Epicycloid

The path traced out by a point $P$ on the Edge of a Circle of Radius $b$ rolling on the outside of a Circle of Radius $a$.

It is given by the equations

$$x = (a+b)\cos\phi-b\cos\left({{a+b\over b}\phi}\right)$$ (1)

$$y = (a+b)\sin\phi-b\sin\left({{a+b\over b}\phi}\right)$$ (2)

$$x^2 = (a+b)^2\cos^2\phi-2b(a+b)\cos\phi\cos\left({{a+b\over b}\phi}\right)+b^2\cos^2\left({{a+b\over b}\phi}\right)$$ (3)

$$y^2 = (a+b)^2\sin^2\phi-2b(a+b)\sin\phi\sin\left({{a+b\over b}\phi}\right)+b^2\sin^2\left({{a+b\over b}\phi}\right),$$ (4)

$r^2=x^2+y^2 = (a+b)^2+b^2$
$- 2b(a+b)\left\{{\cos\left[{\left({{a\over b}+1}\right)\phi }\right]\cos \phi+\sin\left[{\left({{a\over b}+1}\right)\phi}\right]\sin\phi}\right\}.\quad$	(5)

But

$$\cos\alpha\cos\beta+\sin\alpha\sin\beta=\cos(\alpha-\beta),$$ (6)

so

$$r^2 = (a+b)^2+b^2-2b(a+b)\cos\left[{\left({{a\over b}+1}\right)\phi-\phi}\right]$$

$$= (a+b)^2+b^2-2b(a+b)\cos\left({{a\over b}\phi}\right).$$ (7)

Note that $\phi$ is the parameter here, not the polar angle. The polar angle from the center is

$$\tan\theta = {y\over x}={(a+b)\sin\phi-b\sin\left({{a+b\over... ...t)\over (a+b)\cos\phi-b\cos\left({{a+b\over b}\phi}\right)}.$$ (8)

To get $n$ Cusps in the epicycloid, $b=a/n$, because then $n$ rotations of $b$ bring the point on the edge back to its starting position.

$$r^2 = a^2\left[{\left({1+{1\over n}}\right)^2+\left({1\over n}\right)^2-2\left({1\over n}\right)\left({1+{1\over n}}\right)\cos (n\phi )}\right]$$

$$= a^2\left[{1+{2\over n}+{1\over n^2}+{1\over n^2}-\left({2\over n}\right)\left({n+1\over n}\right)\cos (n\phi )}\right]$$

$$= a^2\left[{{n^2+2n+2\over n^2}-{2(n+1)\over n^2}\cos (n\phi )}\right]$$

$$= {a^2\over n^2}\left[{(n^2+2n+2)-2(n+1)\cos (n\phi)}\right],$$ (9)

so

$$\tan\theta = {a\left({n+1\over n}\right)\sin\phi-{a\over n}\sin[(n+1)\phi]\over a\left({n+1\over n}\right)\cos\phi-{a\over n}\cos[(n+1)\phi]}$$

$$= {(n+1)\sin\phi-\sin[(n+1)\phi]\over(n+1)\cos\phi-\cos[(n+1)\phi]}.$$ (10)

An epicycloid with one cusp is called a Cardioid, one with two cusps is called a Nephroid, and one with five cusps is called a Ranunculoid.

$n$-epicycloids can also be constructed by beginning with the Diameter of a Circle, offsetting one end by a series of steps while at the same time offsetting the other end by steps $n$ times as large. After traveling around the Circle once, an $n$-cusped epicycloid is produced, as illustrated above (Madachy 1979).

Epicycloids have Torsion

$$\tau=0$$ (11)

and satisfy

$${s^2\over a^2}+{\rho^2\over b^2}=1,$$ (12)

where $\rho$ is the Radius of Curvature ($1/\kappa$).

See also Cardioid, Cyclide, Cycloid, Epicycloid--1-Cusped, Hypocycloid, Nephroid, Ranunculoid

References

Bogomolny, A. ``Cycloids.'' http://www.cut-the-knot.com/pythagoras/cycloids.html.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 160-164 and 169, 1972.

Lee, X. ``Epicycloid and Hypocycloid.'' http://www.best.com/~xah/SpecialPlaneCurves_dir/EpiHypocycloid_dir/epiHypocycloid.html.

MacTutor History of Mathematics Archive. ``Epicycloid.'' http://www-groups.dcs.st-and.ac.uk/~history/Curves/Epicycloid.html.

Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 219-225, 1979.

Wagon, S. Mathematica in Action. New York: W. H. Freeman, pp. 50-52, 1991.

Yates, R. C. ``Epi- and Hypo-Cycloids.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 81-85, 1952.