K noid - Alchetron, The Free Social Encyclopedia

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 . 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 ] } 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 ) ] } Z ( z ) = ℜ { 1 k − k z k }

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

K-noid Wikipedia

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