Inverse of a curve

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.

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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