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Hypotrochoid

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HypotrochoidDiagram1
Hypotrochoid1
HypotrochoidDiagram2
Hypotrochoid2

A hypotrochoid is a roulette traced by a point P attached to a circle of radius b rolling around the inside of a fixed circle of radius a, where P is a distance h from the center of the interior circle. The parametric equations for a hypotrochoid are

x=(a-b)cost+hcos((a-b)/bt)
(1)
y=(a-b)sint-hsin((a-b)/bt),
(2)

A polar equation can be derived by computing

r^2=x^2+y^2
(3)
=(a-b)^2+h^2+2(a-b)hcos((at)/b).
(4)

Here, the parameter t is not the polar angle theta but is related to it by

tantheta=y/x=((a-b)sint-hsin((a/b-1)t))/((a-b)cost+hcos((a/b-1)t)).
(5)

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

Special cases of the hypotrochoid are summarized in the table below.

curvespecial values
ellipsea=2b
hypocycloidh=b
rose curveh=a-b

The arc length, curvature, and tangential angle are

s(t)=2|(a-b)(b-h)|E((at)/(2b),(2isqrt(bh))/(|b-h|))
(6)
kappa(t)=(b^3-(a-b)h^2+(a-2b)bhcos((at)/b))/(|a-b|[b^2+h^2-2bhcos((at)/b)]^(3/2))
(7)
phi(t)=t(1-a/(2b))+cot^(-1)[(b-h)/(b+h)cot((at)/(2b))],
(8)

where E(x,k) is an elliptic integral of the second kind.

See also

Ellipse, Epitrochoid, Hypocycloid, Hypotrochoid Evolute, Rose Curve, Spirograph, Trochoid

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References

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 165-168, 1972.MacTutor History of Mathematics Archive. "Hypotrochoid." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Hypotrochoid.html.

Cite this as:

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

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