Transfinite Interpolation for Well-Definition in Error Analysis in Solid Modelling

Abstract

An overall approach to the problem of error analysis in the context of solid modelling, analogous to the standard forward/backward error analysis of Numerical Analysis, was described in a recent paper by Hoffmann and Stewart. An important subproblem within this overall approach is the well-definition of the sets specified by inconsistent data. These inconsistencies may come from the use of finite-precision real-number arithmetic, from the use of low-degree curves to approximate boundaries, or from terminating an infinite convergent (subdivision) process after only a finite number of steps.

An earlier paper, by Andersson and the present authors, showed how to resolve this problem of well-definition, in the context of standard trimmed-NURBS representations, by using the Whitney Extension Theorem. In this paper we will show how an analogous approach can be used in the context of trimmed surfaces based on combined-subdivision representations, such as those proposed by Litke, Levin and Schröder.

A further component of the problem of well-definition is ensuring that adjacent patches in a representation do not have extraneous intersections. (Here, ‘extraneous intersections’ refers to intersections, between two patches forming part of the boundary, other than prescribed intersections along a common edge or at a common vertex.) The paper also describes the derivation of a bound for normal vectors that can be used for this purpose. This bound is relevant both in the case of trimmed-NURBS representations, and in the case of combined subdivision with trimming.