Seashell surface

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Seashell surface with parametrization on left

In mathematics, a seashell surface is a surface made by a circle which spirals up the z-axis while decreasing its own radius and distance from the z-axis. Not all seashell surfaces describe actual seashells found in nature.

[edit] Parametrization

The following is a parameterization of one seashell surface:

$\begin{align} x & {} = \frac{5}{4}\left(1-\frac{v}{2\pi}\right)\cos(2v)(1+\cos u)+\cos 2v \\ \\ y & {} = \frac{5}{4}\left(1-\frac{v}{2\pi}\right)\sin(2v)(1+\cos u)+\sin 2v \\ \\ z & {} = \frac{10v}{2\pi}+\frac{5}{4}\left(1-\frac{v}{2\pi}\right)\sin(u)+15 \end{align}$

where $0\le u<2\pi$ and $-2\pi\le v <2\pi$ .

[edit] See also

[edit] References

Eric W. Weisstein, Seashell at MathWorld.
C. Illert (Feb. 1983), "the mathematics of Gnomonic seashells", Mathematical Biosciences 63(1): 21-56.
C. Illert (1987), "Part 1, seashell geometry", Il Nuovo Cimento 9D(7): 702-813.
C. Illert (1989), "Part 2, tubular 3D seashell surfaces", Il Nuovo Cimento 11D(5): 761-780.
C. Illert (Oct 1990),"Nipponites mirabilis, a challenge to seashell theory?", Il Nuovo Cimento 12D(10): 1405-1421.
C. Illert (Dec 1990), "elastic conoidal spires", Il Nuovo Cimento 12D(12): 1611-1632.
C. Illert & C. Pickover (May 1992), "generating irregularly oscillating fossil seashells", IEE Computer Graphics & Applications 12(3):18-22.
C. Illert (July 1995), "Australian supercomputer graphics exhibition", IEEE Computer Graphics & Applications 15(4):89-91.
C. Illert (Editor 1995), "Proceedings of the First International Conchology Conference, 2-7 Jan 1995, Tweed Shire, Australia", publ. by Hadronic Press, Florida USA. 219 pages.
C. Illert & R. Santilli (1995), "Foundations of Theoretical Conchology", publ. by Hadronic Press, Florida USA. 183 pages plus coloured plates.