The Vector
The Vector is an oriented, one dimensional
quantity. Two $\parallel$ Vectors multiply to a Scalar ($\mathbb R$).
Two $\perp$ vectors anti-commute ($e_1e_2=-e_2e_1$)
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 $\bf e_i$
The Scalars $\mathbb R$ are included in
the algebras. every basis $n$-vector
squares
to a Real Number.
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.
The geometric algebra $\mathbb R_2$ of the
2D Vectors has the complex numbers
as its even subalgebra.
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.