SpringerLink - Journal Article

A novel algorithm for explicit optimal multi-degree reduction of triangular surfaces

Hu QianQian ¹ and Wang GuoJin ¹

(1)	Institute of Computer Images and Graphics, State Key Laboratory of CAD & CG, Zhejiang University, Hangzhou, 310027, China

Received: 14 March 2006 Accepted: 15 May 2007

Abstract This paper introduces the algebraic property of bivariate orthonormal Jacobi polynomials into geometric approximation. Based on the latest results on the transformation formulae between bivariate Bernstein polynomials and Jacobi polynomials, we naturally deduce a novel algorithm for multi-degree reduction of triangular Bézier surfaces. This algorithm possesses four characteristics: ability of error forecast, explicit expression, less time consumption, and best precision. That is, firstly, whether there exists a multi-degree reduced surface within a prescribed tolerance is judged beforehand; secondly, all the operations of multi-degree reduction are just to multiply the column vector generated by sorting the series of the control points of the original surface in lexicographic order by a matrix; thirdly, this matrix can be computed at one time and stored in an array before processing degree reduction; fourthly, the multi-degree reduced surface achieves an optimal approximation in the norm L ₂. Some numerical experiments are presented to validate the effectiveness of this algorithm, and to show that the algorithm is applicable to information processing of products in CAD system.

Keywords computer aided design - data compression - triangular Bézier surface - multi-degree reduction - Bernstein polynomial - Jacobi polynomial - L ₂ norm

Supported by the National Grand Fundamental Research 973 Program of China (Grant No. 2004CB719400), the National Natural Science Foundation of China (Grant Nos. 60673031 and 60333010) and the National Natural Science Foundation for Innovative Research Groups (Grant No. 60021201)

Wang GuoJin
Email: wanggj@zju.edu.cn

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