If two points 
and 
are inverse
(sometimes called conjugate) with respect to a circle
(the inversion circle), then the straight line through 
which is perpendicular
to the line of the points 
is called the polar of 
with respect to the circle, and 
is called the inversion pole
of the polar.
An incidence-preserving transformation in which points and lines are transformed into their inversion poles and polars is called reciprocation (a.k.a. constructing the dual).
The concept of poles and polars can also be generalized to arbitrary conic sections. If two tangents to a conic section
at points 
and 
meet at 
,
then 
is called the inversion
pole of the line 
with respect to the conic and 
is said to be the polar of the point 
with respect to the conic (Wells 1991).
In the above figure, let a line through the polar 
meet a conic section at point 
and 
,
and let the line 
intersect the polar line 
at 
.
Then 
form a harmonic
range (Wells 1991).
In the above figure, let two lines through the pole 
meet a conic at points 
, 
and 
, 
. Then 
and 
are concurrent on the polar, as are
the lines 
and 
(Wells 1991).
The concept can be generalized even further to an arbitrary algebraic curve so that every point has a polar with respect to the curve and every line has a pole (Wells 1991).
