Geometric Algebra

Featured Introductions

Geometric Numbers

SIGGRAPH 2019 Geometric Algebra

Geometric Algebra

Clifford’s Geometric Algebra enables a unified, intuitive and fresh perspective
on vector spaces, giving elements of arbitrary dimensionality a natural home.

The Vector

The Vector is an oriented, one dimensional

quantity. Two $parallel$ Vectors multiply to a Scalar ($mathbb R$).

Two $perp$ vectors anti-commute ($e_1e_2=-e_2e_1$)

The Bivector

The Bivector is an oriented, two dimensional

quantity. Bivectors naturally represent
transformations.
Similarly, $n$ vectors combine into an $n$-vector.

The $n$-dimensional geometric algebra $mathbb R_{p,q,r}$ is constructed
from $p$ positive, $q$ negative and $r$ null vectors
called generators, written $bf e_i$

The Scalar

The Scalars $mathbb R$ are included in

the algebras. every basis $n$-vector
squares
to a Real Number.

The Rotor

The product of two vectors, or the

exponentiation of a bivector creates

a rotor. (rotation, translation, ..)

A generic element of the algebra is called a multivector and is
a linear combination of scalar, vector and $n$-vector parts.
$$mathbf X = alpha_0 + alpha_1 mathbf e_1 + .. + alpha_i mathbf e_{12} + .. + alpha_n mathbf e_{12..n} $$

Vectorspace Geometric Algebra

The Geometric Algebras for the 2D and 3D vectors naturally
include 2D and 3D rotations.

$mathbb R_2^+ cong mathbb C$

2D Vectors

The geometric algebra $mathbb R_2$ of the

2D Vectors has the complex numbers

as its even subalgebra.

$mathbb R_3^+ cong mathbb H$

3D Vectors

The geometric algebra $mathbb R_3$ of the

3D Vectors has the quaternions

as its even subalgebra.

Projective Geometric Algebra

Using one extra dimension, we obtain the plane-based Projective Geometric Algebra
for 2 and 3 dimensions. Its elements are points, lines and planes. It
includes elements at infinity and has exception free join and meet operations.
Its even subalgebra provides in rotations & translations and is isomorphic to the dual quaternions

$mathbb R^*_{2,0,1}$ : 2D PGA

In 2D PGA points join $vee$ into

lines and lines meet $wedge$ into points.

Rotations and translations are unified,

with bivectors isomorphic to $mathfrak{SE}(2)$

$mathbb R^*_{3,0,1}$ : 3D PGA

In 3D PGA points and lines join $vee$ into

lines and planes, while lines and planes meet $wedge$

into points and lines. The bivectors are isomorphic

to $mathfrak{SE}(3)$, the dual quaternions

Conformal Geometric Algebra

Using two extra dimensions, we obtain the point-based Conformal Geometric Algebra
for 2 and 3 dimensions. Its elements are points, point pairs, lines, circles, spheres and planes. It
includes an infinite point and has exception free join and meet operations.
Its even subalgebra provides in conformal transformations.

$mathbb R_{3,1}$ : 2D CGA

In 2D CGA, points join ($wedge$) into lines

and circles, and lines and circles meet ($vee$) in points.

Rotations, Translations and Dilations all

come in versor form.

$mathbb R_{4,1}$ : 3D CGA

In 3D CGA, points join ($wedge$) into pairs, lines,

circles, planes and spheres, which meet ($vee$) in points,

pairs, circles and lines. Rotations, Translations

and Dilations all come in versor form.