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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 that touches every member of the family tangentially.

For a curve represented by (f(t,c),g(t,c)), the envelope is found by solving

0=(partialf)/(partialt)(partialg)/(partialc)-(partialf)/(partialc)(partialg)/(partialt).
(2)

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

(partialU)/(partialc)=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 Parabola, Lindelof's Theorem, Negative Pedal Curve

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References

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 33-34, 1972.Yates, R. C. "Envelopes." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 75-80, 1952.

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Envelope

Cite this as:

Weisstein, Eric W. "Envelope." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Envelope.html

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