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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. 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 ...
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 .
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
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]
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
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The Spanish Wikipedia (Spanish: Wikipedia en español) is the Spanish-language edition of Wikipedia, a free online encyclopedia. It has 2,010,226 articles. It has 2,010,226 articles. Started in May 2001, it reached 100,000 articles on 8 March 2006, and 1,000,000 articles on 16 May 2013.