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

$$x = x_0+{k^2(f-x_0])\over(f-x_0)^2+(g-y_0)^2}$$

$$y = y_0+{k^2(g-y_0)\over(f-x_0)^2+(g-y_0)^2}.$$

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

References

Lee, X. ``Inversion.'' http://www.best.com/~xah/SpecialPlaneCurves_dir/Inversion_dir/inversion.html.

Lee, X. ``Inversion Gallery.'' http://www.best.com/~xah/SpecialPlaneCurves_dir/InversionGallery_dir/inversionGallery.html

Yates, R. C. ``Inversion.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 127-134, 1952.