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Mathematics
Differential Equations
Inverting an Equation Containing Elliptic Integrals
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[QUOTE="diegogarcia, post: 6596451, member: 698571"] [B]TL;DR Summary:[/B] 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: [URL]https://maxima.sourceforge.io/docs/manual/maxima_91.html[/URL]) 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. [/QUOTE]
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Inverting an Equation Containing Elliptic Integrals
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