目前，虚拟现实进入了一个高速发展的时期，而数字几何处理作为实现虚拟现实技术体系的关键内容，发挥着越来越重要的作用。本项目重点研究数字几何处理中的核心课题，即曲面上带严格面积约束且分布尽可能均匀的采样。它满足：(1)整个区域被划分成若干块，每个种子点统治一块子区域且与其质心重合，(2)每个种子点统治的子区域满足给定的面积约束。可见，这样的采样分布不仅面积可控，还能保证分布上的相对均匀性。de Geos等人[2012]发现带面积约束的分布问题可通过最优传输理论进行诠释，并给出了一阶收敛的优化算法。在SIGGRAPH ASIA 2016大会上，申请人及其合作者把该问题命名为质心Power图问题，并提出了超线性收敛的优化算法。然而，已有算法仅适用于二维的欧氏空间。本项目拟研究如何在网格曲面上快速计算质心Power图, 并基于测地质心Power图在面积控制上的优点，探讨它在数字几何处理中的重要应用。

Nowadays virtual reality develops rapidly. Digital geometry processing, as the key to construct the technological system of virtual reality, is playing an increasingly important role. In this proposal, the applicant focuses on the hard-core subject, i.e., seeking for an as-uniform-as-possible point distribution with area constraints on curved surfaces. It requires (1) all the sites form a decomposition of the whole domain and each site coincides with the centroid of its dominated cell, and (2) each cell meets the given area constraint. So such a geometric tool is able to easily support area requirements and guarantee the uniformity of point distribution. de Geos et al. found that this problem can be formulated by optimal mass transport (OMT) and they further gave an algorithm with one-order convergence. In the recent SIGGRAPH ASIA 2016, the applicant and his co-authors named this problem as centroidal power diagram and proposed a super-linearly convergent algorithm. However, these existing algorithms are only applicable to the 2D Euclidean space. This proposal focuses on how to compute the centroidal power diagram on polygonal meshes. Furthermore, the applicant will explore its applications in digital geometry processing based on the area-controllable property.