离散Koenigs网和离散等温面
Discrete Koenigs nets and discrete isothermic surfaces
摘要：
我们讨论离散的柯尼希斯网（共轭网与相等的拉普拉斯不变量）和等温表面。我们的离散化是基于对偶四边形的概念：两个平面四边形被称为对偶，如果它们的对应边是平行的，并且它们的非对应对角线是平行的。离散柯尼希斯网定义为平面四边形允许对偶网的网。离散柯尼希斯网的几个新的几何性质被发现;特别是，二维离散柯尼希斯网可以通过与任意顶点相邻的基本四边形的对角线的交点的共面性来表征;这种表征相对于射影变换是不变的。离散等温网被定义为圆形柯尼希斯网。这是一个新的几何特征的离散等温表面先前介绍的圆形网与因式分解的交叉比。
英文摘要：
We discuss discretization of Koenigs nets (conjugate nets with equal Laplace invariants) and of isothermic surfaces. Our discretization is based on the notion of dual quadrilaterals: two planar quadrilaterals are called dual, if their corresponding sides are parallel, and their non-corresponding diagonals are parallel. Discrete Koenigs nets are defined as nets with planar quadrilaterals admitting dual nets. Several novel geometric properties of discrete Koenigs nets are found; in particular, two-dimensional discrete Koenigs nets can be characterized by co-planarity of the intersection points of diagonals of elementary quadrilaterals adjacent to any vertex; this characterization is invariant with respect to projective transformations. Discrete isothermic nets are defined as circular Koenigs nets. This is a new geometric characterization of discrete isothermic surfaces introduced previously as circular nets with factorized cross-ratios.
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