**comp.graphics.algorithms**

## Subject: **Re: General flat-flat intersection**

On Mon, 24 Apr 2006 15:48:09 +0300, Kaba

>1) What is "Geometric algebra" and how it relates to Clifford algebra

>and Grassmann algebra?

A Clifford algebra is like a group, a diverse class of algebraic

structures. Vector spaces come equipped with scalar multiplication,

but only vector addition. A real inner product space is a real vector

space with an inner product, like the dot product, for which the

product of two vectors is a scalar. A quadratic space is almost the

same as an inner product space, but is equipped with a quadratic form,

like the squared length of a vector, instead of an inner product. We

can always make an inner product from a positive definite quadratic

form (signature all positive). Given a quadratic space, we can make a

Clifford algebra by extending the quadratic form (of any signature) to

define a linear and associative (but not commutative) product that

encompasses scalars and vectors and higher grade elements such as

bivectors.

This doesn't read much like geometry does it? As with matrices or real

numbers, the geometry comes in how we use the algebraic technology. So

Geometric Algebra, as discussed in computer graphics, is an assortment

of ways to use Clifford algebras to represent and manipulate geometric

objects and operations. As with matrices and homogeneity, we can add

extra dimensions to incorporate extra abilities.

Grassmann algebra is, on its face, an algebra of linear subspaces of a

vector space. It, too, is a graded algebra with a product, the famous

wedge product. We can find Grassmann algebra within Clifford algebra,

if we like. Of course, we can Clifford algebras with matrices, if we

like; or vice versa.

>2) I am also reading on tensors at the moment. I have a feeling tensors

>are somehow connected to the Clifford algebra. Is there a connection and

>if there is, what is it?

A real tensor (for a mathematician) is merely a multilinear function.

The inputs are taken from either a fixed vector space or its dual. It

has a rank, which is the number of inputs. (Be careful, the rank of a

matrix is something different.) We can represent a rank 2 tensor for

an N-dimensional vector space (and dual) as an NxN matrix. Don't get

too excited; by hook or crook, we can represent almost anything as a

matrix.

Maybe a helpful view is that with enough linear maps, a high enough

dimension, and a little ingenuity we can represent any of these things

within any of the others. There's no general hierarchy or superiority,

despite the claims of proponents. One application may be easy to think

about and efficient to represent with matrices or tensors, another may

have an elegant and compact form in geometric algebra.

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