Abstract: Today's computer graphics is ostensibly based upon insights from projective geometry and computations on homogeneous coordinates. Paradoxically, however, projective spaces and homogeneous coordinates are incompatible with much of the algebra and a good deal of the geometry currently in actual use in computer graphics. To bridge this gulf between theory and practice, Grassmann spaces are proposed here as an alternative to projective spaces. We establish that unlike projective spaces, Grassmann spaces do support all the algebra and geometry needed for contemporary computer graphics. We then go on to explain how to exploit this algebra and geometry for a variety of applications, both old and new, including the graphics pipeline, shading algorithms, texture maps, and overcrown surfaces.