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Differential Geometry of Triangulated Surfaces

When defining curvature and other quantities intrinsic to a surface over a triangulated mesh, there are a variety of ways to proceed. A triangulated surface is composed of planar triangles or more generally, planar polygons, so that a given face of the surface has a constant normal vector and zero curvature except at its boundaries, where the normal vector may change direction from face to face.

For an implicitly defined surface f ( x ) = 0 , the function is constant over the surface so that derivatives lying in the tangent plane are zero. In other words the gradient is orthogonal to any tangential direction and represents a choice of a normal direction and, equivalently, a choice of orientation for the surface. The normal vector is given by

$\displaystyle \hat{{n}}$ = $\displaystyle {\frac{{\nabla f}}{{\vert\nabla f\vert}}}$ . ( 46 )

Nonzero curvature then implies that the tangent space changes from point to point or, equivalently, that the normal vector's direction is changing. The mean curvature can be written as H = ½ ∇ . $ \hat{{n}}$ , which is of course nonzero only at the edges of a triangulated surface where the divergence of $ \hat{{n}}$ is infinite and the mean curvature undefined. The integral of mean curvature along an edge or over a surface is perfectly well-defined, and the square of this quantity can be used in the numerical study of minimal surfaces[ 50 , 26 ]. However, since a surface with zero mean curvature defined in this manner is not necessarily a critical point for the area functional[ 51 ], a different approach that does not neglect this key property of minimal surfaces will be used here.

In this work, H will be approximated as a quantity defined at the vertices of a triangulation[ 52 ]. This most commonly-used approximation to H is calculated in the Surface Evolver using the force at a vertex due to surface tension, γ [ 51 ]. If γ is taken to be 1 , the force due to surface tension, F γ , at vertex i is the negative of the gradient of the area as vertex i is displaced. By the Lapace-Young equation (Equation  1.1 ), the discrete mean curvature h at vertex i can be written as

h i = ½ $\displaystyle {\frac{{F_\gamma}}{{\tilde{A}_i}}}$ , ( 47 )

$ \tilde{{A}}_{i}^{}$ being the area associated with vertex i . This area $ \tilde{{A}}_{i}^{}$ is proportional to the sum of facet areas A j over facets j containing the vertex i

$\displaystyle \tilde{{A}}_{i}^{}$ ≡ ⅓ $\displaystyle \sum_{j}^{}$ A j . ( 48 )

Equation  4.4 implies that a surface with h ≡ 0 is a critical point for the area functional, and with this definition of h , a triangulated surface can be relaxed (hopefully) to a minimal surface by minimizing an approximation to the Willmore energy:

E h = $\displaystyle \sum_{i}^{}$ h i 2 $\displaystyle \tilde{{A}}_{i}^{}$ . ( 49 )

Clearly, a surface with h i = 0 for all i minimizes Equation  4.6 .

The Surface Evolver[ 52 ] is a program developed by Brakke that provides a means to relax a surface with this energy function, or for that matter, most any energy defined over a triangulated surface. This program also provides convenient utilities for manipulating a surface's triangulation. It is anecdotally well known that these approximations to H are numerically unstable; during the course of relaxation, some triangles tend to gain area at the expense of others, and many triangles elongate into skinny triangles that frustrate the approach to an h ≡ 0 surface. The Evolver can alleviate these numerical headaches somewhat by providing facilities to weed out triangles whose area is too small, eliminate short edges, subdivide long edges, and rearrange the edge structure of a triangulation to produce locally equiangular configurations of triangles. The Evolver can minimize the energy as a function of the vertices, faces, and edges of the triangulation by steepest descent and conjugate-gradient minimizations, but it is empirically found that a conjugate-gradient minimization tends to create unstable triangulations in the sense described above unless the surface is reasonably close to an energy minimum.

next up previous
Next: Error in Minimizing Discrete Up: Introduction Previous: Introduction
Chris Kimmer 2011-06-01

cjkimmer -at- ius.edu