De Casteljau Algorithm and Degree Elevation of Toric Surface Patches
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De Casteljau Algorithm and Degree Elevation of Toric Surface Patches∗ LI Jinggai · JI Ye · ZHU Chungang
DOI: 10.1007/s11424-020-9370-y Received: 17 September 2019 c The Editorial Office of JSSC & Springer-Verlag GmbH Germany 2020 Abstract De Casteljau algorithm and degree elevation of B´ezier and NURBS curves/surfaces are two important techniques in computer aided geometric design. This paper presents the de Casteljau algorithm and degree elevation of toric surface patches, which include tensor product and triangular rational B´ezier surfaces as special cases. Some representative examples of toric surface patches with common shapes are illustrated to verify these two algorithms. Moreover, the authors also apply the degree elevation of toric surface patches to isogeometric analysis. And two more examples show the effectiveness of proposed method. Keywords De Casteljau algorithm, degree elevation, depth elevation, isogeometric analysis, toric surface patches.
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Introduction
As we know, de Casteljau algorithm and degree elevation of B´ezier and NURBS curves/surfaces are two well-known common techniques in Computer Aided Geometric Design (CAGD)[1, 2] . The de Casteljau algorithm was firstly proposed by de Casteljau in 1959[3] and it is used to evaluate a point on the curve/surface geometrically[4] . The degree elevation was firstly presented by B´ezier in 1972[5] and it is frequently used in geometric design of composite curves, sweeping and skinning surfaces. Nowadays, the B´ezier and NURBS curves/surfaces become the most basic models in CAGD, and their de Casteljau algorithms and degree elevation formulas also become the hot issues in CAGD. In 1994, Trump and Prautzsch[6] presented some fast algorithms to compute any (n + r)th degree B´ezier representation from an nth degree representation. Zhang and Li[7] developed a recursive algorithm for general T-spline degree elevation LI Jinggai · JI Ye · ZHU Chungang (Corresponding author) School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China; Key Laboratory for Computational Mathematics and Data Intelligence of Liaoning Province, Dalian University of Technology, Dalian 116024, China. Email: [email protected]. ∗ This research was supported by the National Natural Science Foundation of China under Grant Nos. 11671068 and 11801053. This paper was recommended for publication by Editor-in-Chief GAO Xiao-Shan.
LI JINGGAI · JI YE · ZHU CHUNGANG
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and provided two optimized algorithms for T-spline degree elevation. In 1998, Casciola, et al.[8] provided a formula for degree elevation of p-B´ezier curves and described a simple and efficient implementation of it. In 2002, Krasauskas[9] defined a kind of rational multisided patch — Toric surface patch. The tensor product B´ezier surfaces and B´ezier triangles are also special cases of the toric surface patch. Garc´ıa-Puente, et al.[10] studied the degenerations of toric surface patches when all weights tend to infinity. In 2019, Yu, et al.[11] studied the total positivity of a kind of generalized toric
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