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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 ...
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.
The dual can be visualized as a locus in the plane in the form of the polar reciprocal. This is defined with reference to a fixed conic Q as the locus of the poles of the tangent lines of the curve C. [2] The conic Q is nearly always taken to be a circle, so the polar reciprocal is the inverse of the pedal of C.
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
Under any duality, the point P is called the pole of the hyperplane P ⊥, and this hyperplane is called the polar of the point P. Using this terminology, the absolute points of a polarity are the points that are incident with their polars and the absolute hyperplanes are the hyperplanes that are incident with their poles.
If the polar line of C with respect to a point Q is a line L, then Q is said to be a pole of L. A given line has (n−1) 2 poles (counting multiplicities etc.) where n is the degree of C. To see this, pick two points P and Q on L. The locus of points whose polar lines pass through P is the first polar of P and this is a curve of degree n−1.
Polar reciprocation, a concept in geometry also known as polarity; Trilinear polarity, a concept in geometry of the triangle; Polarity of a literal, in mathematical ...