Curvature and the Shape Operator

triangulated brain mesh

Generating a Triangulated Mesh

Now that we have reviewed the first fundamental form, we are ready to describe the actual process in generating a triangulation mesh of the surface. There are many ways to generate this type of mesh. Some methods are designed to be area-preserving while others are distance-preserving.

The difference between these two approaches, as you can probably infer, is that the former is designed to preserve the overall surface area, while the latter is designed to preserve the distance between adjacent points on the surface.

Another way of generating a triangulated mesh is to view each side of each triangle as a spring that has a specified tension. If the points on the surface are too far apart, then the springs will have to stretch beyond this specified tension, and if they are too close together they won't be under enough tension. Thus, the points are spaced so that all of them are equidistant from their neighbors.

Depending on the triangulation method used, the surface triangulation that is rendered will have vertices that are varying distances from one another.