Lie sphere geometry

Summary

Lie sphere geometry studies oriented spheres, oriented hyperplanes and points of $\R^n$ on an equal footing. Its transformations are those that preserve oriented contact. These objects correspond to the points of a quadric in projective space $\mathbb{RP}^{n+2}$, the Lie quadric, given by a quadratic form of signature $(n+1, 2)$. Oriented contact becomes orthogonality for that form, and the Lie sphere transformations are the projective maps preserving the quadric. It contains the Möbius geometry of inversive geometry and Laguerre geometry as subgeometries, and it is named after Sophus Lie.