Clifford algebra
From CGAFaq
Contents |
Introduction
An abstract algebra is one or more sets of objects along with operations satisfying axioms that depend on the kind of algebra. The most familiar example in 3D graphics is the vector space, consisting of scalars (real numbers), vectors (such as translations), and — along with the usual operations on the real numbers — the operations of summing vectors and of multiplying vectors by scalars. A Clifford algebra is a higher-level abstract algebra built on a vector space equipped with a quadratic form (a quadratic space). In many ways Clifford algebra is simpler than vector algebra, because all objects live together, with only one kind of multiplication and one kind of addition. Like vector spaces, Clifford algebras have a dimension; unlike vector spaces, they also have a signature derived from the quadratic form. Certain low-dimensional Clifford algebras are equivalent to ℝ (the real numbers), ℂ (the complex numbers), and ℍ (the quaternions), which hints at their scope. Some computer graphics researchers have begun to employ the technology of Clifford algebra as a substitute for vector algebra and matrices in describing and manipulating geometry; this effort is called geometric algebra.
Quadratic forms
To fully understand Clifford algebras one has to start from quadratic forms. An n-dimensional quadratic form is a homogeneous polynomial of degree 2, which can be thought as a function from an n-dimensional space to the field. Now via a linear change of variables we can express any quadratic form in the diagonal representation:
The scalars
are unique up to multiplication by the square. In short we denote such a form
. Over the reals every number is either a square or a minus square, hence any quadratic form can be expressed as
.
Now given a quadratic form q on the n-dimensional space V, the Clifford algebra Cℓ(q) associated with q is the algebra of dimension 2n containing V and such that for every vector the square of v (in terms of the multiplication in Cℓ(q)) equals q(v) (times the unit of Cℓ(q)).
Over the reals every non-singular quadratic form has a diagonalization . When there are n plus ones and m minus ones we denote the associated Clifford algebra by Cℓn,m. In case
, we shorten the notation to Cℓn := Cℓn,0. (This is not the only convention in use).
In particular such well known structures as complex numbers and quaternions are Clifford algebras. Hence, Clifford algebras can be viewed as the generalization of those.
Brauer-Wall group and 8-periodicity
Beside the examples mentioned above, there is one more which is extremely simple — the algebra of square matrices (of the given dimension) is also a Clifford algebra. This is the trivial Clifford algebra.
Cancelling out those trivial parts from Clifford algebras allows us to consider classes of Clifford algebras. Effectively we embed them into a so called Brauer-Wall group of classes of central simple -graded algebras. The matrix algebras act as the unit element of the Brauer-Wall group.
But sophisticated mathematical theory aside, the notion of the Brauer-Wall group allows us to effectively write down all the classes of Clifford algebras over the reals. There are only eight such classes. The following facts hold:
- for any k,n the algebras Cℓk,n,Cℓk+8,n,Cℓk,n+8 belong to the same class;
- for any k,n the algebra Cℓk,n belongs to the same class as either Cℓk−n,0 (if k > n) or Cℓ0,n−k (if k < n);
- all eight classes of Clifford algebras over the reals are as follows:
Structure of a Clifford algebra
Any Clifford algebra is
-graded in the sense that it is the direct sum
of two subspaces of dimension
. In fact
is an algebra, called even Clifford algebra. For any two elements
their product
lays in
iff both come from the same subspace and in
if they come from different subspaces.
Clifford algebra of R3
From the point of view of computer graphics, the most important case is the Clifford algebra of equipped with the dot product, i.e. the algebra
. It is an eight dimensional algebra containing the following four classes of elements: scalars, vectors, bivectors and 3-vectors (aka volume elements). These classes of elements need a bit of explanation for a novice.
[TODO] Scalars and bivectors make up the even Clifford subalgebra . It is isomorphic to the quaternion algebra. This becomes even more clear if we revert to 8-periodicity, from there we see that
is a direct sum of two copies of the quaternion algebra.
As it was already mentioned, has dimension 8. If we take a basis
of
, then the basis of
is:
.
For short we denote and likewise
.
Observe that the spaces of vectors and bivectors are both of dimension three. Hence their are isomorphic. This isomorphism is the unique property of that makes a cross product work.
What Clifford algebras are good for?
[TODO: Example usages: describing rigid body motion in space, inverse kinematic, connection with O(3) and SO(3), spinnors, rotors, twistors]
External links
- An introduction to Clifford algebras by Alexander Hahn (PDF)
- Another introduction to Clifford algebras, this time by Andrzej Trautman (PDF)
- Geometric Algebra: New Foundations, New Insights, SIGGRAPH 2001 Course (PDF)
- Many counter examples to some common errors about Clifford algebras
- Extensive list of researchers working on Clifford algebras (on their home pages one can find plenty of advanced stuff)
- A text-based program for performing calculations in Clifford algebras
- A visualisation toool for Clifford algebras
- Another visualiser for Clifford algebras
