TOPICS
Search

Trochoid

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook
Trochoid

A trochoid is the locus of a point at a distance b from the center of a circle of radius a rolling on a fixed line. A trochoid has parametric equations

x=aphi-bsinphi
(1)
y=a-bcosphi.
(2)

If b<a, the trochoid is known as a curtate cycloid; if b=a, it is a cycloid; and if b>a, the curve is a prolate cycloid.

The arc length function, curvature, and tangential angle are given by

s(t)=2|a-b|E(t/2,(2isqrt(ab))/(|a-b|))
(3)
kappa(t)=(b(acost-b))/((a^2+b^2-2abcost)^(3/2))
(4)
phi(t)=-t/2+(pi|a-b|)/(2(a-b))-tan^(-1)[(a-b)/(a+b)cot(t/2)]+pi|_t/(2pi)_|,
(5)

where E(t,k) is an incomplete elliptic integral of the second kind and |_x_| is the floor function.

See also

Curtate Cycloid, Cycloid, Epitrochoid, Hypotrochoid, Prolate Cycloid, Roulette

Explore with Wolfram|Alpha

References

Hall, L. "Trochoids, Roses, and Thorns--Beyond the Spirograph." College Math. J. 23, 20-35, 1992.Wagon, S. Mathematica in Action. New York: W. H. Freeman, pp. 46-50, 1991.Yates, R. C. "Trochoids." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 233-236, 1952.

Cite this as:

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

Subject classifications