Envelope

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

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

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

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.

Referenced on Wolfram|Alpha

Envelope

Cite this as:

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

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