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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  and  . Then  form a harmonic range (Wells 1991).
In the above figure, let two lines through the polar  meet a conic at
points  and  and  and  . Then  and  are concurrent
on the polar (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).
Casey, J. "Theory of Poles and Polars, and Reciprocation." §6.7 in A Sequel to the First Six Books of the Elements of Euclid, Containing
an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl.
Dublin: Hodges, Figgis, & Co., pp. 141-148, 1888.
Dörrie, H. 100 Great Problems of Elementary Mathematics: Their History and
Solutions. New York: Dover, p. 157, 1965.
Durell, C. V. "Poles and Polars." Ch. 9 in Modern Geometry: The Straight Line and Circle. London:
Macmillan, pp. 93-97, 1928.
Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the
Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 100-106,
1929.
Lachlan, R. "Poles and Polars." §243-157 in An Elementary Treatise on Modern Pure Geometry. London:
Macmillian, pp. 151-157, 1893.
Wells, D. The Penguin Dictionary of Curious and Interesting Geometry.
London: Penguin, pp. 190-191, 1991.
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