计算机辅助几何设计中极小曲面造型的研究进展

RESEARCH DEVELOPMENT OF MINIMAL SURFACE MODELING IN COMPUTER AIDED GEOMETRIC DESIGN

极小曲面在工程领域有着广泛应用,因此将其引入计算机辅助几何设计领域具有重要意义.详细概述了近年来计算机辅助几何设计领域中极小曲面造型的研究工作,按照造型方法的不同,可将现有工作分为精确造型方法和逼近造型方法两类.精确造型方法主要包括两个部分:某些特殊极小曲面的控制网格表示与构造;等温参数多项式极小曲面的挖掘与性质.逼近造型方法主要包括3个部分:基于数值计算的逼近方法;基于线性偏微分方程的逼近方法;基于能量函数最优化的逼近方法.最后对这些方法进行了分析比较,并讨论了极小曲面造型中有待进一步解决的问题.

Minimal surfaces have been employed in many areas, It is valuable to introduce minimal surfaces into the field of computer aided geometric design. In this paper, the research on modeling of minimal surfaces in CAGD is made a comprehensive survey. According to the modeling methods, the existing work can be clarified into two parts： exact modeling of minimal surfaces and approximating modeling of minimal surface. The exact modedling methods include： representation and construction of control mesh of some special minimal surface; exploration and properties of polynomial parametric minimal surfaces with isothermal parameter. The approximating modeling methods include： approximating methods based on numerical calculations; approximating methods based on linear PDE; approximating methods based on optimization of linear energy function. At last, we analyze and compare the modeling methods, and introduce the research directions in the future.