Pole and polar

"Polar line" redirects here. For the railway line, seePolar Line.

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

The polar lineq to a pointQ with respect to a circle of radiusr centered on the pointO. The pointP is theinversion point ofQ; the polar is the line throughP that is perpendicular to the line containingO,P andQ.

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.

Properties

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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. (La Hire's theorem)
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.

Special case of circles

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Polar reciprocation

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Main article:Correlation (projective geometry)

Illustration of the duality between points and lines, and the double meaning of "incidence". If two linesa andk pass through a single pointQ, then the polarq ofQ joins the polesA andK of the linesa andk, respectively.

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]

General conic sections

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Linep is the polar line to pointP,l toL andm toM

p is the polar line to pointP ;m is the polar line toM

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.

Calculating the polar of a point

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

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

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

$Dx+Ey+F=0\,$

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

${\begin{aligned}D&=A_{xx}\xi +A_{xy}\eta +B_{x}\\E&=A_{xy}\xi +A_{yy}\eta +B_{y}\\F&=B_{x}\xi +B_{y}\eta +C\end{aligned}}$

Calculating the pole of a line

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

First, calculate the numbers x, y and z from

${\begin{bmatrix}x\\y\\z\end{bmatrix}}={\begin{bmatrix}A_{xx}&A_{xy}&B_{x}\\A_{xy}&A_{yy}&B_{y}\\B_{x}&B_{y}&C\end{bmatrix}}^{-1}{\begin{bmatrix}D\\E\\F\end{bmatrix}}$

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

Tables for pole-polar relations

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

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

Via complete quadrangle

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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]

Applications

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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.