---
tag: TFQSF9RA1G
kind: concept
name: Lie sphere geometry
msc: [51B25, 51B10]
sources:
  - url: https://en.wikipedia.org/wiki/Lie_sphere_geometry
same_as:
  wikidata: Q17103254
  wikipedia: 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|Q3Q4FF23VG]] 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|HFBW3SXQVM]] and Laguerre geometry as subgeometries, and it is named after Sophus Lie.

## Explanations

### L1 human

In ordinary geometry, points, lines and circles are different kinds of objects.
Lie sphere geometry treats them all as circles: a point is a circle of radius zero and a line is a circle of infinite radius.
Give each circle an orientation, a direction of travel, and call two oriented circles in contact when they touch with matching directions at the touching point.
The geometry keeps track only of which circles touch which; its transformations may turn points into circles and circles into lines, as long as touching is preserved.