When a surface bends differently in different directions, one curvature value cannot describe the whole point. Slice the surface with planes that contain its normal vector; each slice produces a curve with its own fitting circle and signed curvature. The smallest and largest of these values are the principal curvatures. section 5.2.3
Principal curvatures connect surface shape to familiar curvature measures:
- Their half-sum is the mean curvature, while their product is the Gaussian curvature. section 5.2.3
- The planes containing the corresponding extremal directions are principal planes. section 5.2.3
- A cylinder has zero Gaussian curvature but nonzero mean curvature, showing that these measures capture different aspects of bending. section 5.2.3
Minimal surfaces and 3D printing
A minimal surface has mean curvature zero everywhere. Therefore, when one principal curvature is positive, the other must be negative: the surface has neither concave nor convex bumps. In powder or resin printing, this can help prevent material from becoming trapped and makes printed details easier to clean. section 10.3.1
Figure 10.20 A surface, a tangent plane, its normal vector, and a pair of principal curvatures (Eric Gaba [Sting], CC BY-SA, via Wikimedia Commons)
Use the figure to relate a surface point, its tangent plane, normal vector, and pair of principal curvatures. section 10.3.1
