Blending Bicubic Surfaces Using PHT-spline

In the ?eld of geometric modeling of CAD, combining several surfaces is a simpleand e?cient method of constructing complicated models. So that how to construct ablending surface among several surfaces is an important problem in the CAD. Thatis, for given two or several surfaces and those blending borders, how to construct alow degree blending surface which contacts original surfaces as high order continuityas possible. For this problem, many researches present a lot of methods, like methodof rolling ball, method of partial di?erential equation, method of linear combinationand so on. But there are some disadvantages in these method: like method of rollingball can't choose an arbitrary blending border, method of PDE compute complicated,method of linear combination has to parametrize original surfaces according to theblending borders.In order to overcome the above-mentioned disadvantages, this paper presents anew method of constructing blending surface among bicubic surfaces. The methoduses polynomials over hierarchical t-meshes(PHTSpline) to construct blending surface.Polynomials over hierarchical t-meshes is presented ?rstly by J.S. Deng in 2005. InDeng's paper[8][9], he discussed the dimension of polynomials space over hierarchicalt-meshes S(m,n,α,β,T ) when m 2α+1,n 2β+1. And in this spline space, Dengbuilt a set of basis functions which are all positive and have properties of partition ofunit and local supported set.This paper uses one or several surfaces in the polynomials space over hierarchicalt-meshes S(3, 3,1, 1, T ) to construct blending surface among two or several bicubicsurfaces. Firstly, we choose arbitrary blending borders on original surfaces respectively,then approach blending borders with uv polygonal lines on original surfaces in any giventolerance, and take the uv polygonal lines as new blending borders. In this way, not onlyoriginal surfaces needn't re-parameterize according to blending borders, but also theblending borders'degree is reduces to 3. Secondly, we let blending surface interpolatethe geometric information on uv polygonal lines and optimize some physical energy toconstruct the ?nal blending surface. The blending surface constructed by this methodcontacts original surfaces with no gap and G1 continuity at uv polygonal lines. Andthe shape of the blending surface can be changed easily by choosing di?erent energyweight. what's more, the degree of blending surface, constructed by polynomials overhierarchical t-meshes, is always 3 while the number of original surfaces increases. In this paper, we discuss in detail how to construct a blending surface between twobicubic surfaces by one patch of PHTSpline surface, how to construct a blending surfaceamong three bicubic surfaces with intersections by three patches of PHTSpline surfaces,how to construct a blending surface among three bicubic surfaces without intersectionsby six patches of PHTSpline surfaces. In the case of constructing blending surface withthree patches and six patches of PHTSpline surfaces, we present a simple way to makesure the blending surface has G1 continuity along the contact lines among the three orsix patches of PHTSpline surfaces, and also discuss how to deal with restriction of G1continuity at the singular point with degree three and six.