In
differential geometry, a smooth
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...
in three dimensions has a ridge point 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
In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical. At such points the normal curvatures in all directions are equal, hence, both principal curvatures are ...
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
In mathematics, a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point must reverse direction. A typical example is given in the figure. A cusp is thus a type of singular point of a curve.
For a plane curve de ...
s 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 ...
.
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 s ...
References
*
Differential geometry of surfaces
Surfaces
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