The simple requirement that one should move on the surface of a sphere with constant speed while maintaining a constant angular velocity with respect to a fixed diameter, leads to a path whose cylindrical coordinates turn out to be given by the Jacobian elliptic functions. Many properties of these functions can be derived and visualized using this path, known as Seiffert’s spiral.
REFERENCES
1.
C. G. J. Jacobi, Fundamenta Nova Theoriae Functionum Ellipticarum (Königsberg, 1829).
2.
Alfred Seiffert, “Über eine neue geometrische Einführung in die Theorie der elliptischen Funktionen,” Wissenschaftliche Beilage zum Jahresbericht der Städtischen Realschule zu Charlottenburg, Ostern, 1896.
3.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge U.P., Cambridge, 1958), pp. 527–528.
4.
Instead of using the geographic East and West longitude, we measure φ increasing in the direction opposite to the Earth’s rotation, and keep calling this quantity “longitude.”
5.
For this special case one would have to set but Eq. (2.10) remains valid.
6.
The case represents the trivially closed meridian circle.
7.
L. M. Milne-Thomson, “Jacobian Elliptic Functions and Theta Functions,” in Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (Dover, New York, 1972), 9th printing, p. 573.
8.
One can show that the lengths of sp 2 and s of sp 1 are related by and since s is linearly proportional to φ, is not.
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© 2000 American Association of Physics Teachers.
2000
American Association of Physics Teachers
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