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Reciprocation
Reciprocation is an incidence-preserving transformation in which points are transformed into theirpolars. Aprojective geometry-likeduality principle holds for reciprocation which states that theorems for the original figure can be immediately applied to thereciprocal figure after suitable modification (Lachlan 1893, pp. 174-182). Reciprocation (or "polar reciprocation") is the strictly proper term for duality. Brückner (1900) gave one the first exact definitions of polar reciprocation for constructingdual polyhedra, although the plane geometric version (inversion pole,polar, andcircle power) was considered by none less than Euclid (Wenninger 1983, pp. 1-2).
Lachlan (1893, pp. 257-265) discusses another type of reciprocation he terms "circular reciprocation." However, the circular reciprocal figure is, in general, more complicated than the original, so the method is not as powerful as the usual polar reciprocation.
See also
Canonical Polyhedron,Dual Polyhedron,Duality Principle,Inversion Pole,Midsphere,Polar,ReciprocalExplore with Wolfram|Alpha
References
Brückner, M.Vielecke under Vielflache. Leipzig, Germany: Teubner, 1900.Casey, J. "Theory of Poles and Polars, and Reciprocation." §6.7 inA Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 141-148, 1888.Coxeter, H. S. M. and Greitzer, S. L. "Reciprocation." §6.1 inGeometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 132-136, 1967.Lachlan, R. "Reciprocation" and "Circular Reciprocation." Ch. 11 and §405-414 inAn Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 174-182 and 257-265, 1893.Wenninger, M. J.Dual Models. Cambridge, England: Cambridge University Press, pp. 1-6, 1983.Referenced on Wolfram|Alpha
ReciprocationCite this as:
Weisstein, Eric W. "Reciprocation." FromMathWorld--A Wolfram Web Resource.https://mathworld.wolfram.com/Reciprocation.html