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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 )

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.

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.
 
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© Robert FERRÉOL  2017