Conformal Geometric Algebra

Conformal geometric algebra (CGA) is an elegant projective model of points, lines, circles, planes, spheres, and other objects in $\mathbb{R}^n$ using blades in a geometric algebra over $\mathbb{R}^{n + 2}$.

This package includes a small submodule which defines CGA and a few important operations, including standardform for classifying blades and the functions ipns and opns for obtaining the geometric forms represented by a blade.

You can obtain the standard basis blades for $n$-D CGA using the CGA{n} metric signature.

julia> using GeometricAlgebra.Conformal
julia> basis(CGA{3})5-element Vector{BasisBlade{GeometricAlgebra.Conformal.CGA{3}, 1, Int64}}:
 1 v1
 1 v2
 1 v3
 1 vp
 1 vm

Notice that CGA includes two extra basis vectors: vp and vm squaring to $+1$ and $-1$, respectively. More generally, any metric signature Sig (defining the base space) has a conformalisation CGA{Sig} with two extra dimensions.

The representation of points in $\mathbb{R}^n$

In CGA, points $p \in \mathbb{R}^n$ are represented by null vectors (squaring to zero) in $\mathbb{R}^{n + 2}$ given by the map

\[\begin{align*} \operatorname{up} &: \mathbb{R}^n \to \mathbb{R}^{n + 2} \\ \operatorname{up}(p) &= n_0 + p + \frac12 p^2 n_\infty \end{align*}\]

where $n_0$ and $n_\infty$ are null basis vector satisfying $n_0^2 = n_\infty^2 = 0$ and $n_0 \cdot n_\infty = -1$. In terms of standard basis vectors, they are defined as n0 = (vm - vp)/2 and noo = vm + vp and can be obtained with nullbasis, origin and infinity.

julia> n0, noo = Conformal.nullbasis(3);

Use the up function to lift $1$-vectors from a base space with signature Sig into CGA{Sig}:

julia> @basis 3[ Info: Defined basis blades v1, v2, v3, v12, v13, v23, v123, I in Main
julia> p = v1 + 2v2 + 3v33-component Multivector{3, 1, SVector{3, Int64}}:
 1 v1
 2 v2
 3 v3
julia> up(p)5-component Multivector{GeometricAlgebra.Conformal.CGA{3}, 1, SVector{5, Float64}}:
 1.0 v1
 2.0 v2
 3.0 v3
 6.5 vp
 7.5 vm
julia> ans == n0 + p + 2\p^2*noo # multivectors in Sig and CGA{Sig} are promotedtrue

The overall scale of vectors (and blades) in CGA does not matter; it is a homogeneous model. We can go the other way with the down map, dn.

julia> p == dn(up(p)) == dn(100up(p))true

You may notice that $n_0$ itself is equal to $\operatorname{up}(0)$ and $n_\infty$ is the limit of $\operatorname{up}(x)/\|x\|^2$ as $x$ goes to infinity in any direction. We interpret $n_0$ as the origin and $n_∞$ as the unique point at infinity.

Standard form of CGA blades

It is useful to represent blades in CGA{Sig} in purely terms of $n_0$, $n_∞$ and objects in the base space Sig.

In fact, any blade $X$ in CGA{Sig} is of exactly one of the four forms represented by the subtypes of CGABlade.

Here, $\bm{E}$ is any blade in the base space, $\mathtt{T}_{\bm{p}}$ is the operator which translates by the base space vector $\bm{p}$ and $r$ is a scalar radius parameter.

The standardform method converts any blade in CGA{Sig} to one of these representations.

julia> standardform(v12 + 2v1∧noo)GeometricAlgebra.Conformal.DualFlatBlade{3, 2}:
 2-blade of the form translate(p, E):
  E = 1.0 v12
  p = 2.0 v2

This example shows how $𝐯_1𝐯_2 + 2𝐯_2∧n_∞$ can be written as $𝚃_{2𝐯_2}[𝐯_1𝐯_2]$.

Geometric objects represented by CGA blades

Any CGA blade $X$ can be associated with a subset of the base space $\mathbb{R}^n$ in two ways related by duality, providing clear geometrical meaning. Define the inner and outer product null spaces^[1], obtained with ipns and opns:

\[\begin{align*} \operatorname{ipns}(X) &\coloneqq \{p \in \mathbb{R}^n \cup \{∞\} \mid \operatorname{up}(p) \mathop\rfloor X = 0 \} \\ \operatorname{opns}(X) &\coloneqq \{p \in \mathbb{R}^n \cup \{∞\} \mid \operatorname{up}(p) \wedge X = 0 \} \end{align*}\]

These are related by duality so that ipns(x) == opns(hodgedual(x)) and ipns(hodgedual(x)) == opns(x).

Possible values of $\operatorname{ipns}$ or $\operatorname{opns}$ are the following subsets of the extended base space $\mathbb{R}^n \cup \{∞\}$:

• the empty set,
• an affine $k$-plane (point, line, plane, and so on) which contains $∞$,
• a $k$-sphere (point, point pair, circle, sphere, and so on),
• the point at infinity.

Geometric objects of these kinds are represented by the FlatGeometry and RoundGeometry subtypes of CGAGeometry.

For example, below we find that $\operatorname{opns}$ of the outer product of three conformal points is a circle with a centre, direction and real radius.

julia> p, q, r = up.(randn(Multivector{3,1}, 3));
julia> opns(p∧q∧r)GeometricAlgebra.Conformal.RoundGeometry{3, 2}:
 2-round (circle) around center p spanning E with square radius r2:
  p  = 0.822903 v1 + -4.67489 v2 + 0.651362 v3
  E  = -0.451375 v12 + -0.479854 v13 + -2.03864 v23
  r2 = 20.339048195883557