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The Conformal Model
Why CGA uses a 4,1 signature, the null basis of origin and infinity, and the conformal embedding of Euclidean points.
Our work uses the variant of geometric algebra known as conformal geometric algebra (CGA), written . The choice of signature is what buys the properties that make it practical for robotics: translations become multiplicative transformations of the same kind as rotations, and spheres and circles become first-class elements of the algebra.
Signature
The vector space underlying CGA is . It extends Euclidean — with basis vectors — by two more basis vectors and , where
This is a pseudo-Euclidean space with a Minkowski signature; it can be read as Euclidean space extended by a Minkowski plane, .
A null basis: origin and infinity
In practice the conformal model is set up by a change of basis that replaces with two null vectors:
These behave as a point at the origin and a point at infinity, and both square to zero. The price is a non-orthogonal basis, with the metric
The two null vectors pair with each other () and are orthogonal to the Euclidean directions. Much of the notation in CGA comes down to keeping track of these two.
Since the underlying space is five-dimensional, the algebra has basis blades, of grades through : a scalar, five vectors, ten bivectors, ten trivectors, five quadvectors and the pseudoscalar.
Embedding Euclidean points
A Euclidean point enters the algebra through the conformal embedding
The embedding is nonlinear: the quadratic term bends flat Euclidean space into a parabolic surface — a null cone — inside the larger space. It is closely analogous to the way a vector in is embedded in with homogeneous coordinates, with the quadratic term as the extra ingredient that makes distances and spheres algebraic.
Conformal points are the building blocks: every other primitive is assembled from them with the outer product, which is the subject of geometric primitives.
Two nullspaces
Primitives in CGA are nullspace representations. A primitive is characterised by the set of Euclidean points that multiply with it to give zero, and there are two versions depending on which product is used:
The inner-product nullspace (IPNS) and outer-product nullspace (OPNS) are dual to each other — related by multiplication with the pseudoscalar. Every primitive can be written either way. We treat the OPNS as the primal representation because it is the more convenient to construct with, which makes the IPNS the dual one.
The practical consequence: when a formula looks unfamiliar, check which nullspace it is written in. Constructions are natural in OPNS (span points with ), while constraints and distances are natural in IPNS (test with ).
Next: geometric primitives, built from conformal points.