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

$$x(\psi) = {\textstyle{1\over 2}}r(1+\cos\theta)\cos\psi-{\textstyle{1\over 2}}r(1-\cos\theta)\cos\left({{1+\cos\theta\over 1-\cos\theta}\psi}\right)$$ (1)

$$y(\psi) = {\textstyle{1\over 2}}r(1+\cos\theta)\sin\psi-{\textstyle{1\over 2}}r(1-\cos\theta)\sin\left({{1+\cos\theta\over 1-\cos\theta}\psi}\right)$$ (2)

$$z(\psi) = r\sin\theta\cos\left({{\cos\theta\over 1-\cos\theta}\psi}\right).$$ (3)

The projection on the $xy$-plane is an Epicycloid with Radii

$$a = r\cos\theta$$ (4)

$$b = r\sin^2({\textstyle{1\over 2}}\theta).$$ (5)

See also Helix, Loxodrome, Spherical Spiral

References

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