Karcher JD Saddle tower

karcher JDsaddle Morph 001
“karcher JDsaddle Morph 001”
Another doubly periodic embedded conjugate pair of minimal surfaces. They also were suggested by the Scherk saddle towers. A visible difference to the JE-surfaces is that here the saddle towers are separated by a planar symmetry line. The morphing parameter is the edgelength ratio of the rectangular tori that are the parameter domains for these surfaces.
karcher JDsaddle conjmo 001
“karcher JDsaddle conjmo 001”
These are the conjugate surfaces of the previous sequence. It is a rare coincidence that the surfaces in the previous family and in this conjugate family are the same (the parameter runs in the opposite direction). They look a bit different, because they are assembled from different fundamental pieces, see the following sequence.
karcher JDsaddle assomM 001
“karcher JDsaddle assomM 001”
Associate family morph of the fundamental pieces from which the previous sequences are assembled. The main thing to observe is that planar symmetry lines and straight lines (axes for 180 degree rotation symmetry) change their role: what is a straight line segment on the boundary of one piece is a planar symmetry curve on the the conjugate piece. The associate family deformation does not change the length of curves on the deformed surfaces. It is therefore called an "isometric" deformation. The deformation only bends the surfaces, no part is stretched.
karcher JDsaddle anawire
Karcher JD Saddle tower (Anaglyph wireframe)
karcher JDsaddle conj ana
Karcher JD Saddle tower, conjugate. (Anaglyph wireframe)

These families are parametrized by 4-punctured rectangular tori; they and their conjugates are embedded. We therefore suggest the associate family morphing, and also morphing of the modulus, bb, (0 < bb < 0.5) of the rectangular Torus, which changes the size of the visible holes.

For the visual appearance of these surfaces it is particularly important that the punctures are centers of polar coordinate lines. Formulas are taken from [K1] or [K2]. The Gauss maps for these surfaces are degree 2 elliptic functions. The cases shown are particularly symmetric, the zeros and poles of the Gauss map are half-period points and the punctures are there. In the Jd-case the diagonal of the rectangular fundamental domain joins the two zeros, and in the Je-case it joins a zero and a pole of the Gauss map.

Under suitable choices of the modulus of the Torus these surfaces look like a fence of Scherk Saddle Towers - with a vertical straight line (Je), respectively a planar symmetry line (Jd), separating these towers. The conjugate surfaces look qualitatively the same in the Jd-cases and like a checkerboard array of horizontal handles between vertical planes in the Je-cases.

For a discussion of techniques for creating minimal surfaces with various qualitative features by appropriate choices of Weierstrass data, see either [KWH], or pages 192--217 of [DHKW].