Envelope

The envelope of a one-parameter family of curves given implicitly by

$$U(x,y,c)=0,$$ (1)

or in parametric form by $(f(t,c), g(t,c))$, is a curve which touches every member of the family. For a curve represented by $(f(t,c), g(t,c))$, the envelope is found by solving

$$0={\partial f\over\partial t}{\partial g\over\partial c}-{\partial f\over\partial c}{\partial g\over\partial t}.$$ (2)

For a curve represented implicitly, the envelope is given by simultaneously solving

$${\partial U\over\partial c}=0$$ (3)

$$U(x,y,c)=0.$$ (4)

See also Astroid, Cardioid, Catacaustic, Caustic, Cayleyian Curve, Dürer's Conchoid, Ellipse Envelope, Envelope Theorem, Evolute, Glissette, Hedgehog, Kiepert's Parabola, Lindelof's Theorem, Negative Pedal Curve

References

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 33-34, 1972.

Lee, X. ``Envelope.'' http://www.best.com/~xah/SpecialPlaneCurves_dir/Envelope_dir/envelope.html.

Yates, R. C. ``Envelopes.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 75-80, 1952.