Qualitative and quantitative comparisons of B-spline offset surface approximation methods

Surface offset is one of the most useful operations in Computer Aided Geometric Design (CAGD). However, an implementation of this operator is not trivial primarily because the offset surface, in general, does not have the same representation as the original surface. Hence, it is dif®cult or impossible to represent an exact offset in a system with limited surface forms. For this reason, some CAGD surface offset operators produce results that are, at times, unsatisfactory. In this article, we discuss surface offset approximation methods in B-spline environment: both the original surfaces and their offsets are B-spline surfaces. This article summarizes research contained in the ®rst named author's PhD thesis. The ®rst step in this research was to conduct a survey of known curve and surface offset methods. A set of surface offset methods was compiled. (These methods will be discussed in detail in Section 4.) Some previously developed methods that were just for planar curves or just for cubic curves were extended to surfaces. Some of the studied methods preserve the original smoothness of the base surface, an important criterion in exterior surface modeling, and some do not. Below is a list of the surface offset methods studied in this paper. 1. Based on Hoschek's method for ®tting a cubic Be Âzier curve [6]. 2. Based on Farouki's method for bicubic polynomial representation for a surface patch in terms of bicubic polynomials [5], extended to biquintic surfaces for this research. 3. Two variations of least squares methods. 4. Based on the Tiller±Hanson method for planar curves [9]. This is an implementation of a surface offset method based on offsetting a surface control frame. This method was introduced by Nachman [8]. We considered an extension of Coquilart method [2] (called the Cobb method in Ref. [7]), but found it to be prohibitively expensive in terms of computation. Next, a set of test surfaces and a list of surface offset evaluation criteria were developed. Our major criterion was: how many surface control points are required to achieve the following offset tolerances: 0.1, 0.01, 0.001, 0.0001; for the following general surfaces: surface consisting of elliptic points, surface with some hyperbolic points, surface with boundary curves with large curvature; for the following surface degrees: quadratic, cubic, quintic; and for the following offset distances: 0.5, 1, 1.5, 2, 2.5? During the tests a required offset tolerance was achieved by adding new control points to an original …