极小曲面的Weierstrass表示与建筑造型

极小曲面广泛存在于自然界中，是平均曲率处处为0的一类重要曲面。其上含有多种相容的数学结构，可用不同的数学观点、方法来处理。局部上极小曲面定义为面积泛函的临界点，用变分方程或等价的欧拉—拉格朗日方程(二阶椭圆偏微分方程)表达。可用微分方程求其解析介或近似介，也可用变分方程求其近似介。极小映射是两个黎曼流形间的特殊映射，Weierstrass抛弃面积概念，从另外一个角度给出了极小曲面方程的通解。采用极小曲面Weierstrass表示，借助于计算机绘图技术可以获得各种精美的极小曲面图形。极小曲面在拓扑上可以有随心所欲的复杂，在几何上可以有令人难以琢磨的对称。这些图形在银屏上未显示前，大多无法事先想象出来。作为应用本文绘制了实射影平面在三维欧氏空间的最佳浸入Boy’s曲面的图形。还讨论了几种用极小曲面或调和曲面造型的建筑。

Weierstrass Representation for Minimal Surface and Architectual Modeling

Mininml surfaces are widely exist in nature which is an important type of surface with zero mean curva- ture. Different methods can be adopted to solve because of its many consistent mathematical structures. Minimal surface can be defined as the critical point of the area functional. Therefore it can be expressed by the variational equations or the equivalent Lagrangian-Eulerian equations. The latter is also known as second order elliptic equations. Then the ana- lytical or approximate solutions can be obtained through differential equations and approximate solutions through varia- tional equations. Minimal mapping is a special mapping between two Riemannian manifolds, however Weierstrass presen- ted a general solution of the equation of minimal surfaces from another point of view without area concept. With Weier- strass representation for minimal surface, various kinds of excellent minimal graphes can appear on the computer, which are hard to image before they model because of the complexity in the topology and the elusive symmetry on the geometry. Best immersion Boy＇s surface graphes in 3D Euclindean space are drawn, Moreover, several kinds of architectual model- ing with minimal surfaces or harmonic surfaces are discussed.