Decompositions of the Identity and the Construction of Smooth Surfaces

Geometric modelers typically define surfaces as images of closed polygonal regions under polynomial or rational maps, called patches. The images, also called patches, do not overlap but join along curves in lR . Differential topologists define surfaces as domains of invertible maps, also called patches, from lR to open sets in lR . The patches cover the surface by overlapping in open subsets. This paper develops a surface model that reconciles the apparent discrepancy between the constructive and the analytic approach by defining and characterizing maps that link the domains and ranges of the various types of patches. Of particular interest are families of maps whose composition matches the Taylor expansion of the identity map. Such families are named decompositions of the identity and its members roots of the identity. t Department of Computer Science, Purdue University, W-Lafayette IN 47907-1398 Supported by NSF NY! grant CCR-9457806