Ingeometry, apole andpolar are respectively a point and a line that have a unique reciprocal relationship with respect to a givenconic section.

Polar reciprocation in a given circle is the transformation of each point in the plane into its polar line and each line in the plane into its pole.

Pole and polar have several useful properties:

• If a pointP lies on the linel, then the poleL of the linel lies on the polarp of pointP.
• If a pointP moves along a linel, its polarp rotates about the poleL of the linel.
• If two tangent lines can be drawn from a pole to the conic section, then its polar passes through both tangent points.
• If a point lies on the conic section, its polar is the tangent through this point to the conic section.
• If a pointP lies on its own polar line, thenP is on the conic section.
• Each line has, with respect to a non-degenerated conic section, exactly one pole.

The pole of a lineL in acircleC is a pointQ that is theinversion inC of the pointP onL that is closest to the center of the circle. Conversely, thepolar line (orpolar) of a pointQ in a circleC is the lineL such that its closest pointP to the center of the circle is theinversion ofQ inC.

The relationship between poles and polars is reciprocal. Thus, if a pointA lies on the polar lineq of a pointQ, then the pointQ must lie on the polar linea of the pointA. The two polar linesa andq need not be parallel.

There is another description of the polar line of a pointP in the case that it lies outside the circleC. In this case, there are two lines throughP which aretangent to the circle, and the polar ofP is the line joining the two points of tangency (not shown here). This shows thatpole and polar line are concepts in theprojective geometry of theplane and generalize with anynonsingular conic in the place of the circleC.

The concepts ofa pole and its polar line were advanced inprojective geometry. For instance, the polar line can be viewed as the set ofprojective harmonic conjugates of a given point, the pole, with respect to a conic. The operation of replacing every point by its polar and vice versa is known as a polarity.

Apolarity is acorrelation that is also aninvolution.

For some pointP and its polarp, any other pointQ onp is the pole of a lineq throughP. This comprises a reciprocal relationship, and is one in which incidences are preserved.^[1]

The concepts of pole, polar and reciprocation can be generalized from circles to otherconic sections which are theellipse,hyperbola andparabola. This generalization is possible because conic sections result from a reciprocation of a circle in another circle, and the properties involved, such asincidence and thecross-ratio, are preserved under allprojective transformations.

A generalconic section may be written as a second-degree equation in theCartesian coordinates (x,y) of theplane

$${\displaystyle A_{xx}{x}^2+2A_{xy}xy+A_{yy}{y}^2+2B_{x}x+2B_{y}y+C=0}$$

whereA_xx,A_xy,A_yy,B_x,B_y, andC are the constants defining the equation. For such a conic section, the polar line to a given pole point(ξ,η) is defined by the equation

$${\displaystyle Dx+Ey+F=0}$$

whereD,E andF are likewise constants that depend on the pole coordinates(ξ,η)

The pole of the line${\displaystyle Dx+Ey+F=0}$, relative to the non-degenerated conic section$${\displaystyle A_{xx}{x}^2+2A_{xy}xy+A_{yy}{y}^2+2B_{x}x+2B_{y}y+C=0}$$can be calculated in two steps.

First, calculate the numbers x, y and z from

Now, the pole is the point with coordinates${\displaystyle \left(\frac{x}{z},\frac{y}{z}\right)}$

• Pole-polar relation for an ellipse
• Pole-polar relation for a hyperbola
• Pole-polar relation for a parabola

conic	equation	polar of point${\displaystyle P=(x_0,y_0)}$
circle	${\displaystyle x^2+y^2=r^2}$	${\displaystyle x_{0}x+y_{0}y=r^2}$
ellipse	${\displaystyle {\left(\frac{x}{a}\right)}^2+{\left(\frac{y}{b}\right)}^2=1}$	${\displaystyle \frac{x_{0}x}{a^2}+\frac{y_{0}y}{b^2}=1}$
hyperbola	${\displaystyle {\left(\frac{x}{a}\right)}^2-{\left(\frac{y}{b}\right)}^2=1}$	${\displaystyle \frac{x_{0}x}{a^2}-\frac{y_{0}y}{b^2}=1}$
parabola	${\displaystyle y=a{x}^2}$	${\displaystyle y+y_0=2a{x}_{0}x}$

conic	equation	pole of lineu x +v y =w
circle	${\displaystyle x^2+y^2=r^2}$	${\displaystyle \left(\frac{r^{2}u}{w},\frac{r^{2}v}{w}\right)}$
ellipse	${\displaystyle {\left(\frac{x}{a}\right)}^2+{\left(\frac{y}{b}\right)}^2=1}$	${\displaystyle \left(\frac{a^{2}u}{w},\frac{b^{2}v}{w}\right)}$
hyperbola	${\displaystyle {\left(\frac{x}{a}\right)}^2-{\left(\frac{y}{b}\right)}^2=1}$	${\displaystyle \left(\frac{a^{2}u}{w},-\frac{b^{2}v}{w}\right)}$
parabola	${\displaystyle y=a{x}^2}$	${\displaystyle \left(-\frac{u}{2av},-\frac{w}{v}\right)}$

Inprojective geometry, two lines in a plane always intersect. Thus, given four points forming acomplete quadrangle, the lines connecting the points cross in an additional threediagonal points.

Given a pointZ not on conicC, draw twosecants fromZ throughC crossing at pointsA,B,D, andE. Then these four points form a complete quadrangle, andZ is at one of the diagonal points. The line joining the other two diagonal points is the polar ofZ, andZ is the pole of this line.^[2]

Poles and polars were defined byJoseph Diaz Gergonne and play an important role in his solution of theproblem of Apollonius.^[3]

In planar dynamics a pole is a center of rotation, the polar is the force line of action and the conic is the mass–inertia matrix.^[4] The pole–polar relationship is used to define thecenter of percussion of a planar rigid body. If the pole is the hinge point, then the polar is the percussion line of action as described in planarscrew theory.

• Dual polygon
• Dual polyhedron
• Polar curve
• Brokard's theorem

• Johnson RA (1960).Advanced Euclidean Geometry: An Elementary treatise on the geometry of the Triangle and the Circle. New York: Dover Publications. pp. 100–105.
• Coxeter HSM,Greitzer SL (1967).Geometry Revisited.Washington:MAA. pp. 132–136, 150.ISBN 978-0-88385-619-2.
• Gray J J (2007).Worlds Out of Nothing: A Course in the history of Geometry in the 19th century. London: Springer Verlag. pp. 21.ISBN 978-1-84628-632-2.
• Korn GA,Korn TM (1961).Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. pp. 43–45.LCCN 59014456. The paperback version published by Dover Publications has theISBN 978-0-486-41147-7.
• Wells D (1991).The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. pp. 190–191.ISBN 0-14-011813-6.

Wikimedia Commons has media related toPoles and polars.

• Interactive animation with multiple poles and polars atCut-the-Knot
• Interactive animation with one pole and its polar
• Interactive 3D with coloured multiple poles/polars - open source
• Weisstein, Eric W."Polar".MathWorld.
• Weisstein, Eric W."Reciprocation".MathWorld.
• Weisstein, Eric W."Inversion pole".MathWorld.
• Weisstein, Eric W."Reciprocal curve".MathWorld.
• Tutorial at Math-abundance

• Euclidean plane geometry
• Projective geometry
• Circles

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