Pole and polar

In geometry, a pole and polar are respectively a point and a line that have a unique reciprocal relationship with respect to a given conic 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: The pole of a line L in a circle C is a point Q that is the inversion in C of the point P on L that is closest to the center of the circle.

Conversely, the polar line (or polar) of a point Q in a circle C is the line L such that its closest point P to the center of the circle is the inversion of Q in C. The relationship between poles and polars is reciprocal.

There is another description of the polar line of a point P in the case that it lies outside the circle C. In this case, there are two lines through P which are tangent to the circle, and the polar of P is the line joining the two points of tangency (not shown here).

This shows that pole and polar line are concepts in the projective geometry of the plane and generalize with any nonsingular conic in the place of the circle C. The concepts of a pole and its polar line were advanced in projective geometry.

For instance, the polar line can be viewed as the set of projective harmonic conjugates of a given point, the pole, with respect to a conic.

For some point P and its polar p, any other point Q on p is the pole of a line q through P. 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 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

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

where Axx, Axy, Ayy, Bx, By, 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

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

, 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}

Now, the pole is the point with coordinates

In projective geometry, two lines in a plane always intersect.

[2] Poles and polars were defined by Joseph Diaz Gergonne and play an important role in his solution of the problem 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 the center 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 planar screw theory.

The polar line q to a point Q with respect to a circle of radius r centered on the point O . The point P is the inversion point of Q ; the polar is the line through P that is perpendicular to the line containing O , P and Q .
If a point A lies on the polar line q of another point Q , then Q lies on the polar line a of A . More generally, the polars of all the points on the line q must pass through its pole Q .
Illustration of the duality between points and lines, and the double meaning of "incidence". If two lines a and k pass through a single point Q , then the polar q of Q joins the poles A and K of the lines a and k , respectively.
Line p is the polar line to point P , l to L and m to M
p is the polar line to point P ; m is the polar line to M