#jsDisabledContent { display:none; } My Account |  Register |  Help

# Associate family

Article Id: WHEBN0037016612
Reproduction Date:

 Title: Associate family Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Associate family

Animation showing the deformation of a helicoid into a catenoid as θ changes.

In differential geometry, the associate family (or Bonnet family) of a minimal surface is a one-parameter family of minimal surfaces which share the same Weierstrass data. That is, if the surface has the representation

x_k(\zeta) = \Re \left\{ \int_{0}^{\zeta} \varphi_{k}(z) \, dz \right\} + c_k , \qquad k=1,2,3

the family is described by

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]

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.

Some examples of associate surface families are: the catenoid and helicoid family, the Schwarz P, Schwarz D and gyroid family, and the 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.[2]

There are counterparts to the associate families of minimal surfaces in higher-dimensional spaces and manifolds.[3]

## References

1. ^ 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]
2. ^ 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]
3. ^ J.-H. Eschenburg, The Associated Family, Matematica Contemporanea, Vol 31, 1–12 2006 [3]
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.

Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.