Scherk Surface

Two historically important minimal surfaces was discovered by Heinrich Scherk in 1834. One is doubly periodic (translational symmetry in 2 directions), the other is singly periodic (translational symmetry in 1 direction). The two surfaces are conjugates of each other.

Scherk Surfaces is the third non-trivial example of minimal surfaces. (the first two were the catenoid and helicoid)

[see Catenoid]

[see Helicoid-Catenoid]

Doubly Periodic

scherk 001
Scherk's doubly periodic minimal surface.
The vertical lines, which seperate the saddle pieces, are part of the surface, but they cannot be shown in the graph representation. The graph representation is not conformal, while minimal surfaces which are constructed with the “Weierstrass representation” are always conformal images of their domain.
scherk principal
The principal curvature circles of a minimal surface have equal radii, but lie on different sides of the surface. The mean curvature ( average of the two principal curvatures) is therefore 0.
scherk patch anaglyph
One fundamental domain for the translational symmetries. (anaglyph)
scherk wire anaglyph
One fundamental domain for the translational symmetries. (anaglyph, wireframe)
scherk st
One fundamental domain for the translational symmetries. (anaglyph)
scherk sw crp
One fundamental domain for the translational symmetries. (anaglyph, wireframe)

Singly Periodic

scherk implicit 1 periodic
Scherk's singly periodic minimal surface is conjugate to Scherk doubly periodic surface. It can be described by the implicit equation sin(z) - sinh(x)*sinh(y) = 0 See Saddle Tower

Scherk's singly periodic surface has no vertical lines, so showing with repeatition as “tower” will be artificial. Also keep in mind: These surfaces were an absolute marvel when they where discovered, but after the Weierstrass representation was discovered and people had learnt how to use it, Scherk's formulas (non-conformal graph for the doubly periodic one, implicit equation for the singly periodic one) were no longer important.

To see Scherk's singly periodic surface with translational copies, see Saddle Tower

Scherk Minimal Surface Parametric Equations

  x = u
  y = v
  z = (ln(cos(aa * v) / cos(aa * u))) / aa
scherk 07444
Illustration from 1988. 〔Karcher, H.: Embedded Minimal Surfaces Drived From Scherk's Examples. Manuscripta Math. 62, 83 - 114 (1988)〕
scherk 59454
Illustration from 1988. 〔Karcher, H.: Embedded Minimal Surfaces Drived From Scherk's Examples. Manuscripta Math. 62, 83 - 114 (1988)〕

Related Surface

  1. Catalan Surface
  2. Henneberg Surface
  3. Scherk Surface
  4. Saddle Tower

CatalanHennebergScherk.pdf

Saddle_Tower.pdf

Scherk_w_Handle.pdf