The Geometry of Projective Blending Surfaces

Abstract

Blending surfaces smoothly join two or more primary surfaces that otherwise would intersect in edges. We outline the potential method for deriving blending surfaces, and explain why the method needs to be considered in projective parameter space, concentrating on the case of blending quadrics. Let $W$ be the quadratic polynomial substituted for the homogenizing variable of parameter space. We show that a blending surface derived in projective parameter space is the projective image of a different blending surface derived in affine parameter space, provided that $W = U^{2}$ for some linear $U$. All blending surfaces may therefore be classified on basis of the projective classification of $W$.