插值边界的四边网格离散极小曲面建模方法

如何实现极小曲面的快速三维建模，是几何设计与计算领域中的难点和热点问题.给定一条封闭的边界离散折线，本文研究如何构造以其为边界的四边网格离散极小曲面.首先从曲面的内蕴微分几何度量出发，给出了离散四边网格极小曲面的数学定义；然后利用保长度边界投影、四边网格生成、径向基函数插值映射和非线性优化技术，提出了由给定边界离散折线快速构造离散四边网格极小曲面的一般技术框架.最后通过若干建模实例验证了本文方法的有效性.该方法可实现四边网格极小曲面的高质量建模，在建筑几何领域具有一定的应用价值.

Constructing Discrete Minimal Surfaces with Quadrilateral Meshes from Described Boundary

Efficient modeling of minimal surfaces is a challenging problem and hot topic in the field of geometric design and computation. In this paper, from given boundary closed polylines, we propose a general framework to construct discrete minimal surfaces with quadrilateral meshes. Firstly, the mathematical definition of discerete minimal surface with quadrilateral mesh is given from the intrinsic differential-geometry metric of surfaces. Based on the length-preserving boundary projection method, quad-mesh generation approach and non-linear numerical optimization technique, a novel framework is proposed to construct discrete minimal surfaces with quadrilateral meshes from a described boundary closed discrete polylines. Finally, the effectiveness of the proposed approach is illustrated by several modeling examples. The results show that the proposed method can achieve high-quality modeling of discrete minimal surfaces and have some potential in architecture geometry.