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[[File:Bulle caténoïde.png|thumb|240px|Stretching a soap film between two parallel circular wire loops generates a [[catenoid]]al minimal surface of revolution]]
In [[mathematics]], a '''minimal surface of revolution''' or '''minimum surface of revolution''' is a [[surface of revolution]] defined from two [[point (geometry)|points]] in a [[half-plane]], whose boundary is the axis of revolution of the surface. It is generated by a [[curve]] that lies in the half-plane and connects the two points; among all the surfaces that can be generated in this way, it is the one that [[mathematical optimization|minimizes]] the [[surface area]].<ref name="Mathworld: Minimal Surface of Revolution">{{cite web | url=http://mathworld.wolfram.com/MinimalSurfaceofRevolution.html | title=Minimal Surface of Revolution | last=Weisstein | first=Eric W. | authorlink=Eric W. Weisstein | work=[[Mathworld]] | publisher=[[Wolfram Research]] | accessdate=2012-08-29}}</ref> A basic problem in the [[calculus of variations]] is finding the curve between two points that produces this minimal surface of revolution.<ref name="Mathworld: Minimal Surface of Revolution"/>

==Relation to minimal surfaces==
A minimal surface of revolution is a subtype of [[minimal surface]].<ref name="Mathworld: Minimal Surface of Revolution"/> A minimal surface is defined not as a surface of minimal area, but as a surface with a [[mean curvature]] of 0.<ref name="Mathworld: Minimal Surface">{{cite web | url=http://mathworld.wolfram.com/MinimalSurface.html | title=Minimal Surface | last=Weisstein | first=Eric W. | authorlink=Eric W. Weisstein | work=[[Mathworld]] | publisher=[[Wolfram Research]] | accessdate=2012-08-29}}</ref> Since a mean curvature of 0 is a [[necessary condition]] of a surface of minimal area, all minimal surfaces of revolution are minimal surfaces, but not all minimal surfaces are minimal surfaces of revolution. As a point forms a [[circle]] when [[rotation around a fixed axis|rotated about an axis]], finding the minimal surface of revolution is equivalent to finding the minimal surface passing through two circular [[wireframe model|wireframes]].<ref name="Mathworld: Minimal Surface of Revolution"/> A physical realization of a minimal surface of revolution is [[soap film]] stretched between two parallel circular [[wire]]s: the soap film naturally takes on the shape with least surface area.<ref name="Peter J. Olver">{{cite book | title=Applied Mathematics Lecture Notes | last=Olver | first=Peter J. | chapter=Chapter 21: The Calculus of Variations | year=2012 | url=http://www.math.umn.edu/~olver/am_/cvz.pdf | format=PDF | accessdate=2012-08-29}}</ref><ref name="When Least Is Best-Soap and Solution">
{{cite book | title=When Least Is Best: How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible | last=Nahin | first=Paul J. | publisher=[[Princeton University Press]] | year=2011 | page=265-6 | quote=So what happens to the soap film after it breaks [...]? This discontinuous behavior is called the ''Goldschmidt solution'', after the German mathematician [[C. W. B. Goldschmidt]] (1807-51) who discovered it (on paper) in 1831.}}</ref>

==Catenoid solution==
[[File:Catenoid.svg|thumb|A [[catenoid]]]]
If the half-plane containing the two points and the axis of revolution is given [[Cartesian coordinate]]s, making the axis of revolution into the ''x''-axis of the coordinate system, then the curve connecting the points may be interpreted as the [[graph of a function]]. If the Cartesian coordinates of the two given points are <math>(x_1,y_1)</math>, <math>(x_2,y_2)</math>, then the area of the surface generated by a [[continuous function]] <math>f</math> may be expressed mathematically as
:<math>2\pi\int_{x_1}^{x_2} f(x) \sqrt{1+f'(x)^2} dx</math>
and the problem of finding the minimal surface of revolution becomes one of finding the function that minimizes this integral, subject to the [[boundary conditions]] that <math>f(x_1)=y_1</math> and <math>f(x_2)=y_2</math>.<ref name="sagan"/> In this case, the optimal curve will necessarily be a [[catenary]].<ref name="Mathworld: Minimal Surface of Revolution"/><ref name="sagan"/> The axis of revolution is the directrix of the catenary, and the minimal surface of revolution will thus be a [[catenoid]].<ref name="Mathworld: Minimal Surface of Revolution"/><ref name="Colding and Minicozzi">{{cite book | title=A Course in Minimal Surfaces | series=Graduate Studies in Mathematics | last1=Colding | first1=Tobias Holck | authorlink1=Tobias Colding | last2=Minicozzi II | first2=William P. | chapter=Chapter 1: The Beginning of the Theory | publisher=[[American Mathematical Society]] | year=2011 | url=http://www.ams.org/bookstore/pspdf/gsm-121-prev.pdf | format=PDF | accessdate=2012-08-29}}</ref><ref name="Meeks and Perez">{{cite book | title=A Survey on Classical Minimal Surface Theory | series=University Lectures Series | volume=60 | last1=Meeks III | first1=William H. | last2=Pérez | first2=Joaquín | chapter=Chapter 2.5: Some interesting examples of complete minimal surfaces. | publisher=[[American Mathematical Society]] | year=2012 | url=http://www.ugr.es/~jperez/papers/monograph-book2.pdf | format=PDF | accessdate=2012-08-29}}</ref>

==Goldschmidt solution==
Solutions based on discontinuous functions may also be defined. In particular, for some placements of the two points the optimal solution is generated by a discontinuous function that is nonzero at the two points and zero everywhere else. This function leads to a surface of revolution consisting of two circular disks, one for each point, connected by a degenerate line segment along the axis of revolution. This is known as a '''Goldschmidt solution'''<ref name="sagan">{{citation|contribution=2.6 The problem of minimal surfaces of revolution|title=Introduction to the Calculus of Variations|first=Hans|last=Sagan|publisher=Courier Dover Publications|year=1992|isbn=9780486673660|url=http://books.google.com/books?id=abhS8PgpBskC&pg=PA62|pages=62–66}}</ref><ref name="Mathworld: Goldschmidt Solution">{{cite web | url=http://mathworld.wolfram.com/GoldschmidtSolution.html | title=Goldschmidt Solution | last=Weisstein | first=Eric W. | authorlink=Eric W. Weisstein | work=[[Mathworld]] | publisher=[[Wolfram Research]] | accessdate=2012-08-29}}</ref> after [[German people|German]] mathematician [[Carl Wolfgang Benjamin Goldschmidt]],<ref name="When Least Is Best-Soap and Solution"/> who announced his discovery of it in his 1831 paper "Determinatio superficiei minimae rotatione curvae data duo puncta jungentis circa datum axem ortae" ("Determination of the surface-minimal rotation curve given two joined points about a given axis of origin").<ref>{{cite web | url=http://books.google.com/books/about/Determinatio_superficiei_minimae_rotatio.html?id=bPs-AAAAYAAJ | title=Bibliographic Information: Determinatio superficiei minimae rotatione curvae data duo puncta jungentis circa datum axem ortae | publisher=[[Google Books]] | accessdate=2012-08-27}}</ref>

To continue the physical analogy of soap film given above, these Goldschmidt solutions can be visualized as instances in which the soap film breaks as the circular wires are stretched apart.<ref name="When Least Is Best-Soap and Solution"/> However, in a physical soap film, the connecting line segment would not be present. Additionally, if a soap film is stretched in this way, there is a range of distances within which the catenoid solution is still feasible but has greater area than the Goldschmidt solution, so the soap film may stretch into a configuration in which the area is a [[local minimum]] but not a global minimum. For distances greater than this range, the catenary that defines the catenoid crosses the ''x''-axis and leads to a self-intersecting surface, so only the Goldschmidt solution is feasible.<ref>{{citation|title=The Science of Soap Films and Soap Bubbles|first=Cyril|last=Isenberg|authorlink=Cyril Isenberg|publisher=Courier Dover Publications|year=1992|isbn=9780486269603|page=165|url=http://books.google.com/books?id=PdsVME_LXTYC&pg=PA165}}.</ref>

==References==

{{reflist}}

[[Category:Minimal surfaces]]

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