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:

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

where $0\leq u<2\pi$ and $-2\pi \leq v<2\pi$\\

Various authors have suggested different models for the shape of shell. David M. Raup proposed a model where there is one magnification for the x-y plane, and another for the x-z plane. Chris Illert proposed a model where the magnification is scalar, and the same for any sense or direction with an equation like

    ${\vec {F}}\left({\theta ,\varphi }\right)=e^{\alpha \varphi }\left({\begin{array}{*{20}c}{\cos \left(\varphi \right),}&{-\sin(\varphi ),}&{\rm {0}}\\{\sin(\varphi ),}&{\cos \left(\varphi \right),}&0\\{0,}&{\rm {0,}}&1\\\end{array}}\right){\vec {F}}\left({\theta ,0}\right)$

which starts with an initial generating curve ${\vec {F}}\left({\theta ,0}\right)$ and applies a rotation and exponential magnification.

