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(RECIPROCAL) POLAR OF A CURVE
Other name: dual curve. 
The polar with respect to the circle with centre O
and radius a of the point M_{0}
(x_{0,
}y_{0})
is the line with equation .
If M_{0} is the current point on , then the current point M of the polar of is defined by which yields: in Cartesian coordinates, i.e. in complex parametrization. 
1) Reminder on poles and polars.
Two points M and M_{0} are said to be conjugates of one another with respect to a circle (C) with centre O and radius a when the circle with diameter [M_{0}M] is orthogonal to (C), i.e. ; this comes from the fact that when the line (M_{0}M) cuts the circle at A and B, M and M_{0} are harmonic conjugates with respect to [AB].
The polar of a point M_{0}
with respect to (C) is then the locus of the conjugates M
of the point M_{0} with respect to
the circle, and therefore defined by the relation ;
it is the line orthogonal to the line (OM_{0})
passing by the inverse of M_{0} with
respect to (C); it is also the radical axis of the circle (C)
and of the circle with diameter [OM_{0}].
When M_{0} is outside (C), it
is the line joining the contact points of the tangents to the circle that
pass by M_{0}. Conversely, a line is
the polar of a unique point, called its pole.
Despite its relation to the inversion, the natural space
of this transformation is not the conformal plane, but the projective plane:
the polar of the point O then is the line at infinity, and the polar
of a point at infinity is the line passing by O and perpendicular
to the direction of the point.
2) Definition of the polar of a curve.
The polar of a plane curve
with respect to a circle (C) (the "directrix (C)") is the
envelope of the polars of the points
with respect to (C); it can be proved that it also is the set of
the poles of the tangents to
with respect to (C).
The characteristic point of the polar of M_{0} is also the pole of the tangent to at M_{0}; then, it is the intersection point between the polar and the line perpendicular to this tangent passing by O. Therefore, the point N describes the pedal of , and the point M describes the inverse of the polar. 
This transformation, called reciprocal polar transformation is an involution, that is, the polar of the polar is equal to the initial curve.
The inverse of the polar with respect to the same circle is none other than the pedal of the initial curve; we can sum this up with this diagram:
See two examples opposite: 

The polar of an algebraic curve is an algebraic curve the degree of which is equal to the class of the initial curve (i.e. the degree of the tangential equation).
Examples:
initial curve  position of the centre of (C) with respect to the initial curve  position of the centre of (C) with respect to the polar  polar 
line (polar of the point)  outside of the line  different from the point  point (pole of the line) 
conic  conic  
conic  focus  circle  
conic  inside the conic (i.e. in a domain containing a focus)  ellipse  
conic  outside the conic  hyperbola  
conic  on the conic  parabola  
cardioid  cuspidal point  focus at the 8/9 of the segment line joining the double point to the summit.  Tschirnhausen cubic 
cardioid  centre of the conchoidal circle  focus  Maclaurin trisectrix 
deltoid  centre  vertex  duplicatrix cubic 
astroid  centre  centre  cross curve 
centred cycloid  centre  centre  epispiral 
sinusoidal spiral with parameter .  centre  centre  sinusoidal spiral with parameter . 
See the 3D
generalisation of these notions.
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© Robert FERRÉOL 2017