Seashell surface

Seashell surface

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.

Parametrization

The following is a parameterization of one seashell surface:

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

where 0le u<2pi and -2pile v <2pi.

ee also

*Seashell
*Helix
*Spiral

References

*
* 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.
* Deborah R. Fowler, Hans Meinhardt, and Przemyslaw Prusinkiewicz. Modeling seashells. Proceedings of SIGGRAPH '92 (Chicago, Illinois, July 26-31, 1992), In Computer Graphics, 26, 2, (July 1992), ACM SIGGRAPH, New York, pp. 379-387. [http://algorithmicbotany.org/papers/shells.sig92.html]
* Callum Galbraith, Przemyslaw Prusinkiewicz, and Brian Wyvill. Modeling a Murex cabritii sea shell with a structured implicit surface modeler. The Visual Computer vol. 18, pp. 70-80. http://algorithmicbotany.org/papers/murex.tvc2002.html


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