The ambient spaces of computer graphics and geometric modeling

Four types of ambient mathematical spaces underlie the algebra and geometry of computer graphics and geometric modeling: vector spaces, affine spaces, projective spaces, and Grassmann spaces (H.G. Grassmann, 1894-1911). The author considers at length the theoretical advantages of the coordinate-free approach to understanding geometry. He focuses attention on the operations of addition, subtraction, and scalar multiplication because these are the operations best suited for the construction of freeform curves and surfaces. But there are also spaces with multiplicative structures: structures already coming into vogue in physics (D. Hestenes, 1992) and perhaps of use as well in the fields of computer graphics and computer aided design.