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. 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).[1]
For other uses, see Point reflection.The concept of inversion can be generalized to higher-dimensional spaces.
Axiomatics and generalization[edit]
One of the first to consider foundations of inversive geometry was Mario Pieri in 1911 and 1912.[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. 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.
Anticonformal mapping property[edit]
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 and 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.
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.
