Spherical Spiral

The term "spherical spiral" is used for several spiral curves on a sphere. The curve illustrated above has parametric equations

			(1)
			(2)
			(3)

where

(4)

and is a constant. Substituting gives

			(5)
			(6)
			(7)

This curve is a spherical spiral of the type considered by Bay et al. (2023). Its angle with a meridian varies, so it is not a loxodrome.

A spherical spiral that crosses every meridian at a fixed angle is a loxodrome. On the unit sphere, one such curve has parametric equations

			(8)
			(9)
			(10)

where and . It travels from the south pole toward the north pole, crossing meridians at the constant angle east of north (Gray 1997, pp. 238-240).

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).