Seiffert's spherical spiral is a curve on a sphere made by moving on the sphere with constant speed and angular velocity with respect to a fixed diameter. If the selected diameter is the line from the north pole to the south pole, then the requirement of constant angular velocity means that the longitude of the moving point changes at a constant rate. The cylindrical coordinates of the varying point on this curve are given by the Jacobian elliptic functions.
Formulation
Symbols
Representation via equations The Seiffert's spherical spiral can be expressed in cylindrical coordinates as
r = sn ( s , k ) , θ = k ⋅ s and z = cn ( s , k ) {\displaystyle r=\operatorname {sn} (s,k),\,\theta =k\cdot s{\text{ and }}z=\operatorname {cn} (s,k)}
or expressed as Jacobi theta functions
r = ϑ 3 ( 0 ) ⋅ ϑ 1 ( s ⋅ ϑ 3 − 2 ( 0 ) ) ϑ 2 ( 0 ) ⋅ ϑ 4 ( s ⋅ ϑ 3 − 2 ( 0 ) ) , θ = ϑ 2 2 ( q ) ϑ 3 2 ( q ) ⋅ s and z = ϑ 4 ( 0 ) ⋅ ϑ 3 ( s ⋅ ϑ 3 − 2 ( 0 ) ) ϑ 3 ( 0 ) ⋅ ϑ 4 ( s ⋅ ϑ 3 − 2 ( 0 ) ) {\displaystyle r={\frac {\vartheta _{3}(0)\cdot \vartheta _{1}(s\cdot \vartheta _{3}^{-2}(0))}{\vartheta _{2}(0)\cdot \vartheta _{4}(s\cdot \vartheta _{3}^{-2}(0))}},\,\theta ={\frac {\vartheta _{2}^{2}(q)}{\vartheta _{3}^{2}(q)}}\cdot s{\text{ and }}z={\frac {\vartheta _{4}(0)\cdot \vartheta _{3}(s\cdot \vartheta _{3}^{-2}(0))}{\vartheta _{3}(0)\cdot \vartheta _{4}(s\cdot \vartheta _{3}^{-2}(0))}}} .
See also Rhumb line
References
Seiffert, A. (1896), Ueber eine neue geometrische Einführung in die Theorie der elliptischen Functionen, vol. 127, Wissenschaftliche Beilage zum Jahresbericht der Städtischen Realschule zu Charlottenburg, Ostern, JFM 27.0337.02 Erdös, Paul (2000), "Spiraling the Earth with C. G. J. Jacobi", American Journal of Physics, 88 (10): 888–895, Bibcode:2000AmJPh..68..888E, doi:10.1119/1.1285882
External links Weisstein, Eric W. "Seiffert's Spherical Spiral". MathWorld.