几何偏微分方程和离散曲面设计

摘要

We construct discrete three- and four-sided surface patches with specified C0 or C1 boundary conditions, using several geometric intrinsic curvature driven flows. These flow equations are solved numerically based on discretizations of the involved differential-geometry operators, which are derived from parametric approximations. The constructed surface patches satisfy certain geometric partial differential equations, and therefore have desirable shape. These patches are assembled together for constructing complicated geometric models for shape design. Multi-resolution representations of the models are achieved using repeated subdivision and evolution.%使用若干个几何本质的曲率驱动的偏微分方程来构造符合指定C0或C1边界条件的三边曲面片和四边曲面片,这些方程的数值解由所涉及的微分几何算子的离散化来得到,微分几何算子的离散化则源于参数逼近.所构造的曲面片满足某些特定的几何偏微分方程,故具有理想的形状,将这些曲面片组装起来便构造出复杂的几何模型.通过反复的子分和演化,得到几何模型的多尺度表示.