In
differential geometry, a smooth
surface in three dimensions has a ridge point when a
line of curvature has a local maximum or minimum of
principal curvature
In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends b ...
. 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 points 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 edges on the
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 th ...
.
See also
*
Ridge detection
In image processing, ridge detection is the attempt, via software, to locate ridges in an image, defined as curves whose points are local maxima of the function, akin to geographical ridges.
For a function of ''N'' variables, its ridges are a ...
References
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Differential geometry of surfaces
Surfaces
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