### CHAPTER 21

### THE TOPOLOGY OF CURVY SURFACES

*If others would but reflect on mathematical truths as deeply and as continuously as I have, they would make my discoveries*.—Carl Friedrich Gauss^{1}

One of the most fundamental topics in the geometry of planar curves is curvature. The *curvature* at a point *x* is a number, *k*, that measures the “sharpness” of the turn at x—it measures how quickly the tangent vectors change direction. Roughly speaking, given a normal vector to a curve at *x*, if the curve bends in the direction of , then *k* > 0, if it bends away, then ...