where <math>\Re</math> indicates the real part of a complex number.
For θ = π/2 the surface is called the conjugate of the θ = 0 surface.[1]
The transformation can be viewed as locally rotating the principal curvature directions. The surface normals of a point with a fixed ζ remains unchanged as θ changes; the point itself moves along an ellipse.
Conjugate surfaces have the property that any straight line on a surface maps to a planar geodesic on its conjugate surface and vice versa. If a patch of one surface is bounded by a straight line, then the conjugate patch is bounded by a planar symmetry line. This is useful for constructing minimal surfaces by going to the conjugate space: being bound by planes is equivalent to being bound by a polygon.[2]
There are counterparts to the associate families of minimal surfaces in higher-dimensional spaces and manifolds.[3]
↑Matthias Weber, Classical Minimal Surfaces in Euclidean Space by Examples, in Global Theory of Minimal Surfaces:
Proceedings of the Clay Mathematics Institute 2001 Summer School, Mathematical Sciences Research Institute, Berkeley, California, June 25–July 27, 2001. American Mathematical Soc., 2005 [1]
↑Hermann Karcher, Konrad Polthier, "Construction of Triply Periodic Minimal Surfaces", Phil. Trans. R. Soc. Lond. A 16 September 1996 vol. 354 no. 1715 2077–2104 [2]