# Catenoid > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Catenoid > Markdown URL: https://mediated.wiki/source/Catenoid.md > Source: https://en.wikipedia.org/wiki/Catenoid > Source revision: 1315789636 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) {{Short description|Surface of revolution of a catenary}} thumb|right|alt=three-dimensional diagram of a catenoid|A catenoid thumb|right|alt=animation of a catenary sweeping out the shape of a catenoid as it rotates about a central point|A catenoid obtained from the rotation of a catenary In [geometry](/source/geometry), a '''catenoid''' is a type of [surface](/source/Surface_(mathematics)), arising by rotating a [catenary](/source/catenary) curve about an axis (a [surface of revolution](/source/surface_of_revolution)).{{cite book|last1=Dierkes|first1=Ulrich|last2=Hildebrandt|first2=Stefan|last3=Sauvigny|first3=Friedrich|title=Minimal Surfaces|date=2010|publisher=[Springer Science & Business Media](/source/Springer_Science_%26_Business_Media)|isbn=9783642116988|page=141|url=https://books.google.com/books?id=9YhBOg6vO-EC&pg=PA141|language=en}} It is a [minimal surface](/source/minimal_surface), meaning that it occupies the least area when bounded by a closed space.{{cite book|last1=Gullberg|first1=Jan|title=Mathematics: From the Birth of Numbers|date=1997|publisher=[W. W. Norton & Company](/source/W._W._Norton_%26_Company)|isbn=9780393040029|page=[https://archive.org/details/mathematicsfromb1997gull/page/538 538]|url=https://archive.org/details/mathematicsfromb1997gull|url-access=registration|language=en}} It was formally described in 1744 by the mathematician [Leonhard Euler](/source/Leonhard_Euler). [Soap film](/source/Soap_film) attached to twin circular rings will take the shape of a catenoid. Because they are members of the same [associate family](/source/associate_family) of surfaces, a catenoid can be bent into a portion of a [helicoid](/source/helicoid), and vice versa. == Geometry == The catenoid was the first non-trivial minimal [surface](/source/surface_(topology)) in 3-dimensional Euclidean space to be discovered apart from the [plane](/source/plane_(geometry)). The catenoid is obtained by rotating a catenary about its [directrix](/source/Directrix_(conic_section)). It was found and proved to be minimal by [Leonhard Euler](/source/Leonhard_Euler) in 1744.{{cite book |last1=Euler |first1=Leonhard |author-link=Leonard Euler |editor-last=Carathéodory |editor-first=Constantin |editor-link=Constantin Carathéodory |title=Methodus inveniendi lineas curvas: maximi minimive proprietate gaudentes sive solutio problematis isoperimetrici latissimo sensu accepti |date=1952 |orig-year=reprint of 1744 edition |publisher=Springer Science & Business Media |isbn=3-76431-424-9 |language=Latin |url=https://books.google.com/books?id=zNDdVFZalSAC}}{{cite journal|last1=Colding|first1=T. H.|last2=Minicozzi|first2=W. P.|title=Shapes of embedded minimal surfaces|journal=Proceedings of the National Academy of Sciences|date=17 July 2006|volume=103|issue=30|pages=11106–11111|doi=10.1073/pnas.0510379103|pmc=1544050|pmid=16847265|bibcode=2006PNAS..10311106C|doi-access=free}} Early work on the subject was published also by [Jean Baptiste Meusnier](/source/Jean_Baptiste_Meusnier).{{cite book|url=https://archive.org/details/mmoiresurlathor00salvgoog|format=PDF|last1=Meusnier|first1=J. B.|title=Mémoire sur la courbure des surfaces|trans-title=Dissertation on the curvature of surfaces |date=1881|publisher=F. Hayez, Printer of the Royal Academy of Belgium|location=Brussels|language=fr|isbn=9781147341744|pages=477–510}}{{rp|11106}} There are only two [minimal surfaces of revolution](/source/minimal_surfaces_of_revolution) ([surfaces of revolution](/source/surfaces_of_revolution) which are also minimal surfaces): the [plane](/source/plane_(geometry)) and the catenoid.{{cite web|title=Catenoid|url=http://mathworld.wolfram.com/Catenoid.html|website=Wolfram MathWorld|accessdate=15 January 2017|language=en}} The catenoid may be defined by the following parametric equations: {{NumBlk|::|\begin{align} x &= c \cosh \frac{v}{c} \cos u \\ y &= c \cosh \frac{v}{c} \sin u \\ z &= v \end{align}|{{EquationRef|1}}}} where u \in [-\pi, \pi) and v \in \mathbb{R} and c is a non-zero real constant. In cylindrical coordinates: \rho =c \cosh \frac{z}{c}, where c is a real constant. A physical model of a catenoid can be formed by dipping two [circular](/source/circle) rings into a soap solution and slowly drawing the circles apart. The catenoid may be also defined approximately by the [stretched grid method](/source/stretched_grid_method) as a facet 3D model. == Helicoid transformation == [[Image:helicatenoid.gif|thumb|right|256px|alt=Continuous animation showing a right-handed helicoid deforming into a catenoid, a left-handed helicoid, and back again|Deformation of a right-handed [helicoid](/source/helicoid) into a left-handed one and back again via a catenoid]] Because they are members of the same [associate family](/source/associate_family) of surfaces, one can bend a catenoid into a portion of a [helicoid](/source/helicoid) without stretching. In other words, one can make a (mostly) [continuous](/source/continuous_function) and [isometric](/source/Isometry) deformation of a catenoid to a portion of the [helicoid](/source/helicoid) such that every member of the deformation family is [minimal](/source/Minimal_surface) (having a [mean curvature](/source/mean_curvature) of zero). A [parametrization](/source/Parametric_equation) of such a deformation is given by the system \begin{align} x(u,v) &= \sin \theta \,\cosh v \,\cos u + \cos \theta \,\sinh v \,\sin u \\ y(u,v) &= \sin \theta \,\cosh v \,\sin u - \cos \theta \,\sinh v \,\cos u \\ z(u,v) &= v \sin \theta + u \cos \theta \end{align} for (u,v) \in (-\pi, \pi] \times (-\infty, \infty), with deformation parameter -\pi < \theta \le \pi, where: * \theta = \pi corresponds to a right-handed helicoid, * \theta = \pm \pi / 2 corresponds to a catenoid, and * \theta = 0 corresponds to a left-handed helicoid. == The critical catenoid conjecture == A ''critical'' catenoid is a catenoid in the unit ball that meets the boundary sphere orthogonally. Up to rotation about the origin, it is given by rescaling {{EquationNote|Eq. 1}} with c=1 by a factor (\rho_0\cosh\rho_0)^{-1} , where \rho_0\tanh\rho_0=1 . It is an embedded annular solution of the [free boundary problem](/source/free_boundary_problem) for the area functional in the unit ball and the ''critical catenoid conjecture'' states that it is the unique such annulus. The similarity of the critical catenoid conjecture to [Hsiang-Lawson's conjecture](/source/Hsiang-Lawson's_conjecture) on the Clifford torus in the 3-sphere, which was proven by [Simon Brendle](/source/Simon_Brendle) in 2012,{{cite journal |last=Brendle |first=Simon |title=Embedded minimal tori in S3 and the Lawson conjecture | journal = Acta Mathematica | volume = 211 | pages = 177–190 | year = 2013 |issue=2 | doi=10.1007/s11511-013-0101-2|s2cid=119317563 |doi-access=free |arxiv=1203.6597 }} has driven interest in the conjecture,{{cite journal|last=Devyver |first=B.|title=Index of the critical catenoid |journal= Geometriae Dedicata |date=2019|volume=199|pages=355–371|doi=10.1007/s10711-018-0353-2}} as has its relationship to the Steklov eigenvalue problem.{{cite journal|last1= Fraser |first1=Ailana |authorlink1=Ailana Fraser |last2= Schoen |first2=Richard |authorlink2=Richard Schoen|title= The first Steklov eigenvalue, conformal geometry, and minimal surfaces |journal=[Advances in Mathematics](/source/Advances_in_Mathematics) |date=2011|volume=226|issue=5|pages=4011–4030|doi= 10.1016/j.aim.2010.11.007|doi-access=free |arxiv=0912.5392}} Nitsche proved in 1985 that the only immersed minimal disk in the unit ball with free boundary is an equatorial totally geodesic disk.{{cite journal|last=Nitsche|first=J. C. C.|title= Stationary partitioning of convex bodies |journal= Archive for Rational Mechanics and Analysis |date=1985|volume=89|issue=1 |pages=1–19|doi= 10.1007/BF00281743 |bibcode=1985ArRMA..89....1N }} Nitsche also claimed without proof in the same paper that any free boundary constant mean curvature annulus in the unit ball is rotationally symmetric, and hence a catenoid or a parallel surface. Non-embedded counterexamples to Nitsche’s claim have since been constructed.{{cite conference |first=H. C. |last=Wente |title=Tubular capillary surfaces in a convex body |page=288 |editor1-first=P. |editor1-last=Concus |editor2-first=K. |editor2-last=Lancaster |book-title=Advances in Geometric Analysis and Continuum Mechanics |conference=Proceedings of a conference held at Stanford University on August 2–5, 1993, in honor of the seventieth birthday of Robert Finn |publisher=International Press |year=1993}}{{cite journal|last1= Fernández |first1=I. |last2= Hauswirth |first2=L.|last3=Mira |first3=P.|title= Free boundary minimal annuli immersed in the unit ball |journal= Archive for Rational Mechanics and Analysis |date=2023|volume=247|issue=6|pages=108|doi= 10.1007/s00205-023-01943-z|doi-access=free |arxiv=2208.14998|bibcode=2023ArRMA.247..108F }} The critical catenoid conjecture is stated in the embedded case by Fraser and Li{{cite journal|last1= Fraser |first1=A. |last2=Li |first2=M. M.|title= Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary |journal= Journal of Differential Geometry |date=2014|volume=96|issue=6|pages=183–200|doi=10.4310/jdg/1393424916|doi-access=free |arxiv=1204.6127}} and has been proven by McGrath with the extra assumption that the annulus is reflection invariant through coordinate planes,{{cite journal|last= McGrath |first=P.| title= A characterization of the critical catenoid |journal= Indiana University Mathematics Journal |date=2018|volume=67|issue=2|pages=889–897|doi=10.1512/iumj.2018.67.7251 |jstor=26769410 |url= https://www.jstor.org/stable/26769410|arxiv=1603.04114}} and by Kusner and McGrath when the annulus has antipodal symmetry.{{cite journal|last1= Kusner |first1=R.|last2= McGrath |first2=P.|title= On Steklov eigenspaces for free boundary minimal surfaces in the unit ball |journal= American Journal of Mathematics |date=2024|volume=146|issue=5|pages=1275–1293|doi=10.1353/ajm.2024.a937942 |arxiv=2011.06884}} As of 2025 the full conjecture remains open. == References == {{reflist}} == Further reading == * {{cite book |last1=Krivoshapko |first1=Sergey |last2=Ivanov |first2=V. N. |title=Encyclopedia of Analytical Surfaces |date=2015 |publisher=Springer |isbn=9783319117737 |chapter=Minimal Surfaces |chapter-url=https://books.google.com/books?id=cXTdBgAAQBAJ&pg=PA427 |language=en}} == External links == * {{springer|title=Catenoid|id=p/c020800}} * [http://www.princeton.edu/~rvdb/WebGL/catenoid.html Catenoid – WebGL model] * [http://posner.library.cmu.edu/Posner/books/book.cgi?call=517.4_E88M_1744 Euler's text describing the catenoid] at Carnegie Mellon University * [https://www.youtube.com/watch?v=31Om4VrSzb8 Calculating the surface area of a Catenoid] * [https://mathworld.wolfram.com/MinimalSurfaceofRevolution.html Minimal Surface of Revolution] {{Minimal surfaces}} Category:Minimal surfaces [de:Minimalfläche#Das Katenoid](/source/de%3AMinimalfl%C3%A4che) --- Adapted from the Wikipedia article [Catenoid](https://en.wikipedia.org/wiki/Catenoid) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Catenoid?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.