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Parabola Evolute


ParabolaEvolute

Given a parabola with parametric equations

x=at^2
(1)
y=at,
(2)

the evolute is given by

x_e=1/2a(1+6t^2)
(3)
y_e=-4at^3.
(4)

Eliminating x and y gives the implicit equation

 (2x-a)^3=(27)/2ay^2.
(5)

With a=1, this can be solved for x to give

 x=1/2+3/4(2y)^(2/3),
(6)

which is known as the semicubical parabola.

From a point above the evolute, three normals can be drawn to the parabola, while only one normal can be drawn to the parabola from a point below the evolute.


See also

Evolute, Parabola, Semicubical Parabola

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Cite this as:

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

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