Seiffert's spiral

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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.

Contents

Formulation

Symbols

cylindrical radius
angle of curve from beginning of spiral to a particular point on the spiral

basic Jacobi Elliptic Function [2]
Jacobi Theta Functions (where the kind of Theta Functions show) [3]
elliptic modulus (any positive real constant) [4]

Representation via equations

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

or expressed as Jacobi theta functions

. [5]

See also

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References

  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.