Pole (geometry) - 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.

Read more about this topic: Pole (geometry)

Famous quotes containing the words general and/or sections:

“It has been the struggle between privileged men who have managed to get hold of the levers of power and the people in general with their vague and changing aspirations for equality, for justice, for some kind of gentler brotherhood and peace, which has kept that balance of forces we call our system of government in equilibrium.”
—John Dos Passos (1896–1970)

“Childhood lasts all through life. It returns to animate broad sections of adult life.... Poets will help us to find this living childhood within us, this permanent, durable immobile world.”
—Gaston Bachelard (1884–1962)