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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.
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
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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The assertion that "if a point Q is on the polar line A of a point P, then the point P must lie on the polar line B of the point Q" is sometimes called the reciprocity theorem. Does it deserve its own page Reciprocity theorem (projective geometry) ? Tkuvho 14:55, 14 September 2010 (UTC)
The dual of a convex polyhedron can be obtained by the process of polar reciprocation. [21] Dual polyhedra exist in pairs, and the dual of a dual is just the original polyhedron again. Some polyhedra are self-dual, meaning that the dual of the polyhedron is congruent to the original polyhedron. [22]
Reciprocating motion, also called reciprocation, is a repetitive up-and-down or back-and-forth linear motion. It is found in a wide range of mechanisms, including reciprocating engines and pumps . The two opposite motions that comprise a single reciprocation cycle are called strokes .