> typically, vector multiplication is treated as either dot-product or > cross-product (with cross-product only existing in certain numbers of > dimensions, such as 3 and 7, and "sort of" in 2).
This is exactly why I said "vector algebra considered harmful". The cross product is actually a shadow of a more general product called the outer product, which is defined in every dimension and not just for vectors. The dot product is the inner product. The geometric product is a combination of the inner and outer products. The geometric product of two 3d vectors is a quaternion. > for 2D and 3D, vector math works fine. > > and in 3D, vector math is much nicer than 3D trigonometry. > To see what's possible when you don't confine yourself to vector algebra, have a look at A Treatise of Plane Geometry Through Geometric Algebra[1]. Saying that vector algebra works fine to my mind is like saying that programming in assembly works fine. > quaternions also exist, and can be useful, but for many practical tasks > (such as performing rotations) share an ongoing battle with matrix-math, and > in some cases with angle-based systems. > > may as well include quaternions as well though, since as I see it there is > no fundamental conflict between them and vectors. > There's no conflict but if you confine yourself to vector math, you're losing a large chunk of the higher level structure available. Culturally, we've been indoctrinated into thinking that matrix math and vector algebra is the be-all-end-all of spatial computation when it's really the lowest level representation. There are much richer languages out there that deal with hierarchies of subspaces [2]. [1] http://www.lomont.org/Math/GeometricAlgebra/GA%20Treatise%20A%20-%20Calvet.pdf [2] http://faculty.luther.edu/~macdonal/laga/ _______________________________________________ fonc mailing list [email protected] http://vpri.org/mailman/listinfo/fonc
