Associate family

[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.

The Enneper surface is conjugate to itself: it is left invariant as θ changes.

If a patch of one surface is bounded by a straight line, then the conjugate patch is bounded by a planar symmetry line.

[2] There are counterparts to the associate families of minimal surfaces in higher-dimensional spaces and manifolds.