Curvature and the Shape Operator

Summary and Conclusions

We have finally come to a point where we can summarize everything we have learned in this tutorial and relate it to the generation of 3D models of the brain.

The importance of our triangulation lies in the fact that it creates a coordinate system of vertices that defines a local patch. This patch can be defined rather arbitrarily depending on the degree of accuracy that you wish to have in your curvature calculations.

Often, a coordinate patch can be defined as the space surrounding a point that includes 8-12 of its neighboring points. This generates a patch that is large enough to allow for a useful calculation of the local shape operator, while small enough to remain accurate.

From this shape operator, then, we are able to calculate the mean and Gaussian curvatures, which are able to provide us with useful information regarding the properties of the surface we are examining.