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 = [画像:(a^2-b^2)/acos^3t]
(3)
y_e = [画像:(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 = [画像:(3pi(a^2-b^2)^2)/(8ab),]
(10)
and the curvature, and tangential angle are
kappa(t) = [画像:(a^2b^2)/(3|(a^2-b^2)costsint|(b^2cos^2t+a^2sin^2t)^(3/2))]
(11)
phi(t) = [画像:tan^(-1)((atant)/b).]
(12)
See also
Astroid, Ellipse, Ellipse Involute, EvoluteExplore with Wolfram|Alpha
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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.Referenced on Wolfram|Alpha
Ellipse EvoluteCite this as:
Weisstein, Eric W. "Ellipse Evolute." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EllipseEvolute.html