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Ellipse Evolute


EllipseEvolute

The evolute of an ellipse specified parametrically by

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

is given by the parametric equations

x_e=(a^2-b^2)/acos^3t
(3)
y_e=(b^2-a^2)/bsin^3t.
(4)

Eliminating t allows this to be written

(ax)^(2/3)+(by)^(2/3)=[(a^2-b^2)cos^3t]^(2/3)+[(b^2-a^2)sin^3t]^(2/3)
(5)
=(a^2-b^2)^(2/3)(sin^2t+cos^2t)
(6)
=(a^2-b^2)^(2/3)
(7)
=c^(4/3),
(8)

which is a stretched astroid sometimes known as the Lamé curve.

From a point inside the evolute, four normal vectors can be drawn to the ellipse, from a point on the evolute precisely, three normals can be drawn, and from a point outside, only two normal vectors can be drawn.

The arc length and area enclosed are

s=(4b^2)/a
(9)
A=(3pi(a^2-b^2)^2)/(8ab),
(10)

and the curvature, and tangential angle are

kappa(t)=(a^2b^2)/(3|(a^2-b^2)costsint|(b^2cos^2t+a^2sin^2t)^(3/2))
(11)
phi(t)=tan^(-1)((atant)/b).
(12)

See also

Astroid, Ellipse, Ellipse Involute, Evolute

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 217, 1987.Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 99-101, 1997.

Cite this as:

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

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