The necessary and sufficient condition that an algebraic curve has an algebraic
involute is that the arc length
is a two-valued algebraic function of the coordinates of the extremities. Furthermore,
this function is a root of a quadratic
equation whose coefficients are rational functions
of 
and 
.
Humbert's Theorem
See also
Algebraic Curve, InvoluteExplore with Wolfram|Alpha
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 195, 1959.Referenced on Wolfram|Alpha
Humbert's TheoremCite this as:
Weisstein, Eric W. "Humbert's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HumbertsTheorem.html