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 = [画像:(a-b)cost+hcos((a-b)/bt)]

(1)

y = [画像:(a-b)sint-hsin((a-b)/bt),]

(2)

A polar equation can be derived by computing

r^2 = x^2+y^2

(3)

= [画像:(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

[画像: 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.

curve special values

ellipse a=2b

hypocycloid h=b

rose curve h=a-b

The arc length, curvature, and tangential angle are

s(t) = [画像:2|(a-b)(b-h)|E((at)/(2b),(2isqrt(bh))/(|b-h|))]

(6)

kappa(t) = [画像:(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) = [画像: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.

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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." https://mathshistory.st-andrews.ac.uk/Curves/Hypotrochoid/.

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Hypotrochoid

Cite this as:

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

Hypotrochoid

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