For a smooth surface

Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...

 in three dimensions a ridge point occurs when a line of curvature has a local maximum or minimum of principal curvature. The set of ridge points form curves on the surface called ridges.

The ridges of a given surface fall into two families, typically designated red and blue, depending on which of the two principal curvatures has an extremum.

At umbilical point

Umbilical point

In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points that are locally spherical. At such points both principal curvatures are equal, and every tangent vector is a principal direction....

s the colour of a ridge will change from red to blue. There are two main cases: one has three ridge lines passing through the umbilic, and the other has one line passing through it.

Ridge lines correspond to cuspidal edge

Cusp (singularity)

In the mathematical theory of singularities a cusp is a type of singular point of a curve. Cusps are local singularities in that they are not formed by self intersection points of the curve....

s on the focal surface

Focal surface

For a surface in three dimension the focal surface, surface of centers or evolute is formed by taking the centers of the curvature spheres, which are the tangential spheres whose radii are the reciprocals of one of the principal curvatures at the point of tangency...

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