Conical Spiral

$\begin{figure}\begin{center}\BoxedEPSF{ConicalSpiral1.epsf scaled 700}\quad\BoxedEPSF{ConicalSpiral2.epsf scaled 1200}\end{center}\end{figure}$

A surface modeled after the shape of a Seashell. One parameterization (left figure) is given by

$\displaystyle x$	$\textstyle =$	$\displaystyle 2[1 - e^{u/(6\pi)}]\cos u\cos^2({\textstyle{1\over 2}}v)$	(1)
$\displaystyle y$	$\textstyle =$	$\displaystyle 2[-1 + e^{u/(6\pi)}]\cos^2({\textstyle{1\over 2}}v)\sin u$	(2)
$\displaystyle z$	$\textstyle =$	$\displaystyle 1 - e^{u/(3\pi)} - \sin v + e^{u/(6\pi)}\sin v,$	(3)

where $v\in [0,2\pi)$ , and $u\in [0,6\pi)$ (Wolfram). Nordstrand gives the parameterization

$\displaystyle x$	$\textstyle =$	$\displaystyle \left[{\left({1-{v\over 2\pi}}\right)(1+\cos u)+c}\right]\cos(nv)$	(4)
$\displaystyle x$	$\textstyle =$	$\displaystyle \left[{\left({1-{v\over 2\pi}}\right)(1+\cos u)+c}\right]\sin(nv)$	(5)
$\displaystyle z$	$\textstyle =$	$\displaystyle {bv\over 2\pi}+a\sin u\left({1-{v\over 2\pi}}\right)$	(6)

for $u,v\in [0,2\pi]$ (right figure with

, and

References

Gray, A. ``Sea Shells.'' §11.6 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 223-223, 1993.

Nordstrand, T. ``Conic Spiral or Seashell.'' http://www.uib.no/people/nfytn/shelltxt.htm.