Misplaced Pages

k-noid

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
"Trinoid" redirects here. For the entities from the television show Bakuryū Sentai Abaranger, see Wicked Lifeforms Evolien § Trinoids.
Trinoid
7-noid

In differential geometry, a k-noid is a minimal surface with k catenoid openings. In particular, the 3-noid is often called trinoid. The first k-noid minimal surfaces were described by Jorge and Meeks in 1983.

The term k-noid and trinoid is also sometimes used for constant mean curvature surfaces, especially branched versions of the unduloid ("triunduloids").

k-noids are topologically equivalent to k-punctured spheres (spheres with k points removed). k-noids with symmetric openings can be generated using the Weierstrass–Enneper parameterization f ( z ) = 1 / ( z k − 1 ) 2 , g ( z ) = z k − 1 {\displaystyle f(z)=1/(z^{k}-1)^{2},g(z)=z^{k-1}\,\!} . This produces the explicit formula

X ( z ) = 1 2 ℜ { ( − 1 k z ( z k − 1 ) ) [ ( k − 1 ) ( z k − 1 ) 2 F 1 ( 1 , − 1 / k ; ( k − 1 ) / k ; z k ) − ( k − 1 ) z 2 ( z k − 1 ) 2 F 1 ( 1 , 1 / k ; 1 + 1 / k ; z k ) − k z k + k + z 2 − 1 ] } {\displaystyle {\begin{aligned}X(z)={\frac {1}{2}}\Re {\Bigg \{}{\Big (}{\frac {-1}{kz(z^{k}-1)}}{\Big )}{\Big }{\Bigg \}}\end{aligned}}}
Y ( z ) = 1 2 ℜ { ( i k z ( z k − 1 ) ) [ ( k − 1 ) ( z k − 1 ) 2 F 1 ( 1 , − 1 / k ; ( k − 1 ) / k ; z k ) + ( k − 1 ) z 2 ( z k − 1 ) 2 F 1 ( 1 , 1 / k ; 1 + 1 / k ; z k ) − k z k + k − z 2 − 1 ) ] } {\displaystyle {\begin{aligned}Y(z)={\frac {1}{2}}\Re {\Bigg \{}{\Big (}{\frac {i}{kz(z^{k}-1)}}{\Big )}{\Big }{\Bigg \}}\end{aligned}}}
Z ( z ) = ℜ { 1 k − k z k } {\displaystyle Z(z)=\Re \left\{{\frac {1}{k-kz^{k}}}\right\}}

where 2 F 1 ( a , b ; c ; z ) {\displaystyle _{2}F_{1}(a,b;c;z)} is the Gaussian hypergeometric function and ℜ { z } {\displaystyle \Re \{z\}} denotes the real part of z {\displaystyle z} .

It is also possible to create k-noids with openings in different directions and sizes, k-noids corresponding to the platonic solids and k-noids with handles.

References

  1. L. P. Jorge and W. H. Meeks III, The topology of complete minimal surfaces of finite total Gaussian curvature, Topology 22 (1983)
  2. N Schmitt (2007). "Constant Mean Curvature n-noids with Platonic Symmetries". arXiv:math/0702469.
  3. Matthias Weber (2001). "Classical Minimal Surfaces in Euclidean Space by Examples" (PDF). Indiana.edu. Retrieved 2012-10-05.
  4. H. Karcher. "Construction of minimal surfaces, in "Surveys in Geometry", University of Tokyo, 1989, and Lecture Notes No. 12, SFB 256, Bonn, 1989, pp. 1-96" (PDF). Math.uni-bonn-de. Retrieved 2012-10-05.
  5. Jorgen Berglund, Wayne Rossman (1995). "Minimal Surfaces with Catenoid Ends". Pacific J. Math. 171 (2): 353–371. arXiv:0804.4203. Bibcode:2008arXiv0804.4203B. doi:10.2140/pjm.1995.171.353. S2CID 11328539.

External links

Minimal surfaces
...
Categories: