level of output. In the typical case shown in the top figure, with smoothly curved isoquants, a firm with fixed unit costs of the inputs will have isocost curves that are linear and downward sloped; any point of tangency between an isoquant and an isocost curve represents the cost-minimizing input combination for producing the output level associated with that isoquant.

In a homogeneous medium (also called a uniform medium), all the secondary wavefronts that expand from a given primary wavefront in a given time are congruent and similarly oriented, so that their envelope may be considered as the envelope of a single secondary wavefront which preserves its orientation while its center (source) moves over . If is its center while is its point of tangency with , then moves parallel to , so that the plane tangential to at is parallel to the plane tangential to at . Let another (congruent and similarly orientated) secondary wavefront be centered on , moving with , and let it meet its envelope at point . Then, by the same reasoning, the plane tangential to at is parallel to the other two planes.

Circle packings, as studied in this book, are systems of circles that touch at tangent points but do not overlap, according to a combinatorial pattern of adjacencies specifying which pairs of circles should touch. The circle packing theorem states that a circle packing exists if and only if the pattern of adjacencies forms a planar graph; it was originally proved by Paul Koebe in the 1930s, and popularized by William Thurston, who rediscovered it in the 1970s and connected it with the theory of conformal maps and conformal geometry. As a topic, this should be distinguished from sphere packing, which considers higher dimensions (here, everything is two dimensional) and is more focused on packing density than on combinatorial patterns of tangency. The book is divided into four parts, in progressive levels of difficulty.

Edelsbrunner and Welzl (1986) first studied the problem of constructing all k-sets of an input point set, or dually of constructing the k-level of an arrangement. The k-level version of their algorithm can be viewed as a plane sweep algorithm that constructs the level in left-to-right order. Viewed in terms of k-sets of point sets, their algorithm maintains a dynamic convex hull for the points on each side of a separating line, repeatedly finds a bitangent of these two hulls, and moves each of the two points of tangency to the opposite hull. Chan (1999) surveys subsequent results on this problem, and shows that it can be solved in time proportional to Dey's O(nk1/3) bound on the complexity of the k-level.

Let C and D be two plane conics. If it is possible to find, for a given n > 2, one n-sided polygon that is simultaneously inscribed in C (meaning that all of its vertices lie on C) and circumscribed around D (meaning that all of its edges are tangent to D), then it is possible to find infinitely many of them. Each point of C or D is a vertex or tangency (respectively) of one such polygon. If the conics are circles, the polygons that are inscribed in one circle and circumscribed about the other are called bicentric polygons, so this special case of Poncelet's porism can be expressed more concisely by saying that every bicentric polygon is part of an infinite family of bicentric polygons with respect to the same two circles.

In the first approach, the given circles are shrunk or swelled (appropriately to their tangency) until one given circle is shrunk to a point P. In that case, Apollonius' problem degenerates to the CCP limiting case, which is the problem of finding a solution circle tangent to the two remaining given circles that passes through the point P. Inversion in a circle centered on P transforms the two given circles into new circles, and the solution circle into a line. Therefore, the transformed solution is a line that is tangent to the two transformed given circles. There are four such solution lines, which may be constructed from the external and internal homothetic centers of the two circles. Re-inversion in P and undoing the resizing transforms such a solution line into the desired solution circle of the original Apollonius problem.

The precise shape of any Doyle spiral can be parameterized by a pair of natural numbers describing the number of spiral arms for each of the three ways of grouping circles by their opposite points of tangency. If the numbers of arms of two of the three types of spiral arm are p and q, with p < q and with fewer than q arms of the third type, then the number of arms of the third type is necessarily q-p. As special cases of this formula, when p=q the arms of the third type degenerate to circles, and there are infinitely many of them. And when p=q/2 the two types of arms with the smaller number p of copies are mirror reflections of each other and the arms with q copies degenerate to straight lines.

When the consumer's preference set is non-convex, then (for some prices) the consumer's demand is not connected. A disconnected demand implies some discontinuous behavior by the consumer as discussed by Hotelling: > If indifference curves for purchases be thought of as possessing a wavy > character, convex to the origin in some regions and concave in others, we > are forced to the conclusion that it is only the portions convex to the > origin that can be regarded as possessing any importance, since the others > are essentially unobservable. They can be detected only by the > discontinuities that may occur in demand with variation in price-ratios, > leading to an abrupt jumping of a point of tangency across a chasm when the > straight line is rotated. But, while such discontinuities may reveal the > existence of chasms, they can never measure their depth.

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 shaft and pulleys share a common centerline. The constraints of the key are set in relation to the keyseat. In engineering design, particularly in the use of computer-aided drafting and design, in the creation of 3D assemblies and multibody systems, the plural term "constraints" refers to demarcations of geometrical characteristics between two or more entities or solid modeling bodies; these delimiters are intentional in defining diverse properties of theoretical physical position and motion, or displacement. In addition, 2D sketches -including the ones used to create extrusions and solid bodies- can also be constrained. There are several constraints that may be applied between the entities or bodies depending much on their actual natural geometry; sometimes these are also referred to as ’’mates’’ and include: collinearity, perpendicularity, tangency, symmetry, coincidency and parallelity among other ways of establishing the orientation of the entity.

If income is altered in response to the price change such that a new budget line is drawn passing through the old consumption bundle but with the slope determined by the new prices and the consumer's optimal choice is on this budget line, the resulting change in consumption is called the Slutsky substitution effect. The idea is that the consumer is given enough money to purchase her old bundle at the new prices, and her choice changes are seen. If instead, a new budget line is found with the slope determined by the new prices but tangent to the indifference curve going through the old bundle, the difference between the new point of tangency and the old bundle is the Hicks substitution effect. The idea now is that the consumer is given just enough income to achieve her old utility at the new prices, and how her choice changes is seen.

He shows that the two arguments can be presented in the same terms, since the PPF plays the same role as the mirror-image indifference curve in an Edgeworth box. He also mentions that there’s no need for the curves to be differentiable, since the same result obtains if they touch at pointed corners. His definition of optimality was equivalent to Pareto’s: > If... it is possible to move one individual into a preferred position > without moving another individual into a worse position... we may say that > the relative optimum is not reached... The optimality condition for production is equivalent to the pair of requirements that (i) price should equal marginal cost and (ii) output should be maximised subject to (i). Lerner thus reduces optimality to tangency for both production and exchange, but does not say why the implied point on the PPF should be the equilibrium condition for a free market.

Property 1 is obvious from the definition. Property 2 is also clear: for any circle of radius r, and any point P on it, the circle of radius 2r centered at P is tangent to the circle in its point opposite to P; this applies in particular to P=H, giving the anticomplementary circle C. Property 3 in the formulation of the homothety immediately follows; the triangle of points of tangency is known as the anticomplementary triangle. For properties 4 and 5, first observe that any two of the three Johnson circles are interchanged by the reflection in the line connecting H and their 2-wise intersection (or in their common tangent at H if these points should coincide), and this reflection also interchanges the two vertices of the anticomplementary triangle lying on these circles. The 2-wise intersection point therefore is the midpoint of a side of the anticomplementary triangle, and H lies on the perpendicular bisector of this side.

Instead of passing through points, a different condition on a curve is being tangent to a given line. Being tangent to five given lines also determines a conic, by projective duality, but from the algebraic point of view tangency to a line is a quadratic constraint, so naive dimension counting yields 25 = 32 conics tangent to five given lines, of which 31 must be ascribed to degenerate conics, as described in fudge factors in enumerative geometry; formalizing this intuition requires significant further development to justify. Another classic problem in enumerative geometry, of similar vintage to conics, is the Problem of Apollonius: a circle that is tangent to three circles in general determines eight circles, as each of these is a quadratic condition and 23 = 8\. As a question in real geometry, a full analysis involves many special cases, and the actual number of circles may be any number between 0 and 8, except for 7\.

The Greek mathematician Menaechmus solved problems and proved theorems by using a method that had a strong resemblance to the use of coordinates and it has sometimes been maintained that he had introduced analytic geometry. Apollonius of Perga, in On Determinate Section, dealt with problems in a manner that may be called an analytic geometry of one dimension; with the question of finding points on a line that were in a ratio to the others. Apollonius in the Conics further developed a method that is so similar to analytic geometry that his work is sometimes thought to have anticipated the work of Descartes by some 1800 years. His application of reference lines, a diameter and a tangent is essentially no different from our modern use of a coordinate frame, where the distances measured along the diameter from the point of tangency are the abscissas, and the segments parallel to the tangent and intercepted between the axis and the curve are the ordinates.

From there they could agree to a mutually beneficial trade to anywhere in the lens formed by these indifference curves. But the only points from which no mutually beneficial trade exists are the points of tangency between the two people's indifference curves, such as point E. The contract curve is the set of these indifference curve tangencies within the lens—it is a curve that slopes upward to the right and goes through point E. right In microeconomics, the contract curve is the set of points representing final allocations of two goods between two people that could occur as a result of mutually beneficial trading between those people given their initial allocations of the goods. All the points on this locus are Pareto efficient allocations, meaning that from any one of these points there is no reallocation that could make one of the people more satisfied with his or her allocation without making the other person less satisfied. The contract curve is the subset of the Pareto efficient points that could be reached by trading from the people's initial holdings of the two goods.

The power of a point P outside of a given circle If C is a circle and P is a point outside C, then the power of P with respect to C is the square of the length of a line segment from P to a point T of tangency with C. Equivalently, if P has distance d from the center of the circle, and the circle has radius r, then (by the Pythagorean theorem) the power is d2 − r2. The same formula d2 − r2 may be extended to all points in the plane, regardless of whether they are inside or outside of C: points on C have zero power, and points inside C have negative power. The power diagram of a set of n circles Ci is a partition of the plane into n regions Ri (called cells), such that a point P belongs to Ri whenever circle Ci is the circle minimizing the power of P. The radical axis of two intersecting circles. The power diagram of the two circles is the partition of the plane into two halfplanes formed by this line.