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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 ).
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) 
noncircular conic  outside the conic  real singular point  rational bicircular quartic 
noncircular 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 2019