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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 $OP\cdot OQ = k^2$. 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 $O$). If the Polar equation of $C$ is $r(\theta)$, then the inverse curve has polar equation

r={k^2\over r(\theta)}.

If $O=(x_0,y_0)$ and $P=(f(t),g(t))$, then the inverse has equations
$\displaystyle x$ $\textstyle =$ $\displaystyle x_0+{k^2(f-x_0])\over(f-x_0)^2+(g-y_0)^2}$  
$\displaystyle y$ $\textstyle =$ $\displaystyle y_0+{k^2(g-y_0)\over(f-x_0)^2+(g-y_0)^2}.$  

Curve Inversion Center Inverse Curve
Archimedean Spiral Origin Archimedean Spiral
Cardioid Cusp Parabola
Circle any point another Circle
Cissoid of Diocles Cusp Parabola
Cochleoid Origin Quadratrix of Hippias
Epispiral Origin Rose
Fermat's Spiral Origin Lituus
Hyperbola center Lemniscate
Hyperbola Vertex Right Strophoid
Hyperbola with $a=\sqrt{3}$ Vertex Maclaurin Trisectrix
Lemniscate center Hyperbola
Lituus Origin Fermat's Spiral
Logarithmic Spiral Origin Logarithmic Spiral
Maclaurin Trisectrix Focus Tschirnhausen's Cubic
Parabola Focus Cardioid
Parabola Vertex Cissoid of Diocles
Quadratrix of Hippias   Cochleoid
Right Strophoid Origin the same Right Strophoid Inverse Curve
Sinusoidal Spiral Origin Sinusoidal Spiral
Tschirnhausen Cubic   Sinusoidal Spiral

See also Inversion, Inversion Center, Inversion Circle


Lee, X. ``Inversion.''

Lee, X. ``Inversion Gallery.''

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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© 1996-9 Eric W. Weisstein