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- TL;DR Summary
- Is it possible to analytically invert this elliptic integral equation?

Hello,

For my own amusement, I am deriving the eqations for various roulettes, i.e. a the trace of a curve rolling on another curve.

When considering rolling ellipses, I encounter equations containing elliptic integrals (of the second kind) that need to be inverted.

For example, here is one such equation:

t = a * elliptic_e(u, E)

where a, E are positive, real contants and t, u are the real variables of concern.

(The notation is from Maxima: https://maxima.sourceforge.io/docs/manual/maxima_91.html)

In other words, I need to express u as a function of t.

Can this equation be analytically inverted?

For specific values of t, I can easily find a value for u by using a numerical root finding method but an exact, analytical answer would be preferable.

Another such equation is:

elliptic_e(u, Er) = a/ar * elliptic_e(t, Ef)

Again, I need to invert this equation to find u as a function of t (a, ar, Er, Ef are all positive real constants).

I know that the inverse of an elliptic integral is an elliptic function but I don't know how to invert these equations.

For my own amusement, I am deriving the eqations for various roulettes, i.e. a the trace of a curve rolling on another curve.

When considering rolling ellipses, I encounter equations containing elliptic integrals (of the second kind) that need to be inverted.

For example, here is one such equation:

t = a * elliptic_e(u, E)

where a, E are positive, real contants and t, u are the real variables of concern.

(The notation is from Maxima: https://maxima.sourceforge.io/docs/manual/maxima_91.html)

In other words, I need to express u as a function of t.

Can this equation be analytically inverted?

For specific values of t, I can easily find a value for u by using a numerical root finding method but an exact, analytical answer would be preferable.

Another such equation is:

elliptic_e(u, Er) = a/ar * elliptic_e(t, Ef)

Again, I need to invert this equation to find u as a function of t (a, ar, Er, Ef are all positive real constants).

I know that the inverse of an elliptic integral is an elliptic function but I don't know how to invert these equations.