中文摘要：

从极小曲面上平均曲率处处为零出发求解三角域上的Plateau-Bézier问题.首先提出了一种新的线性能量函数,称之为平均曲率平方能量.基于该能量函数的极小化,推导出了内部控制顶点应满足的充要条件.通过造型实例,与基于Dirichlet能量极小化的求解方法进行了比较,发现两者各有千秋.特别地,若给定的边界曲线恰巧为三角域上的调和Bézier曲面的边界曲线,则按照该方法所构造出的曲面便为调和曲面;若给定的边界曲线恰好为等温参数多项式极小曲面的边界曲线,则按照该方法便可重构出该极小曲面.

英文摘要：

This paper solves the Plateau-Bézier problem over triangular domain from the fact that mean curvature of minimal surface equals zero at every point. A new kind of linear energy function called squared mean curvature energy is firstly proposed. From the minimization of the new energy function, it derives the sufficient and necessary condition that inner control points should satisfy. The method is compared with the method based on the minimization of Dirichlet energy through modeling examples. In particular, if the given boundary curves are the boundary curves of the harmonic Bézier surface over triangular domain, then the surface constructed by this method is just the harmonic surface; if the given boundary curves are the boundary curves of the parametric polynomial minimal surface with isothermal parameter, then we can reconstruct the minimal surface by this method.