Inverse Curve

Given a circle C with center O and radius k, then two points P and Q are inverse with respect to C if OP·OQ=k^2. If P describes a curve C_1, then Q describes a curve C_2 called the inverse of C_1 with respect to the circle C (with inversion center O). The Peaucellier inversor can be used to construct an inverse curve from a given curve.

If the polar equation of C is r(theta), then the inverse curve has polar equation


If O=(x_0,y_0) and P=(f(t),g(t)), then the inverse has equations


See also

Inversion, Inversion Center, Inversion Circle, Peaucellier Inversor, Reciprocal, Reciprocal Curve, Reciprocation

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Lawrence, J. D. "Inversion." §2.3 in A Catalog of Special Plane Curves. New York: Dover, pp. 43-46 and 203, 1972.Welke, S. "Inversion of Elementary Algebraic Curves with Respect to a Circle." Mathematica Educ. Res. 4, 16-22, 1995.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 120, 1991.Yates, R. C. "Inversion." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 127-134, 1952.

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Inverse Curve

Cite this as:

Weisstein, Eric W. "Inverse Curve." From MathWorld--A Wolfram Web Resource.

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