Envelope Surfaces

We construct a class of envelope surfaces in Rd, more precisely envelopes of balls. An envelope surface is a closed C1 (tangent continuous) manifold wrapping tightly around the union of a set of balls. Such a manifold is useful in modeling since the union of a finite set of balls can approximate any closed smooth manifold arbitrarily close. The theory of envelope surfaces generalizes the theoretical framework of skin surfaces earlier developed for molecular modeling. However, envelope surfaces are more flexible: where a skin surface is controlled by a single parameter, envelope surfaces can be adapted locally. We show that a special subset of envelope surfaces is piecewise quadratic and derive conditions under which the envelope surface is C1. These conditions can be verified automatically. We give examples of envelope surfaces to demonstrate their flexibility in surface design.