AN ALGEBRAIC APPROACH TO CURVES AND SURFACES ON THE SPHERE AND ON OTHER QUADRICS

An explicit representation for any irreducible rational Bezier curve a nd Bezier surface patch on the unit sphere is given. The extension to general quadrics (ellipsoids, hyperboloids, paraboloids) is outlined. The construction is based on an algebraic result concerning Pythagorea n quadruples in polynomial rings and can be additionally interpreted a s a generalized stereographic projection onto the sphere. This project ion is shown to be the composition of a hyperbolic projection (a speci al net projection) with a stereographic projection. The investigation of its properties leads to new results for the biquadratic Bezier patc h on the sphere. Further attention is payed to the interpolation of a given point set with a spherical rational curve. The results are exten ded to rational B-spline curves and tensor product B-spline surfaces.