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SCHERK SURFACES
Surfaces studied by Scherk in 1834.
Heinrich Ferdinand Scherk (17981885): German mathematician. The minimal helicoids are also sometimes called Scherk surfaces. 
First Scherk surface
Cartesian equation: ,
i.e.
, .
Equivalent form: . Included lines: , and . Weierstrass parametrization: with and , which gives: 
The first Scherk surface is the only minimal surface that is a translation surface. It is obtained by translation of the curve of the log cosine (which is also the catenary of equal strength) along itself.

View made with Povray by Alain Esculier 
picture by JeanMarie Dendoncker. 
Compare to the Enneper surface, another minimal surface.
Second Scherk surface
Cartesian equation: .
Simply periodic minimal surface. 
Engraving of the first Scherk surface, by Patrice Jeener, with his kind authorization.
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© Robert FERRÉOL 2017