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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 $a=0.2$, $b=1$, $c=0.1$, and $n=2$).


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.''

mathematica.gif Wolfram Research ``Mathematica Version 2.0 Graphics Gallery.''

© 1996-9 Eric W. Weisstein