Constrained polynomial approximation of quadric surfaces

Abstract

Quadric surfaces are one of the geometric elements most commonly used for shape expression and mechanical accessory cartography in three dimensions. Since quadric surfaces cannot be represented exactly using polynomials, it is not suitable to deal with most problems encountered in engineering application. For this reason, we present a new method for polynomial approximation of quadric surfaces. The approximation tensor product surface obtained by the proposed method has no limit on its degree, i.e., arbitrary degree. It has C(k,h) (k,h=0,1,2) continuity at four corners with the original quadric surface. Meanwhile, the method has an optimal approximation result in the L2-norm, and it can also be applied to piecewise continuous quadric surface patches or a quadric surface patch combined with surface subdivision scheme. And the resulting piecewise approximation patches are globally C0 continuous. Finally, error estimation for the approximation method is given and numerical examples are presented to validate the feasibility and effectiveness of our method.