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

seashell A seashell or sea shell, also known simply as a shell, is a hard, protective outer layer usually created by an animal or organism that lives in the sea. The shell is part of the body of the animal. Empty seashells are often found washe ...

s found in nature.

Parametrization

The following is a parameterization of one seashell surface: :

\begin
x &  = \frac\left(1-\frac\right)\cos(2v)(1+\cos u)+\cos 2v \\  \\
y &  = \frac\left(1-\frac\right)\sin(2v)(1+\cos u)+\sin 2v \\  \\
z &  = \frac+\frac\left(1-\frac\right)\sin(u)+15
\end

where

0\le u<2\pi

and

-2\pi\le v <2\pi

\\ Various authors have suggested different models for the shape of shell.

David M. Raup David M. Raup (April 24, 1933 – July 9, 2015) was a University of Chicago paleontologist. Raup studied the fossil record and the diversity of life on Earth. Raup contributed to the knowledge of extinction events along with his colleague Jack S ...

proposed a model where there is one magnification for the x-y plane, and another for the x-z plane. Chris IllertDr Chris Illert was awarded his Ph.D. on 26 September 2013 at the University of Western Sydney http://www.uws.edu.au/__data/assets/image/0004/547060/2013_ICS_Graduates.jpg. proposed a model where the magnification is scalar, and the same for any sense or direction with an equation like :

\vec\left(  \right) = e^ \left(  \right)\vec\left(  \right)

which starts with an initial generating curve

\vec\left(  \right)

and applies a rotation and exponential magnification.

Parametrization

See also