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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. 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.
In Euclidean space, the dual of a polyhedron is often defined in terms of polar reciprocation about a sphere. Here, each vertex (pole) is associated with a face plane (polar plane or just polar) so that the ray from the center to the vertex is perpendicular to the plane, and the product of the distances from the center to each is equal to the square of the radius.
In the complex number approach, where reciprocation is the apparent operation, this procedure leads to the complex projective line, often called the Riemann sphere. It was subspaces and subgroups of this space and group of mappings that were applied to produce early models of hyperbolic geometry by Beltrami , Cayley , and Klein .
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Reciprocation may refer to: Reciprocating motion , a type of oscillatory motion, as in the action of a reciprocating saw Reciprocation (geometry) , an operation with circles that involves transforming each point in plane into its polar line and each line in the plane into its pole
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Under Polar reciprocation it says: If r 0 is the radius of the sphere, and r 1 and r 2 respectively the distances from its centre to the pole and its polar, then: r 1 2 · r 2 2 = r 0 2. Shouldn't the formula read: r 1 · r 2 = r 0 2, in accordance with Inversive geometry#Circle inversion?--Episcophagus 17:25, 2 December 2014 (UTC) Yes indeed ...
Polar point group, a symmetry in geometry and crystallography; Pole and polar (a point and a line), a construction in geometry Polar cone; Polar coordinate system, uses a central point and angles; Polar curve (a point and a curve), a generalization of a point and a line; Polar set, with respect to a bilinear pairing of vector spaces