过测地线的调和Bézier曲面设计

Harmonic Bézier Surfaces Passing Given Geodesic

极小曲面和测地线是建筑几何领域中2类重要的几何元素,而调和曲面常作为极小曲面的一种线性近似.文中将测地线和调和曲面结合起来,提出一种过给定测地线的调和Bézier曲面设计方法.首先给出了一种构造调和Bézier曲面的新方法,证明了调和Bézier曲面的形状由第一层和第二层的控制顶点完全决定;然后根据测地线的性质,证明了过给定测地线的调和Bézier曲面的形状由该测地线和第二层的首末2个控制顶点完全决定.最后通过若干实例验证了该方法的有效性.文中方法充分利用了测地线和调和曲面的特殊几何性质,对张拉膜建筑结构的几何设计具有一定的实用价值.

Minimal surfaces and geodesics are two important kinds of elements in the field of architecture geometry, and the harmonic surface is often considered as a linear approximation of minimal surface. In this paper, by the combination of geodesic and harmonic surface, we propose a new method which generates harmonic B6zier surfaces determined by a given geodesic. We originally prove that the control points on the first and second layers of the control mesh are the only factors that determine the shape of a certain harmonic B6zier surface. With this property of harmonic surfaces, we further prove that the shape of harmonic B6zier surface through a given geodesic is entirely determined by the given geodesic and two control points on opposite end of the second layer..At the end of this paper, the effectiveness of the proposed method is illustrated by several examples with data analysis. The proposed construction method is derived from the geometric properties of geodesic and harmonic surfaces, which is valuable for the design of tension membrane structure.