Gaussian Curvature * 1 Geometric Interpretation of Principal Curvatures

We have learned that the two principal curvatures (and vectors) determine the local shape of a point on a surface. One characterizes the rate of maximum bending of the surface and the tangent direction in which it occurs, while the other characterizes the rate and tangent direction of minimum bending. The rate of surface bending along any tangent direction at the same point is determined by the two principal curvatures according to Euler’s formula. In this lecture, we will first look at how the local shape at a surface point can be approximated using its principal curvatures and direction. Then we will look at how to characterizes the rate of change of a vector defined on a surface with respect to a tangent vector. Our main focus will nevertheless be on two new measures of the curving a surface — its Gaussian and mean curvatures — that turn out to have greater geometrical significance than the principal curvatures.