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.
Inversion seems to have been discovered by a number of people contemporaneously, including Steiner (1824), Quetelet (1825), Bellavitis (1836), Stubbs and Ingram (1842–3) and Kelvin (1845).
The proof roughly goes as below: Invert with respect to the incircle of triangle ABC.
Then the inversive distance (usually denoted δ) is defined as the natural logarithm of the ratio of the radii of the two concentric circles.
It provides an exact solution to the important problem of converting between linear and circular motion.
The inversion of a cylinder, cone, or torus results in a Dupin cyclide.
(north pole) of the sphere onto the tangent plane at the opposite point
, green in the picture), then it will be mapped by the inversion at the unit sphere (red) onto the tangent plane at point
[7] Edward Kasner wrote his thesis on "Invariant theory of the inversion group".
[8] 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.
These Möbius planes can be described axiomatically and exist in both finite and infinite versions.
The invariant is: According to Coxeter,[9] the transformation by inversion in circle was invented by L. I. Magnus in 1831.
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.
where: Reciprocation is key in transformation theory as a generator of the Möbius group.
The other generators are translation and rotation, both familiar through physical manipulations in the ambient 3-space.
Introduction of reciprocation (dependent upon circle inversion) is what produces the peculiar nature of Möbius geometry, which is sometimes identified with inversive geometry (of the Euclidean plane).
Möbius group elements are analytic functions of the whole plane and so are necessarily conformal.
becomes As mentioned above, zero, the origin, requires special consideration in the circle inversion mapping.
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.
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.
In a real n-dimensional Euclidean space, an inversion in the sphere of radius r centered at the point
: The transformation by inversion in hyperplanes or hyperspheres in En can be used to generate dilations, translations, or rotations.
When two parallel hyperplanes are used to produce successive reflections, the result is a translation.
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.
Inversive geometry has been applied to the study of colorings, or partitionings, of an n-sphere.
[11] 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.
Computing the Jacobian in the case zi = xi/‖x‖2, where ‖x‖2 = x12 + ... + xn2 gives JJT = kI, with k = 1/‖x‖4n, and additionally det(J) is negative; hence the inversive map is anticonformal.
These together with the subspace hyperplanes separating hemispheres are the hypersurfaces of the Poincaré disc model of hyperbolic geometry.
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.
