Envelope Theorem

Relates Evolutes to single paths in the Calculus of Variations. Proved in the general case by Darboux and Zermelo (1894) and Kneser (1898). It states: ``When a single parameter family of external paths from a fixed point $O$ has an Envelope, the integral from the fixed point to any point $A$ on the Envelope equals the integral from the fixed point to any second point $B$ on the Envelope plus the integral along the envelope to the first point on the Envelope, $J_{OA}=J_{OB}+J_{BA}$.''

References

Kimball, W. S. Calculus of Variations by Parallel Displacement. London: Butterworth, p. 292, 1952.