Inversive geometry
In geometry, inversive geometry is the study of "inversion", a transformation of the Euclidean plane that maps circles or lines to other circles or lines and that preserves the angles between crossing curves. Many difficult problems in geometry become much more tractable when an inversion is applied. The concept of inversion can be generalized to higher-dimensional spaces. Inversion in a circle. Inverse of a point. To invert a number in arithmetic usually means to take its reciprocal. A closely related idea in geometry is that of "inverting" a point. In the plane, the inverse of a point  P with respect to a "reference circle (Ø)" with center "O" and radius "r" is a point "P", lying on the ray from "O" through "P" such that This is called circle inversion or plane inversion. The inversion taking any point "P" (other than "O") to its image "P" also takes "P" back to "P", so the result of applying the same inversion twice is the identity transformation on all the points of the plane other than "O" (self-inversion). To make inversion an involution it is necessary to introduce a point at infinity, a single point placed on all  the lines, and extend the inversion, by definition, to interchange the center "O" and this point at infinity. It follows from the definition that the inversion of any point inside the reference circle must lie outside it, and vice versa, with the center and the point at infinity changing positions, whilst any point on the circle is unaffected (is "invariant" under inversion). In summary, the nearer a point to the center, the further away its transformation, and vice versa. To construct the inverse "P" of a point "P" outside a circle "Ø": To construct the inverse "P" of a point  P inside a circle "Ø": Dutta's construction. There is a construction of the inverse point to "A" with respect to a circle "P" that is "independent" of whether "A" is inside or outside "P". Consider a circle "P" with center "O" and a point "A" which may lie inside or outside the circle "P". Properties. The inversion of a set of points in the plane with respect to a circle is the set of inverses of these points. The following properties make circle inversion useful. Additional properties include: Application. For a circle not passing through the center of inversion , the center of the circle being inverted and the center of its image under inversion are collinear with the center of the reference circle. This fact can be used to prove that the Euler line of the intouch triangle of a triangle coincides with its OI line. The proof roughly goes as below: Invert with respect to the incircle of triangle "ABC". The medial triangle of the intouch triangle is inverted into triangle "ABC", meaning the circumcenter of the medial triangle, that is, the nine-point center of the intouch triangle, the incenter and circumcenter of triangle "ABC" are collinear . Any two non-intersecting circles may be inverted into concentric circles. Then the inversive distance (usually denoted δ) is defined as the natural logarithm of the ratio of the radii of the two concentric circles. In addition, any two non-intersecting circles may be inverted into congruent circles, using circle of inversion centered at a point on the circle of antisimilitude. The Peaucellier–Lipkin linkage is a mechanical implementation of inversion in a circle. It provides an exact solution to the important problem of converting between linear and circular motion. Pole and polar. If point "R" is the inverse of  point "P" then the lines perpendicular to the line "PR" through one of the points is the polar of the other point (the pole). Poles and polars have several useful properties: In three dimensions. Circle inversion is generalizable to sphere inversion in three dimensions. The inversion of a point "P" in 3D with respect to a reference sphere centered at a point "O" with radius "R" is a point "P" ' on the ray with direction "OP" such that formula_3. As with the 2D version, a sphere inverts to a sphere, except that if a sphere passes through the center "O  of the reference sphere, then it inverts to a plane. Any plane passing through O", inverts to a sphere touching at "O". A circle, that is, the intersection of a sphere with a secant plane, inverts into a circle, except that if the circle passes through "O" it inverts into a line. This reduces to the 2D case when the secant plane passes through "O", but is a true 3D phenomenon if the secant plane does not pass through "O". Examples in three dimensions. Sphere. The simplest surface (besides a plane) is the sphere. The first picture shows a non  trivial inversion (the center of the sphere is not the center of inversion) of a sphere together with two orthogonal intersecting pencils of circles. Cylinder, cone, torus. The inversion of a cylinder, cone, or torus results in a Dupin cyclide. Spheroid. A spheroid is a surface of revolution and contains a pencil of circles which is mapped onto a pencil of circles (see picture). The inverse image of a spheroid is a surface of degree 4. Hyperboloid of one sheet. A hyperboloid of one sheet, which is a surface of revolution contains a pencil of circles which is mapped onto  a pencil of circles. A hyperboloid of one sheet contains additional two pencils of lines, which are mapped onto pencils of circles. The picture shows one such line (blue) and its inversion. Stereographic projection as the inversion of a sphere. A stereographic projection usually projects a sphere from a point formula_4 (north pole) of the sphere onto the tangent plane at the opposite point formula_5 (south pole). This mapping can be performed by an inversion of the sphere onto its tangent plane. If the sphere (to be projected) has the equation formula_6 (alternately written formula_7; center formula_8, radius formula_9, green  in the picture), then it will be mapped by the inversion at the unit sphere (red) onto the tangent plane at point formula_10. The lines through the center of inversion (point formula_4) are mapped onto themselves. They are the projection lines of the stereographic projection. 6-sphere coordinates. The 6-sphere coordinates are a coordinate system for three-dimensional space obtained by inverting the Cartesian coordinates. Axiomatics and generalization. One of the first to consider foundations of inversive geometry was Mario Pieri in 1911 and 1912. Edward Kasner wrote his thesis on "Invariant theory of the inversion group". More recently  the mathematical structure of inversive geometry has been interpreted as an incidence structure where the generalized circles are called "blocks": In incidence geometry, any affine plane together with a single point at infinity forms a Möbius plane, also known as an "inversive plane". The point at infinity is added to all the lines. These Möbius planes can be described axiomatically and exist in both finite and infinite versions. A model for the Möbius plane that comes from the Euclidean plane is the Riemann sphere. Invariant. The cross-ratio between 4 points formula_12 is invariant under an inversion. In particular if  O is the centre of the inversion and formula_13 and formula_14 are distances to the ends of a line L, then length of the line formula_15 will become formula_16 under an inversion with centre O. The invariant is: Relation to Erlangen program. According to Coxeter, the transformation by inversion in circle was invented by L. I. Magnus in 1831. Since then this mapping has become an avenue to higher mathematics. Through some steps of application of the circle inversion map, a student of transformation geometry soon appreciates the significance of Felix Klein’s Erlangen program, an outgrowth of certain models  of hyperbolic geometry Dilation. The combination of two inversions in concentric circles results in a similarity, homothetic transformation, or dilation characterized by the ratio of the circle radii. When formula_19 the equation for formula_20 becomes Higher geometry. As mentioned above, zero, the origin, requires special consideration in the circle inversion mapping. The approach is to adjoin a point at infinity designated ∞ or 1/0 . In the complex number approach, where reciprocation is the apparent operation, this procedure leads to the complex projective line, often called the Riemann sphere. It was subspaces and subgroups of this space and group of mappings  that were applied to produce early models of hyperbolic geometry by Beltrami, Cayley, and Klein. Thus inversive geometry includes the ideas originated by Lobachevsky and Bolyai in their plane geometry. Furthermore, Felix Klein was so overcome by this facility of mappings to identify geometrical phenomena that he delivered a manifesto, the Erlangen program, in 1872. Since then many mathematicians reserve the term "geometry" for a space together with a group of mappings of that space. The significant properties of figures in the geometry are those that are invariant under this group. For example, Smogorzhevsky develops several theorems of inversive geometry  before beginning Lobachevskian geometry. In higher dimensions. In "n"-dimensional space where there is a sphere of radius "r", inversion in the sphere is given by The transformation by inversion in hyperplanes or hyperspheres in E can be used to generate dilations, translations, or rotations. Indeed, two concentric hyperspheres, used to produce successive inversions, result in a dilation or contraction on the hyperspheres' center. Such a mapping is called a similarity. When two parallel hyperplanes are used to produce successive reflections, the result is a translation. When two hyperplanes intersect in an ("n"–2)-flat, successive reflections produce a rotation where every point  of the ("n"–2)-flat is a fixed point of each reflection and thus of the composition. All of these are conformal maps, and in fact, where the space has three or more dimensions, the mappings generated by inversion are the only conformal mappings. Liouville's theorem is a classical theorem of conformal geometry. The addition of a point at infinity to the space obviates the distinction between hyperplane and hypersphere; higher dimensional inversive geometry is frequently studied then in the presumed context of an "n"-sphere as the base space. The transformations of inversive geometry are often referred to as Möbius transformations . Inversive geometry has been applied to the study of colorings, or partitionings, of an "n"-sphere. Anticonformal mapping property. The circle inversion map is anticonformal, which means that at every point it preserves angles and reverses orientation (a map is called conformal if it preserves "oriented" angles). Algebraically, a map is anticonformal if at every point the Jacobian is a scalar times an orthogonal matrix with negative determinant: in two dimensions the Jacobian must be a scalar times a reflection at every point. This means that if "J" is the Jacobian, then formula_23 and formula_24 Computing the Jacobian in the case  , where gives , with , and additionally det("J") is negative; hence the inversive map is anticonformal. In the complex plane, the most obvious circle inversion map (i.e., using the unit circle centered at the origin) is the complex conjugate of the complex inverse map taking "z" to 1/"z". The complex analytic inverse map is conformal and its conjugate, circle inversion, is anticonformal. In this case a homography is conformal while an anti-homography is anticonformal. Inversive geometry and hyperbolic geometry. The ("n" − 1)-sphere with equation will have a positive radius if "a" + ... + "a" is greater than "c", and on  inversion gives the sphere Hence, it will be invariant under inversion if and only if "c" = 1. But this is the condition of being orthogonal to the unit sphere. Hence we are led to consider the ("n" − 1)-spheres with equation which are invariant under inversion, orthogonal to the unit sphere, and have centers outside of the sphere. These together with the subspace hyperplanes separating hemispheres are the hypersurfaces of the Poincaré disc model of hyperbolic geometry. Since inversion in the unit sphere leaves the spheres orthogonal to it invariant, the inversion maps the points inside the unit sphere  to the outside and vice versa. This is therefore true in general of orthogonal spheres, and in particular inversion in one of the spheres orthogonal to the unit sphere maps the unit sphere to itself. It also maps the interior of the unit sphere to itself, with points outside the orthogonal sphere mapping inside, and vice versa; this defines the reflections of the Poincaré disc model if we also include with them the reflections through the diameters separating hemispheres of the unit sphere. These reflections generate the group of isometries of the model, which tells us that the isometries are  conformal. Hence, the angle between two curves in the model is the same as the angle between two curves in the hyperbolic space.