Graphical Models and Image Processing Generalized Subdivision of Bé Zier Surfaces
Graphical Models and Image Processing Generalized Subdivision of Bé Zier Surfaces
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通讯作者：
Shi Hu;Guozhao Wang;T. Jin
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作者：
Shi Hu;Guozhao Wang;T. Jin
2. MAIN RESULTS In this paper, subdivision methods for rectangular Bézier A rectangular Bé zier surface of degree n ϫ m can be surfaces are generalized to subdivide a rectangular Bézier surface patch of degree n ؋ m into two rectangular Bézier sur-represented by face patches of degree n ؋ (m ؉ n), while the parameter domain of the Bézier surface is decomposed into two trapezoids. As an application, a conversion from rectangular Bézier sur-P(u, v) ϭ n iϭ0 m jϭ0 P ij B n i (u) B m j (v), 0 Յ u, v Յ 1, faces to triangular Bézier surfaces is presented. where B n i (u) ϭ (n i)u i (1 Ϫ u) nϪi are univariate Bernstein polynomials of degree n, and P ij (0 Յ i Յ n, 0 Յ j Յ m) are control points of P(u, v). Without loss of generality, 1. INTRODUCTION we consider the trimmed surface patch defined on the The Bé zier surface is a very useful tool in surface model-domain D 1 (see Fig. 1). ing [1, 2]. Subdivision algorithms for Bé zier surfaces are First of all, we introduce some operator symbols for a very important in rendering and curve–curve, curve– Bé zier curve P(t) with control points P i (0 Յ i Յ n). surface, and surface–surface intersection calculations 1. The invariant operator I: IP i ϭ P i , [3, 4]. It is well known that, subdivision algorithms for rectangular Bé zier surfaces, which decompose a rec-2. The shifting operator E: EP i ϭ P iϩ1 , tangular patch into two rectangular patches of the same 3. The difference operator ⌬: ⌬P i ϭ (E Ϫ I) P i ϭ degree, splitting its parametric domain into two rec-P iϩ1 Ϫ P i , tangles, are based on the subdivision of Bé zier curves. If we split the parametric domain of a rectangular 4. The degree elevation operator A n : Bé zier surface into two trapezoids, the surface is again decomposed into two surface patches. Can these two trimmed surface patches be represented as rectangular Bé zier surface patches? How do we obtain the new control points from those of the original surface patch? These questions are considered in the second section. By using parameter transformations, we show that the control points of these two rectangular Bé zier patches can be obtained from those of …
