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Seiffert's Spherical Spiral


SeiffertsSpiral

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

r=sn(s,k)
(1)
theta=ks
(2)
z=cn(s,k),
(3)

where k is a positive constant and sn(s) and cn(s) 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.


See also

Spherical Curve, Spherical Spiral

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References

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. https://mathworld.wolfram.com/SeiffertsSphericalSpiral.html

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