---
tag: 4S4Z7SQ47G
kind: concept
name: Conformal geometric algebra
msc: [15A66, 51B10]
specializes: [N8H460W3E2]
sources:
  - url: https://en.wikipedia.org/wiki/Conformal_geometric_algebra
same_as:
  wikidata: Q5160252
  wikipedia: Conformal geometric algebra
---

## Summary

Conformal geometric algebra is the [[geometric algebra|N8H460W3E2]] of $\R^{n+1,1}$, used as a model of Euclidean space $\R^n$.
In it, points, spheres and planes are all represented by vectors.
A point $x \in \R^n$ is embedded as a null vector $X$, and the inner product of two embedded points is $-\tfrac{1}{2}$ times their squared distance.
Spheres and planes are the vectors orthogonal to the points that lie on them.
All Möbius transformations of $\R^n \cup \set{\infty}$ act by conjugation with elements of the algebra.
It is used in computer graphics and robotics, where rigid motions, intersections and tangencies become algebraic operations.

## Explanations

### L1 human

The trick is to add two extra dimensions to space, one that squares to $+1$ and one that squares to $-1$.
In this bigger space a point becomes a vector of length zero, and the distance between two points is read from their inner product.
A sphere is then also a single vector, and asking whether a point lies on it is just asking whether an inner product is zero.
Rotations, translations, scalings and inversions in spheres all become the same kind of algebraic move.