In
differential geometry, the associate family (or
Bonnet
A Bonnet is a variety of headgear, hat or cap
Specific types of headgear referred to as "bonnets" may include
Scottish
*Blue bonnet, a distinctive woollen cap worn by men in Scotland from the 15th-18th centuries And its derivations:
** Feat ...
family) of a
minimal surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).
The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
is a one-parameter family of minimal surfaces which share the same
Weierstrass data. That is, if the surface has the representation
:
the family is described by
:
where
indicates the real part of a
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
.
For ''θ'' = ''π''/2 the surface is called the conjugate of the ''θ'' = 0 surface.
The transformation can be viewed as locally rotating the
principal curvature
In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends by ...
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
In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis (a surface of revolution). It is a minimal surface, meaning that it occupies the least area when bounded by a closed space. It was formally describe ...
and
helicoid
The helicoid, also known as helical surface, after the plane and the catenoid, is the third minimal surface to be known.
Description
It was described by Euler in 1774 and by Jean Baptiste Meusnier in 1776. Its name derives from its similar ...
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.
There are counterparts to the associate families of minimal surfaces in higher-dimensional spaces and manifolds.
[J.-H. Eschenburg, The Associated Family, Matematica Contemporanea, Vol 31, 1–12 200]
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References
{{Minimal surfaces
Differential geometry
Minimal surfaces