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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

$\displaystyle x(\psi)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}r(1+\cos\theta)\cos\psi-{\textstyle{1\over 2}}r(1-\cos\theta)\cos\left({{1+\cos\theta\over 1-\cos\theta}\psi}\right)$ (1)
$\displaystyle y(\psi)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}r(1+\cos\theta)\sin\psi-{\textstyle{1\over 2}}r(1-\cos\theta)\sin\left({{1+\cos\theta\over 1-\cos\theta}\psi}\right)$ (2)
$\displaystyle z(\psi)$ $\textstyle =$ $\displaystyle r\sin\theta\cos\left({{\cos\theta\over 1-\cos\theta}\psi}\right).$ (3)

The projection on the $xy$-plane is an Epicycloid with Radii
$\displaystyle a$ $\textstyle =$ $\displaystyle r\cos\theta$ (4)
$\displaystyle b$ $\textstyle =$ $\displaystyle r\sin^2({\textstyle{1\over 2}}\theta).$ (5)

See also Helix, Loxodrome, Spherical Spiral


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

© 1996-9 Eric W. Weisstein