In mathematics, a translation plane is a projective plane which admits a certain group of symmetries (described below). Along with the Hughes planes and the Figueroa planes, translation planes are among the most well-studied of the known non-Desarguesian planes, and the vast majority of known non-Desarguesian planes are either translation planes, or can be obtained from a translation plane via successive iterations of dualization and/or derivation.[1]
In a projective plane, let P represent a point, and l represent a line. A central collineation with centerP and axisl is a collineation fixing every point on l and every line through P. It is called an elation if P is on l, otherwise it is called a homology. The central collineations with center P and axis l form a group.[2] A line l in a projective plane Π is a translation line if the group of all elations with axis l acts transitively on the points of the affine plane obtained by removing l from the plane Π, Πl (the affine derivative of Π). A projective plane with a translation line is called a translation plane.
The affine plane obtained by removing the translation line is called an affine translation plane. While it is often easier to work with projective planes, in this context several authors use the term translation plane to mean affine translation plane.[3][4]
^Eric Moorhouse has performed extensive computer searches to find projective planes. For order 25, Moorhouse has found 193 projective planes, 180 of which can be obtained from a translation plane by iterated derivation and/or dualization. For order 49, the known 1349 translation planes give rise to more than 309,000 planes obtainable from this procedure.
^Geometry Translation Plane Retrieved on June 13, 2007
from the plane Π, Πl (the affine derivative of Π). A projective plane with a translation line is called a translationplane. The affine plane obtained...
points of the plane not on the line. A translationplane is Moufang if every line of the plane is a translation line. A Moufang plane can also be described...
mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect...
then P {\displaystyle {\mathcal {P}}} is a translationplane, or a dual translationplane, or a Hughes plane. The latter can be characterized as follows:...
(1899), The foundations of geometry, translation by E. J. Townsend, 1902, Chicago Knarr, N. (1995), Translationplanes. Foundations and construction principles...
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are four types: translations, rotations, reflections, and glide reflections (see below § Classification). The set of Euclidean plane isometries forms...
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same up to an affine transformation of the plane. Thus e.g. a translation of the plane (hence a translation of the mirrors and centres of rotation) does...
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a translation, or squeeze mapping combined with a homothety and a translation. To visualise the general affine transformation of the Euclidean plane, take...
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