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Circle Involute


CircleInvolute

The involute of the circle was first studied by Huygens when he was considering clocks without pendula for use on ships at sea. He used the circle involute in his first pendulum clock in an attempt to force the pendulum to swing in the path of a cycloid. For a circle of radius a,

x=acost
(1)
y=asint
(2)

the parametric equation of the involute is given by

x_i=a(cost+tsint)
(3)
y_i=a(sint-tcost).
(4)

The arc length, curvature, and tangential angle are

s(t)=1/2at^2
(5)
kappa(t)=1/(at)
(6)
phi(t)=t.
(7)

The Cesàro equation is

 rho^2=2as.
(8)

See also

Circle, Circle Evolute, Circle Involute Pedal Curve, Ellipse Involute, Goat Problem, Involute

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 220, 1987.Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 105, 1997.Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagination. New York: Chelsea, pp. 6-7, 1999.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 190-191, 1972.MacTutor History of Mathematics Archive. "Involute of a Circle." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Involute.html.

Cite this as:

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

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