Spherical Spiral

The spherical curve 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

			(1)
			(2)
			(3)

where

(4)

and is a constant. Plugging in therefore gives

			(5)
			(6)
			(7)

It is a special case of a loxodrome.

The arc length, curvature, and torsion are all slightly complicated expressions.

A series of spherical spirals are illustrated in Escher's woodcuts "Sphere Surface with Fish" (Bool et al. 1982, pp. 96 and 318) and "Sphere Spirals" (Bool et al. 1982, p. 319; Forty 2003, Plate 67).