### The Vector

The **Vector** is an oriented, **one dimensional**

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

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

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** 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** 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 $\bf e_i$

$\mathbb R$

The **Scalars $\mathbb R$** are included in

the algebras. every basis **$n$-vector**

squares
to a Real Number.

$e^{\mathbf e_{ij}}$

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.**
$$\mathbf X = \alpha_0 + \alpha_1 \mathbf e_1 + .. + \alpha_i \mathbf e_{12} + .. + \alpha_n \mathbf e_{12..n} $$
**

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

$\mathbb R_2^+ \cong \mathbb C$

The geometric algebra $\mathbb R_2$ of the

**2D Vectors** has the **complex numbers**

as its even subalgebra.

$\mathbb R_3^+ \cong \mathbb H$

The geometric algebra $\mathbb R_3$ of the

**3D Vectors** has the **quaternions**

as its even subalgebra.

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**

In **2D PGA** points **join** $\vee$ into

lines and lines **meet** $\wedge$ into points.

**Rotations** and **translations** are unified,

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

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**

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**.

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.

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.