Question

Find the evolute of hyperbola ​

Answer 1

Ànswer:

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.