INVERSE OF A CURVE WITH RESPECT TO A POINT

 Notion studied by Steiner in 1824.

 Polar equation: inverse of the curve with respect to O (reference circle ). In cartesian coordinates: , inverse of the curve .

The inverse of a curve with respect to a point is the image of this curve by an inversion with pole the considered point.

The inverse of an algebraic curve of degree n, with the cyclic points as points of order p, with respect to an inversion pole of order q (for the initial curve), is a curve of degree 2n  2p  q, with the cyclic points as points of order n  p  q and the pole as a point of order n  2p.

 The inversion in the plane of center O and of radius a can be realized by the composition of the three following transformations of the space: 1) the central projection of center N (0, 0, a) on the sphere of center O and radius a: (blue curve >   red curve opposite).  2) the reflection in respect of xOy: (red curve > black curve opposite). 3) the central projection of center N (0, 0, a) on the xOy plane: (black curve > green curve opposite). Examples:

 initial curve inversion centre (position with respect to the initial line) inversion centre (position with respect to the inverse curve) inverse curve line outside of the line on the circle circle circle outside of the circle outside of the circle circle conic on the conic singular point rational circular cubic (line if the pole is at a vertex of the conic) hyperbola on the hyperbola double point circular cubic with a double point rectangular hyperbola on the hyperbola double point strophoid (line if the pole is at a vertex of the hyperbola) hyperbola with eccentricity 2 vertex double point Maclaurin trisectrix parabola on the parabola cuspidal point cissoid (line if the pole is at a vertex of the parabola) ellipse on the ellipse isolated point rational circular cubic with an isolated point (including the visiera) non-circular conic outside the conic real singular point rational bicircular quartic non-circular conic focus pole limaçon of Pascal parabola focus cuspidal point cardioid centred conic centre centre Booth curve rectangular hyperbola centre centre lemniscate of Bernoulli sinusoidal spiral pole pole sinusoidal spiral Tschirnhausen cubic focus summit of the loop Cayley sextic rose centre centre epispiral trisectrix limaçon (which is a rose) summit of the interior loop double point Maclaurin trisectrix (which is an epispiral) nodal curve centre centre same nodal curve (with a rotation) lemniscate of Bernoulli focus ? limaçon of Pascal with a double point where the tangents are perpendicular lemniscate of Bernoulli on the curve ? strophoid simple folium "point" vertex isolated point duplicatrix cubic Kampyle of Eudoxus centre centre double egg

See also the anallagmatic curves, which are their own inverses.

© Robert FERRÉOL  2019