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Epicycloid

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An epicycloid is 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 therefore an epitrochoid with h=b. Epicycloids are given by the parametric equations

x=(a+b)cosphi-bcos((a+b)/bphi)
(1)
y=(a+b)sinphi-bsin((a+b)/bphi).
(2)

A polar equation can be derived by computing

x^2=(a+b)^2cos^2phi-2b(a+b)cosphicos((a+b)/bphi)+b^2cos^2((a+b)/bphi)
(3)
y^2=(a+b)^2sin^2phi-2b(a+b)sinphisin((a+b)/bphi)+b^2sin^2((a+b)/bphi),
(4)

so

r^2=x^2+y^2=(a+b)^2+b^2-2b(a+b){cos[(a/b+1)phi]cosphi+sin[(a/b+1)phi]sinphi}.
(5)

But

cosalphacosbeta+sinalphasinbeta=cos(alpha-beta),
(6)

so

r^2=(a+b)^2+b^2-2b(a+b)cos[(a/b+1)phi-phi]
(7)
=(a+b)^2+b^2-2b(a+b)cos(a/bphi).
(8)

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

tantheta=y/x=((a+b)sinphi-bsin((a+b)/bphi))/((a+b)cosphi-bcos((a+b)/bphi)).
(9)

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[(1+1/n)^2+(1/n)^2-2(1/n)(1+1/n)cos(nphi)]
(10)
=a^2[1+2/n+1/(n^2)+1/(n^2)-(2/n)((n+1)/n)cos(nphi)]
(11)
=a^2[(n^2+2n+2)/(n^2)-(2(n+1))/(n^2)cos(nphi)]
(12)
=(a^2)/(n^2)[(n^2+2n+2)-2(n+1)cos(nphi)],
(13)

so

tantheta=(a((n+1)/n)sinphi-a/nsin[(n+1)phi])/(a((n+1)/n)cosphi-a/ncos[(n+1)phi])
(14)
=((n+1)sinphi-sin[(n+1)phi])/((n+1)cosphi-cos[(n+1)phi]).
(15)

For positive integer n=a/b, the enclosed area and total arc length are

A=pi(a+b)(a+2b)=(pia^2(n+1)(n+2))/(n^2)
(16)
L=8(a+b)=(8a(n+1))/n.
(17)

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.

When the two circles have equal radii, the traced epicycloid is a cardioid. During one circuit of the fixed circle, the rolling circle turns once relative to the fixed circle and its center makes one turn about the fixed center. Its orientation therefore makes two complete turns relative to the plane, which is the coin paradox (Veritasium 2024).

EpicycloidConstruction

Epicycloids can also be constructed by beginning with the diameter of a circle and offsetting one end by a series of steps of equal arc length along the circumference while at the same time offsetting the other end along the circumference by steps n+1 times as large. After traveling around the circle once, the envelope of an n-cusped epicycloid is produced, as illustrated above (Madachy 1979).

Epicycloids have torsion

tau=0
(18)

and satisfy

(s^2)/(a^2)+(rho^2)/(b^2)=1,
(19)

where rho is the radius of curvature (1/kappa).

See also

Cardioid, Cyclide, Cycloid, Epicycloid Evolute, Epicycloid Involute, Epicycloid Pedal Curve, Epitrochoid, Hypocycloid, Nephroid, Ranunculoid

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 217, 1987.Bogomolny, A. "Cycloids." https://www.cut-the-knot.org/pythagoras/cycloids.shtml.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 160-164 and 169, 1972.Lemaire, J. Hypocycloïdes et epicycloïdes. Paris, France: Albert Blanchard, 1967.MacTutor History of Mathematics Archive. "Epicycloid." https://mathshistory.st-andrews.ac.uk/Curves/Epicycloid/.Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 219-225, 1979.Smith, D. E. History of Mathematics, Vol. 2: Special Topics of Elementary Mathematics. New York: Dover, p. 328, 1958.Veritasium. "The SAT Question Everyone Got Wrong." Nov. 30, 2023. https://www.youtube.com/watch?v=FUHkTs-Ipfg.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.

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Cite this as:

Weisstein, Eric W. "Epicycloid." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Epicycloid.html

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