---
tag: XVBZJVQ3WA
kind: definition
name: Conformal geometric algebra
msc: [15A66, 51B10]
defines: 4S4Z7SQ47G
uses: [N8H460W3E2]
sources:
  - url: https://en.wikipedia.org/wiki/Conformal_geometric_algebra
---

## Definition

Let $n \ge 1$, and let $\R^{n+1,1} = \R^n \oplus \R e_+ \oplus \R e_-$, where $\R^n$ has the dot product, $e_+ \cdot e_+ = 1$, $e_- \cdot e_- = -1$, and the three summands are pairwise orthogonal.
Put $n_0 = \tfrac{1}{2}(e_- - e_+)$ and $n_\infty = e_- + e_+$.
The *conformal geometric algebra* of $\R^n$ is the [[geometric algebra|N8H460W3E2]] $\mathcal{G}(\R^{n+1,1})$, together with the embedding
$$x \mapsto X = x + \tfrac{1}{2} \abs{x}^2 n_\infty + n_0, \qquad x \in \R^n.$$

## Remarks

From $e_\pm \cdot e_\pm = \pm 1$ one gets $n_0 \cdot n_0 = n_\infty \cdot n_\infty = 0$ and $n_0 \cdot n_\infty = -1$.
Hence $X \cdot X = \abs{x}^2 - \abs{x}^2 = 0$, so points become null vectors, and $X \cdot Y = x \cdot y - \tfrac{1}{2}(\abs{x}^2 + \abs{y}^2) = -\tfrac{1}{2} \abs{x - y}^2$.
Example: for $c \in \R^n$ and $r > 0$, put $S = C - \tfrac{1}{2} r^2 n_\infty$ with $C$ the image of $c$.
Since $X \cdot n_\infty = -1$, one gets $X \cdot S = \tfrac{1}{2}(r^2 - \abs{x - c}^2)$, which vanishes exactly when $x$ lies on the sphere with centre $c$ and radius $r$.
Likewise, for a unit vector $m \in \R^n$ and $\delta \in \R$, the vector $m + \delta n_\infty$ satisfies $X \cdot (m + \delta n_\infty) = x \cdot m - \delta$, and it represents the plane $x \cdot m = \delta$.
Non-example: the vector $e_- \in \R^{n+1,1}$ is not the image of a point, since $e_- \cdot e_- = -1 \ne 0$.
The versors of the algebra, the products of invertible vectors, act on $\R^{n+1,1}$ by conjugation, up to sign.
They induce exactly the Möbius transformations of $\R^n \cup \set{\infty}$, the maps generated by translations, rotations, dilations and inversions in spheres.