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Spherical Helix

The tangent indicatrix of a curve of constant precession is a spherical helix. The equation of a spherical helix on a sphere with radius r making an angle theta with the z-axis is

x(psi)=1/2r(1+costheta)cospsi-1/2r(1-costheta)cos((1+costheta)/(1-costheta)psi)
(1)
y(psi)=1/2r(1+costheta)sinpsi-1/2r(1-costheta)sin((1+costheta)/(1-costheta)psi)
(2)
z(psi)=rsinthetacos((costheta)/(1-costheta)psi).
(3)

The projection on the xy-plane is an epicycloid with radii

a=rcostheta
(4)
b=rsin^2(1/2theta).
(5)

See also

Helix, Loxodrome, Spherical Spiral

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References

Scofield, P. D. "Curves of Constant Precession." Amer. Math. Monthly 102, 531-537, 1995.

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Spherical Helix

Cite this as:

Weisstein, Eric W. "Spherical Helix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SphericalHelix.html

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