Seiffert's spiral
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.[1] The cylindrical coordinates of the varying point on this curve are given by the Jacobian elliptic functions.
Formulation
Symbols
| [math]\displaystyle{ r }[/math] | cylindrical radius |
| [math]\displaystyle{ \theta }[/math] | angle of curve from beginning of spiral to a particular point on the spiral |
| [math]\displaystyle{ \operatorname{sn}(s, k) }[/math]
[math]\displaystyle{ \operatorname{cn}(s, k) }[/math] |
basic Jacobi Elliptic Function[2] |
| [math]\displaystyle{ \vartheta_{i}(s) }[/math] | Jacobi Theta Functions (where [math]\displaystyle{ i }[/math] the kind of Theta Functions show)[3] |
| [math]\displaystyle{ k }[/math] | elliptic modulus (any positive real constant)[4] |
Representation via equations
The Seiffert's spherical spiral can be expressed in cylindrical coordinates as
[math]\displaystyle{ r = \operatorname{sn}(s, k),\, \theta = k \cdot s \text{ and } z = \operatorname{cn}(s, k) }[/math]
or expressed as Jacobi theta functions
[math]\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))} }[/math].[5]
See also
References
- ↑ Bowman, F (1961). Introduction to Elliptic Functions with Applications. New York: Dover. https://archive.org/details/introductiontoel0000bowm.
- ↑ Weisstein, Eric W.. "Jacobi Elliptic Functions" (in en). https://mathworld.wolfram.com/.
- ↑ Weisstein, Eric W.. "Jacobi Theta Functions" (in en). https://mathworld.wolfram.com/.
- ↑ W., Weisstein, Eric. "Elliptic Modulus -- from Wolfram MathWorld" (in en). https://mathworld.wolfram.com/EllipticModulus.html.
- ↑ Weisstein, Eric W.. "Seiffert's Spherical Spiral" (in en). https://mathworld.wolfram.com/.
- Seiffert, A. (1896), Ueber eine neue geometrische Einführung in die Theorie der elliptischen Functionen, 127, Wissenschaftliche Beilage zum Jahresbericht der Städtischen Realschule zu Charlottenburg, Ostern
- Erdös, Paul (2000), "Spiraling the Earth with C. G. J. Jacobi", American Journal of Physics 88 (10): 888–895, doi:10.1119/1.1285882
External links
- Weisstein, Eric W.. "Seiffert's Spherical Spiral". http://mathworld.wolfram.com/SeiffertsSphericalSpiral.html.
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