Constructing Minimal Triangular Mesh Based on Discrete Mean Curvature

In this paper, a so-called Plateau-Mesh problem is proposed, that is, to find a triangular mesh with the boundary formed by a given spacial closed polygon, whose area is minimum among all triangular meshes with the same boundary. To solve this problem, the former work by minimizing a function describing the mesh area directly, cannot obtain the global minimum of the function, only obtain its local minimum. In order to overcome this shortcoming, a new method to minimize the objective function which is measured by discrete mean curvatures is presented. As a base of the algorithm, the partial derivatives of discrete mean curvatures of the triangular mesh are evaluated. Numerical examples and error analysis are also given and the results show that our algorithm is correct and effective. Keywords-Plateau-Mesh problem; triangular mesh; least squares; discrete mean curvature; minimal surface