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.

Seashell surface with parametrization on left

Wheel-like Star Shell Astralium calcar, Diameter 3,5 cm; Originating from the Philippines

Parametrization

The following is a parameterization of one seashell surface:

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

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

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^[1] proposed a model where the magnification is scalar, and the same for any sense or direction with an equation like

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

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

See also

• Helix
• Seashell
• Spiral