Inversive geometry

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Steiner chain

Given two circles with one interior to the other, if small tangent circles can be inscribed around the region between the two circles such that the final circle is tangent to the first, the circles form a Steiner chain.The simplest way to construct a Steiner chain is to perform an inversion on a symmetrical arrangement on circles packed between a central circle of radius and an outer concentric circle of radius (Wells 1991). In this arrangement,(1)so the ratio of the radii for the small and large circles is(2)In addition, the radii of the circles in the ring are(3)and their centers are located at a distance(4)from the origin.To transform the symmetrical arrangement into a Steiner chain, take an inversion center which is a distance from the center of the symmetrical figure. Then the radii and of the outer and center circles become(5)(6)respectively. Equivalently, a Steiner chain results whenever the inversivedistance between the two original..

Stammler circles radical circle

The radical circle of the Stammler circles has center at the nine-point center , which is Kimberling center . The radius is given by(1)(2)(3)(P. Moses and J.-P. Ehrmann, pers. comm., Jan. 28, 2004), where is the circumradius, is the circumcenter, is the orthocenter, is the nine-point center, and is the area of the reference triangle.Its circle function is given by(4)which corresponds to Kimberling center .No Kimberling centers lie on it.The radical line of the circumcircle and Stammler circles radical circle passes through the circumcenter (i.e., bisects the circumcircle) and is perpendicular to the Euler line (P. Moses, pers. comm., Jan. 28, 2005).The radical line of the Stammler circle and Stammler circles radical circle passes through the Kimberling center and is perpendicular to the Euler line (P. Moses, pers. comm., Jan. 28, 2005).The radical line of the first Droz-Farny circle and..

Midsphere

The midsphere is the sphere with respect to which the polyhedron vertices of a polyhedron are the inversion poles of the planes of the faces of the dual polyhedron (and vice versa), also called the intersphere, reciprocating sphere, or inversion sphere. The radius of the midsphere is called the midradius.The midsphere touches all polyhedron edges of a semiregular or regular polyhedron, as well as the edges of the dual of that solid (Cundy and Rollett 1989, p. 117). Note that the midsphere does not necessarily pass through the midpoints of the edges a polyhedron dual, but is rather only tangent to the edges at some point along their lengths.The figure above shows the Platonic solids and their duals, with the circumsphere of the solid, midsphere, and insphere of the dual superposed.

Midradius

The radius of the midsphere of a polyhedron, also called the interradius. Let be a point on the original polyhedron and the corresponding point on the dual. Then because and are inverse points, the radii , , and satisfy(1)The above figure shows a plane section of a midsphere.Let be the inradius the dual polyhedron, circumradius of the original polyhedron, and the side length of the original polyhedron. For a regular polyhedron with Schläfli symbol , the dual polyhedron is . Then(2)(3)(4)Furthermore, let be the angle subtended by the polyhedron edge of an Archimedean solid. Then(5)(6)(7)so(8)(Cundy and Rollett 1989).For a Platonic or Archimedean solid, the midradius of the solid and dual can be expressed in terms of the circumradius of the solid and inradius of the dual gives(9)(10)and these radii obey(11)..

Salmon's theorem

There are at least two theorems known as Salmon's theorem. This first states that if and are two points, and are the perpendiculars from and to the polars of and , respectively, with respect to a circle with center , then (Durell 1928; Salmon 1954, §101, p. 93).The second Salmon's theorem states that, given a track bounded by two confocal ellipses, if a ball is rolled so that its trajectory is tangent to the inner ellipse, the ball's trajectory will be tangent to the inner ellipse following all subsequent caroms as well (Salmon 1954, §189, pp. 181-182).

Reciprocation

An incidence-preserving transformation in which points are transformed into their polars. A projective geometry-like duality principle holds for reciprocation which states that theorems for the original figure can be immediately applied to the reciprocal figure after suitable modification (Lachlan 1893, pp. 174-182). Reciprocation (or "polar reciprocation") is the strictly proper term for duality. Brückner (1900) gave one the first exact definitions of polar reciprocation for constructing dual polyhedra, although the plane geometric version (inversion pole, polar, and circle power) was considered by none less than Euclid (Wenninger 1983, pp. 1-2).Lachlan (1893, pp. 257-265) discusses another type of reciprocation he terms "circular reciprocation." However, the circular reciprocal figure is, in general, more complicated than the original, so the method is not as powerful..

Inversive distance

The inversive distance is the natural logarithm of the ratio of two concentric circles into which the given circles can be inverted. Let be the distance between the centers of two nonintersecting circles of radii and . Then the inversive distance is(1)(Coxeter and Greitzer 1967).The inversive distance between the Soddy circlesis given by(2)and the circumcircle and incircle of a triangle with circumradius and inradius are at inversive distance(3)(Coxeter and Greitzer 1967, pp. 130-131).

Radical center

The radical lines of three circles are concurrent in a point known as the radical center (also called the power center). This theorem was originally demonstrated by Monge (Dörrie 1965, p. 153). It is a special case of the three conics theorem (Evelyn et al. 1974, pp. 13 and 15).The point of concurrence of the three radical lines of three circles is the point(Kimberling 1998, p. 225).

Inversion sphere

The sphere with respect to which inverse points are computed (i.e., with respect to which geometrical inversion is performed). For example, the cyclides are inversions in a sphere of tori. The center of the inversion sphere is called the inversion center, and its radius is called the inversion radius. When dual polyhedra are being considered, the inversion sphere is commonly called the midsphere (or intersphere, or reciprocating sphere).In two dimensions, the inversion sphere collapses to an inversioncircle.

Circle inverse curve

The inverse curve of the circle with parametric equations(1)(2)with respect to an inversion circle with center and radius is given by(3)(4)which is another circle.

Inversion pole

If two points and are inverse with respect to a circle (the inversion circle), then the straight line through which is perpendicular to the line of the points is called the polar of the point with respect to the circle, and is called the pole of the polar.An incidence-preserving transformation in which points and lines are transformedinto their poles and polars is called a reciprocation.The concept of poles and polars can also be generalized to arbitrary conic sections. If two tangents to a conic section at points and meet at , then is called the pole of the line with respect to the conic and is said to be the polar of the point with respect to the conic (Wells 1991). Let a line through meet a conic at points and and its polar at . Then , , , and are a harmonic range (Wells 1991). Furthermore, if two lines through a pole meet a conic at points and and points and , then the lines and meet on the polar, as do the lines and .The concept can be generalized even further to an arbitrary..

Chordal theorem

The locus of the point at which two given circles possess the same circle power is a straight line perpendicular to the line joining the midpoints of the circle and is known as the chordal (or, more commonly, the radical line) of the two circles.

Poncelet transverse

Let a circle lie inside another circle . From any point on , draw a tangent to and extend it to . From the point, draw another tangent, etc. For tangents, the result is called an -sided Poncelet transverse.If, on the circle of circumscription there is one point of origin for which a four-sided Poncelet transverse is closed, then the four-sided transverse will also close for any other point of origin on the circle (Dörrie 1965).

Poncelet's porism

If an -sided Poncelet transverse constructed for two given conic sections is closed for one point of origin, it is closed for any position of the point of origin. Specifically, given one ellipse inside another, if there exists one circuminscribed (simultaneously inscribed in the outer and circumscribed on the inner) -gon, then any point on the boundary of the outer ellipse is the vertex of some circuminscribed -gon. If the conic is taken as a circle (Casey 1888, pp. 124-126) , then a polygon which has both an incenter and a circumcenter (and for which the transversals would therefore close) is called a bicentric polygon.Amazingly, this problem is isomorphic to Gelfand'squestion (King 1994).For an even-sided polygon, the diagonals are concurrent at the limiting point of the two circles, whereas for an odd-sided polygon, the lines connecting the vertices to the opposite points of tangency are concurrent at the limiting point.Inverting..

Inversion

Inversion is the process of transforming points to a corresponding set of points known as their inverse points. Two points and are said to be inverses with respect to an inversion circle having inversion center and inversion radius if is the perpendicular foot of the altitude of , where is a point on the circle such that .The analogous notation of inversion can be performed in three-dimensional space withrespect to an inversion sphere.If and are inverse points, then the line through and perpendicular to is sometimes called a "polar" with respect to point , known as the "inversion pole". In addition, the curve to which a given curve is transformed under inversion is called its inverse curve (or more simply, its "inverse"). This sort of inversion was first systematically investigated by Jakob Steiner.From similar triangles, it immediately follows that the inverse points and obey(1)or(2)(Coxeter 1969, p. 78),..

Polar

If two points and are inverse (sometimes called conjugate) with respect to a circle (the inversion circle), then the straight line through which is perpendicular to the line of the points is called the polar of with respect to the circle, and is called the inversion pole of the polar.An incidence-preserving transformation in which points and lines are transformed into their inversion poles and polars is called reciprocation (a.k.a. constructing the dual).The concept of poles and polars can also be generalized to arbitrary conic sections. If two tangents to a conic section at points and meet at , then is called the inversion pole of the line with respect to the conic and is said to be the polar of the point with respect to the conic (Wells 1991).In the above figure, let a line through the polar meet a conic section at point and , and let the line intersect the polar line at . Then form a harmonic range (Wells 1991).In the above figure, let two lines through..

Inverse points

Points, also called polar reciprocals, which are transformed into each other through inversion about a given inversion circle (or inversion sphere). The points and are inverse points with respect to the inversion circle if(Wenninger 1983, p. 2). In this case, is called the inversion pole and the line through and perpendicular to is called the polar. In the above figure, the quantity is called the circle power of the point relative to the circle .Inverse points with respect to a triangle are generally understood to use the triangle's circumcircle as the inversion circle (Gallatly 1913).The point which is the inverse point of a given point with respect to an inversion circle may be constructed geometrically using a compass only (Coxeter 1969, p. 78; Courant and Robbins 1996, pp. 144-145).Inverse points can also be taken with respect to an inversion sphere, which is a natural extension of geometric inversion from the plane..

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