Computing Discrete Minimal Surfaces Using a Nonlinear Spring Model

摘要

The Plateau's problem-named after Joseph Plateau (1801-1883), who experimented with soap films-is to prove the. existence of a minimal surface with a given boundary. If we take a wire and dip it in soap water, we obtain a soap film with the wire as boundary, and the film forms a minimal surface.rnJesse Douglas1 and Tibor Rad62 independently found the problem's general solutions by around 1931. However, the Plateau's problem persists because, although the minimal surfaces exist for an arbitrary boundary, they're difficult to define and compute. Consider all curves passing through a point on the surface. Each curve has an associated curvature. Intuitively, curvature is the amount by which a geometric object deviates from being flat or straight. A point's mean curvature is the average of all curvatures at the point. A surface is a minimal surface if and only if the mean curvature is zero at any point on the surface. For a surface defined in 3D, the mean curvature is related to the target point's unit normal. The mean curvature is positive if the surface curves away from the normal; it's negative if the surface curves toward the normal. Therefore, our basic idea is to dynamically modify the surface points (vertices) along the mean curvature normals at the points so that the mean curvatures converge to zero.