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Fix CS1 deprecated date parameter errors

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[[Image:helicatenoid.gif|thumb|right|256px|Animation showing the deformation of a helicoid into a catenoid as ''θ'' changes.]]

In [[differential geometry]], the '''associate family''' (or [[Pierre Ossian Bonnet|Bonnet]] family) of a [[minimal surface]] is a one-parameter family of minimal surfaces which share the same [[Weierstrass–Enneper parameterization|Weierstrass data]]. That is, if the surface has the representation
:<math>x_k(\zeta) = \Re \left\{ \int_{0}^{\zeta} \varphi_{k}(z) \, dz \right\} + c_k , \qquad k=1,2,3</math>
the family is described by

:<math>x_k(\zeta,\theta) = \Re \left\{ e^{i \theta} \int_0^\zeta \varphi_{k}(z) \, dz \right\} + c_k , \qquad \theta \in [0,2\pi] </math>

For ''θ'' = ''π''/2 the surface is called the conjugate of the ''θ'' = 0 surface.<ref>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 [http://www.indiana.edu/~minimal/research/claynotes.pdf]</ref>

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.

Some examples of associate surface families are: the [[catenoid]] and [[helicoid]] family, the [[Schwarz_minimal_surface#Schwarz_P_.28.22Primitive.22.29|Schwarz P]], [[Schwarz_minimal_surface#Schwarz_D_.28.22Diamond.22.29|Schwarz D]] and [[gyroid]] family, and the [[Scherk surface|Scherk's first and second surface]] family. The [[Enneper surface]] is conjugate to itself: it is left invariant as ''θ'' changes.

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.<ref>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 [http://www.polthier.info/articles/triply/triply_withoutApp.pdf]</ref>

There are counterparts to the associate families of minimal surfaces in higher-dimensional spaces and manifolds.<ref>J.-H. Eschenburg, The Associated Family, Matematica Contemporanea, Vol 31, 1–12 2006 [http://www.mat.unb.br/matcont/31_1.pdf]</ref>

==References==
{{reflist}}

[[Category:Differential geometry]]
[[Category:Minimal surfaces]]

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