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).

See also

Apollonius' Problem, Dual Polyhedron, Inverse Points, Inversion Circle, Inversion Pole, Polar Equation, Polar Plot, Polarity, Reciprocal, Reciprocation, Salmon's Theorem, Trilinear Polar

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References

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, Ireland: 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, England: 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, England: Macmillian, pp. 151-157, 1893.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London, England: Penguin, pp. 190-191, 1991.

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Polar

Cite this as:

Weisstein, Eric W. "Polar." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Polar.html

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