Spherical Spiral

The path taken by a ship which travels from the south pole to the north pole of a Sphere while keeping a fixed (but not Right) Angle with respect to the meridians. The curve has an infinite number of loops since the separation of consecutive revolutions gets smaller and smaller near the poles. It is given by the parametric equations

$\displaystyle x$	$\textstyle =$	$\displaystyle \cos t\cos c$
$\displaystyle y$	$\textstyle =$	$\displaystyle \sin t\cos c$
$\displaystyle z$	$\textstyle =$	$\displaystyle -\sin c,$

where

$\begin{displaymath} c\equiv\tan^{-1}(at) \end{displaymath}$

and

is a constant.

References

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, p. 162, 1993.

Lauwerier, H. ``Spherical Spiral.'' In Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 64-66, 1991.