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Ellipse Involute

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EllipseInvolute

The involute of an ellipse specified parametrically by

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

is given by the parametric equations

x_i=a[cost+(bE(t,e)sint)/(sqrt(a^2sin^2t+b^2cos^2t))]
(3)
y_i=b[sint+(bE(t,e)cost)/(sqrt(a^2sin^2t+b^2cos^2t))],
(4)

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

e=sqrt(1-(a^2)/(b^2)).
(5)

is the eccentricity.

The curvature and tangential angle are given by

kappa(t)=1/(bE(t,e))
(6)
phi(t)=tan^(-1)((atant)/b),
(7)

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

See also

Ellipse, Ellipse Evolute, Involute

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Cite this as:

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

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