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{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 $$-2\pi\le v <2\pi$$\\

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,} &  & 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.