The extremal curvatures and their directions
At each point on a surface, the principal curvatures k1 and k2 are the maximum and minimum values of normal curvature over all tangent directions.
The corresponding directions are called principal directions, and they are always perpendicular (except at umbilical points where k1 = k2).
Each cross shows the two principal directions at that point. Red/blue colors indicate the sign of the curvature. Yellow circles mark umbilical points where the principal directions are undefined (because all directions have equal curvature).
The principal curvatures are the eigenvalues of the shape operator S, and the principal directions are the corresponding eigenvectors.
Curvature lines (or lines of curvature) are curves on the surface that always point in a principal direction. They form an orthogonal network that reveals the intrinsic geometry of the surface.
Curvature lines are curves on the surface that always point in principal directions. They form an orthogonal network (except at umbilical points) and reveal the intrinsic geometry of the surface.
Euler's formula tells us the normal curvature in any direction: it's a weighted average of the principal curvatures, with weights determined by the angle to the principal directions.
Euler's formula: normal curvature in any direction is a weighted average of the principal curvatures
Both curvatures same sign (bowl-like). K > 0.
Opposite signs (saddle-like). K < 0.
One curvature is zero (cylinder-like). K = 0.
Equal curvatures (sphere-like). All directions are principal.
The principal curvatures are the fundamental building blocks. All other curvature quantities can be expressed in terms of k1 and k2:
Next: Geodesics — the straightest possible curves on a curved surface.