Seiffert's_spiral Knowpia

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 edit

Symbols edit


$r$	cylindrical radius
$\theta$	angle of curve from beginning of spiral to a particular point on the spiral
$\operatorname {sn} (s,k)$ $\operatorname {cn} (s,k)$	basic Jacobi Elliptic Function^[2]
$\vartheta _{i}(s)$	Jacobi Theta Functions (where $i$ the kind of Theta Functions show)^[3]
$k$	elliptic modulus (any positive real constant)^[4]

Representation via equations edit

The Seiffert's spherical spiral can be expressed in cylindrical coordinates as

$r=\operatorname {sn} (s,k),\,\theta =k\cdot s{\text{ and }}z=\operatorname {cn} (s,k)$

or expressed as Jacobi theta functions

$r={\frac {\vartheta _{3}(0)\cdot \vartheta _{1}(s\cdot \vartheta _{3}^{-2}(0))}{\vartheta _{2}(0)\cdot \vartheta _{4}(s\cdot \vartheta _{3}^{-2}(0))}},\,\theta ={\frac {\vartheta _{2}^{2}(q)}{\vartheta _{3}^{2}(q)}}\cdot s{\text{ and }}z={\frac {\vartheta _{4}(0)\cdot \vartheta _{3}(s\cdot \vartheta _{3}^{-2}(0))}{\vartheta _{3}(0)\cdot \vartheta _{4}(s\cdot \vartheta _{3}^{-2}(0))}}$ .^[5]

References edit

^ Bowman, F (1961). Introduction to Elliptic Functions with Applications. New York: Dover.
^ Weisstein, Eric W. "Jacobi Elliptic Functions". mathworld.wolfram.com. Retrieved 2023-01-31.
^ Weisstein, Eric W. "Jacobi Theta Functions". mathworld.wolfram.com. Retrieved 2023-01-31.
^ W., Weisstein, Eric. "Elliptic Modulus -- from Wolfram MathWorld". mathworld.wolfram.com. Retrieved 2023-01-31.{{cite web}}: CS1 maint: multiple names: authors list (link)
^ Weisstein, Eric W. "Seiffert's Spherical Spiral". mathworld.wolfram.com. Retrieved 2023-01-31.

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, doi:10.1119/1.1285882

External links edit

Weisstein, Eric W. "Seiffert's Spherical Spiral". MathWorld.

[1] Bowman, F (1961). Introduction to Elliptic Functions with Applications. New York: Dover.

[2] Weisstein, Eric W. "Jacobi Elliptic Functions". mathworld.wolfram.com. Retrieved 2023-01-31.

[3] Weisstein, Eric W. "Jacobi Theta Functions". mathworld.wolfram.com. Retrieved 2023-01-31.

[4] W., Weisstein, Eric. "Elliptic Modulus -- from Wolfram MathWorld". mathworld.wolfram.com. Retrieved 2023-01-31.{{cite web}}: CS1 maint: multiple names: authors list (link)

[5] Weisstein, Eric W. "Seiffert's Spherical Spiral". mathworld.wolfram.com. Retrieved 2023-01-31.

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Summary

Formulation edit

Symbols edit

Representation via equations edit

See also edit

References edit

External links edit