When a surface bends differently in different directions, a single sphere may not describe its local shape. Instead, slice the surface through its normal vector and fit a circle to each resulting curve. The smallest and largest signed curvatures are the principal curvatures; mean curvature is their half-sum. section 5.2.3
Useful comparisons:
- A plane has zero mean curvature everywhere. section 5.2.3
- A sphere has constant mean curvature. section 5.2.3
- A cylinder has zero Gaussian curvature but nonzero mean curvature, because its two principal curvatures differ. section 5.2.3
Reading a cylinder
A cylinder curves around its circular cross-section but remains straight along its axis. Its principal curvatures therefore combine to produce nonzero mean curvature, even though their product—the Gaussian curvature—is zero. section 5.2.3
