王相海, 黄俊英, 李 明. GC\+1约束的多三角Bézier曲面混合降阶逼近研究[J]. 计算机研究与发展, 2013, 50(5): 1012-1020.
 引用本文: 王相海, 黄俊英, 李 明. GC\+1约束的多三角Bézier曲面混合降阶逼近研究[J]. 计算机研究与发展, 2013, 50(5): 1012-1020.
Wang Xianghai, Huang Junying, Li Ming. Approximate Degree Reduction Method by Blending of Multi-Triangular Bézier Surfaces with GC\+1 Constraint[J]. Journal of Computer Research and Development, 2013, 50(5): 1012-1020.
 Citation: Wang Xianghai, Huang Junying, Li Ming. Approximate Degree Reduction Method by Blending of Multi-Triangular Bézier Surfaces with GC\+1 Constraint[J]. Journal of Computer Research and Development, 2013, 50(5): 1012-1020.

## Approximate Degree Reduction Method by Blending of Multi-Triangular Bézier Surfaces with GC\+1 Constraint

• 摘要: 三角曲面的降阶问题一直是CAGD领域的一个难点问题，近年来受到关注.对L2范数下多三角Bézier曲面在拼接边界满足GC\+1约束的降阶逼近问题进行研究，包括:1)给出了一种L2范数下单一三角Bézier曲面的一次降多阶的逼近算法;2)对两个三角Bézier曲面在拼接边界上满足GC\+1约束的降阶逼近算法进行研究，提出一种通过调整两个三角Bézier曲面片距离拼接边界的第2排内部控制点来满足GC\+1约束的降阶逼近算法;3)研究基于调整三角Bézier曲面片内部控制点的多三角曲面片在各拼接边界满足GC\+1约束的曲面降阶算法.算法首先按照2)中的方法，确定每两个三角Bézier曲面片在公共边界满足GC\+1约束的降阶逼近所需要调整的内部控制点，然后构造blending函数.通过将每个三角Bézier曲面所对应的多组控制点进行混合，形成新的混合降阶曲面的三角Bézier格式，并在理论上证明该混合三角Bézier降阶曲面片与其周边的各降阶曲面片仍保持GC\+1约束.实验结果表明，所提方法简单实用，逼近效果好.

Abstract: Recently, the problem of the approximate degree reduction for triangular surface attracts much attention,and is always a hotspot in the field of CAGD. This paper investigates the approximate multi-degree reduction of triangular Bézier surface by minimizing the defined distance function with GC\+1 constraint on boundary, which includes the following: 1) A kind of algorithm for the degree reduction of triangular Bézier surface is given by minimizing the defined distance function; 2) The approximate degree reduction problem of two triangular Bézier surfaces with GC\+1 constraint is studied, an algorithm of the degree reduction of two surfaces with GC\+1 constraint is proposed by adjusting the second line control vertices nearby the boundary and; 3) The approximate degree reduction of multi-triangular Bézier surface with GC\+1 constraint is studied by adjusting the internal control points. Firstly, we confirm some groups of internal control points after each two triangular Bézier surfaces approximate degree reduction with GC\+1 constraint, and then structure blending function and constructing a new blending format for approximating multi-degree reduction surface. Finally, It is proved in theory that the new triangular Bézier surface and its surrounding surfaces still keep GC\+1 constraint. The simulation results prove that the proposed algorithm is practical and efficient.

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