The spherical curve obtained when moving along the surface of a sphere with constant speed, while maintaining a constant angular velocity with respect to a fixed diameter (Erdős 2000). This curve is given in cylindrical coordinates by the parametric equations

			(1)
			(2)
			(3)

where is a positive constant and and are Jacobi elliptic functions (Whittaker and Watson 1990, pp. 527-528).

Erdős (2000) provides a derivation of the equations of this curve, as well as an analysis of its properties, including conditions for obtaining periodic orbits.

Bowman, F. Introduction to Elliptic Functions, with Applications. New York: Dover, p. 34, 1961.

Erdős, P. "Spiraling the Earth with C. G. J. Jacobi." Amer. J. Phys. 68, 888-895, 2000.

Seiffert. "Über eine neue geometrische Einführung in die Theorie der elliptischen Funktionen." Wissensch. Beiträge Jahresber. Städtischen Realschule zu Charlottenburg, Ostern. 1896.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.

CITE THIS AS:

Weisstein, Eric W. "Seiffert's Spherical Spiral." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SeiffertsSphericalSpiral.html