Pole and Polar - General Conic Sections

General Conic Sections

The concepts of pole, polar and reciprocation can be generalized from circles to other conic sections which are the ellipse, hyperbola and parabola. This generalization is possible because conic sections result from a reciprocation of a circle in another circle, and the properties involved, such as incidence and the cross-ratio, are preserved under all projective transformations.

A general conic section may be written as a second-degree equation in the Cartesian coordinates (x, y) of the plane

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

where A_xx, A_xy, A_yy, B_x, B_y, and C 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

$D x + E y + F = 0\,$

where D, E and F are likewise constants that depend on the pole coordinates (ξ, η)

$D = A_{xx} \xi + A_{xy} \eta + B_{x}\,$

$E = A_{xy} \xi + A_{yy} \eta + B_{y}\,$

$F = B_{x} \xi + B_{y} \eta + C\,$

If the pole lies on the conic section, its polar is tangent to the conic section. However, the pole need not lie on the conic section.

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