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Hyperbola Evolute


HyperbolaEvolute

The evolute of a hyperbola with parametric equations

x=acosht
(1)
y=bsinht
(2)

is

x_e=((a^2+b^2))/acosh^3t
(3)
y_e=-((a^2+b^2))/bsinh^3t,
(4)

which is similar to a Lamé curve, but with a minus sign. Eliminating t gives the implicit Cartesian equation for the evolute as

 (ax)^(2/3)-(by)^(2/3)=(a^2+b^2)^(2/3).
(5)

From a point between the two branches of the evolute, two normals can be drawn to the hyperbola. However, from a point beyond the evolute, four normals can be drawn.


See also

Evolute, Hyperbola

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Cite this as:

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

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