Curvature in Triangle Meshes

Abstract

In many cases notions from differential geometry can be usefully extended to piecewise planar surfaces, and this chapter covers curvature measures on triangle meshes. A frequently used principle is to obtain a smooth surface approximation and to estimate the curvature from this approximation. Alternatively, the integral of some curvature measures can be computed from a small region of the mesh and then normalized by dividing by the area of that region.

Following these principles, we first discuss how to extend the definition of a surface normal to the edges and vertices of a triangle mesh. Next, we cover the estimation of Gaußian and mean curvature on a triangle mesh. Finally, techniques for computing the shape operator are discussed—followed by a discussion on how to obtain the principal curvatures from the shape operator.

Appendix

In this appendix, we derive the Cotan formula for the gradient of the area of a triangle. Given a triangle (shown in Fig. 8.9) whose vertex p _i is movable, compute the gradient of the area of the triangle as a function of p _i .

Fig. 8.9

A triangle with a vertex p _i

It is clear that the gradient is perpendicular to the plane of the triangle since moving p _i either in the positive or negative direction along the triangle normal will increase the area. Hence, the present position is a minimum. Moving p _i parallel to the base line b will not change the area. It follows that the gradient is in the plane of the triangle and orthogonal to b. It is trivial to find the length of the gradient, and this leads to the first line of the equation below. After a number of steps, we reach the expression used in (8.4).

Some of the steps may be a little tricky. The bottom line is that we need to use the fact that the cotangent of an angle between two vectors a and b is equal to $\frac{\mathbf {a}^{T} \mathbf {b}}{\|\mathbf {a} \times \mathbf {b}\|}$:

$$\begin{array}{*{20}l} {\nabla A({\mathbf p}_i )} & { = \frac{{({\mathbf b} \times {\mathbf a}) \times {\mathbf b}}}{{2\left\| {{\mathbf b} \times {\mathbf a}} \right\|}}} \\ {} & { = \frac{{({\mathbf b}^t {\mathbf b}){\mathbf a} - ({\mathbf b}^t {\mathbf a}){\mathbf b}}}{{2\left\| {{\mathbf b} \times {\mathbf a}} \right\|}}} \\ {} & { = \frac{{({\mathbf b}^t {\mathbf b}){\mathbf a} - ({\mathbf b}^t {\mathbf a}){\mathbf a} + ({\mathbf b}^t {\mathbf a}){\mathbf a} - ({\mathbf b}^t {\mathbf a}){\mathbf b}}}{{2\left\| {{\mathbf b} \times {\mathbf a}} \right\|}}} \\ {} & { = \frac{{ - ({\mathbf c}^t {\mathbf b}){\mathbf a}}}{{2\left\| {{\mathbf c} \times - {\mathbf b}} \right\|}} + \frac{{({\mathbf b}^t {\mathbf a}){\mathbf c}}}{{2\left\| {{\mathbf c} \times - {\mathbf b}} \right\|}}} \\ {} & { = \frac{{ - ({\mathbf c}^t - {\mathbf b}){\mathbf a}}}{{2\left\| {{\mathbf c} \times - {\mathbf b}} \right\|}} + \frac{{({\mathbf b}^t {\mathbf a}){\mathbf c}}}{{2\left\| {{\mathbf b} \times {\mathbf a}} \right\|}}} \\ {} & { = \frac{1}{2}({\mathbf a}\,\cot \beta + {\mathbf c}\,\cot \alpha ).} \\ \end{array}$$