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Surfaces are called Weingarten surfaces (or just Weingarten for short), if for the Gaussian
curvature K and the mean curvature function H
d K ∧ d H = 0
holds on all the surface. In what follows, the principle curvatures are functional related, i.e locally
one of the principle curvature can be derived as a function of the other or vice versa.
The animation below shows a deformation that
acts on the set of Bianchi surfaces which are
Weingarten surfaces.
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Such surfaces can be described in terms of the solutions
of a Painleve V equation and the deformation parameter controlling
the shown deformation is a parameter in this equation.
In particular all Bianchi surfaces of revolution with non-constant
curvature are either the catenoid ( or homotheisis of it)
or can be described in terms of Painleve V functions.
The corresponding associated family of each surface you see in the animation
is given by a scaling of only one surfaces, which you do not see.
Here is an example of such a family:
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