Removal of Gaps among Compound C 1 Bi-Cubic Parametric B-spline Surfaces

Manipulation of control points is a standard procedure in parametric B-spline surface design. Unfortunately, this tool alone is not sufficient to achieve certain goals such as removal of gaps among compound parametric B-spline surfaces. In this paper, we introduce a new approach for removing gaps but maintaining at least the geometrical smoothness condition. If the underlying knot sequences along the connecting boundaries of two parametric B-spline surfaces are proportional, they can be connected in a G fashion without changing the structure of B-spline representation. Otherwise, they can still be connected in a G fashion, but with one requirement, namely, by sacrificing the true B-spline representation along a boundary strip of one of the two B-spline surfaces. Our method involves manipulation of Bézier coefficients along a boundary strip, and we demonstrate the feasibility of this approach by considering the C cubic setting. This procedure is generalized to connecting up to four B-spline surfaces with a common corner. §0. Prelude This paper represents a joint work with Charles Chui under the sponsorship of an NSF GOALI grant with matching support by McDonnell Douglas Aerospace (now Boeing Company, St. Louis). The research problem was motivated by the need of connecting surface patches that are designed independently. The basic assumption is that the boundary of each surface patch is supposed to be determined by a given finite set of points. However, certain gaps, though invisible to the human eyes, were usually found to exist when these surface patches are connected by using the CAD tools from Unigraphics. The objective of the paper was, therefore, to develop a design criterion to avoid such gaps, while yielding a reasonably smooth Wavelets and Splines: Athens 2005 287 Guanrong Chen and Ming-Jung Lai (eds.), pp. 287–313. Copyright oc 200x by Nashboro Press, Brentwood, TN. ISBN 0-9728482-6-6 All rights of reproduction in any form reserved. 288 Lai, Lian & Cassidy compound surface. We use the G smoothness criterion as the goal in our research. The paper [4], with Charles Chui as a co-author, was completed and submitted in April, 1998, to the Journal of Computer Aided Geometric Design (CAGD) for publication. A revision of the paper was re-submitted in 2003, but was later rejected. To our surprise, however, the same journal, CAGD, has recently published a similar work [12]. Since our paper [4] has not appeared in print, the intention of this writing is to publish its exact content and to point out the similarity with the recently published work [12] in CAGD. In [12], certain necessary and sufficient conditions for two NURBS patches R1(u, v) and R2(ũ, ṽ) to join in the G 1 fashion are given under the assumption that R1(u, 0) = R2(u, 0) (cf. (3) in [12, p.289]). This is a very restrictive assumption. Although the G condition implies that R1(u, 0) and R2(u, 0) have a common boundary curve, namely, R1(u, 0)= R2(w(u), 0) with re-parametrization, say w(u), yet the assumption R1(u, 0) = R2(u, 0) rarely happens in practice, and indeed does not even apply to true NURBS. As mentioned above, two adjacent patches are usually determined only by imposing a finite set of common (interpolating) points on the boundaries. That is, it is more practical to assume R1(u, 0) = R2(ũ, 0), only for some finite discrete values of the parameters u and ũ, in the usual design process. In addition, the weaker condition of common interpolating points (as opposed to a common boundary curve) allows us to extend the study to connecting true NURBS patches, as in our later published work [3]. Our study in [4] is based on the practical setting, and an effective algorithm is developed to modify the spline coefficients near the (common) boundary of only one of two spline surface patches to achieve a G compound patch, without gaps. In this regard, we like to mention that although our condition is only sufficient (but not necessary in general), yet it applies to practical applications and can add to the capability of the Unigraphics CAD tools. In the following, the content of [4] is presented without change. Since this volume is dedicated to Professor Charles K. Chui on the occasion of his 65th birthday, he declines to be a co-author of this paper. We are happy to honor and respect his wish. §