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7.2 Central Collineations

Let φ be a collineation of a projective plane which fixes one point block-wise (the so-called centre) and one block point-wise (the so-called axis). If the centre is contained in the axis, φ is called elation. Otherwise, φ is called homology. The group of elations with given axis is called translation group of the plane (relative to the chosen axis). A projective plane with transitive translation group is called translation plane. Here transitivity is on the points outside the axis.

  • ElationsByPairs( centre, axis, pairs, data ) O
  • ElationsByPairs( centre, axis, pairs, blocks ) O
  • ElationsByPairsSmall( centre, axis, pairs, data ) O

    Let centre be a point and axis a block of a projective plane defined by blocks (or by data as returned by ElationPrecalc). The list pairs must contain pairs of points outside axis. ElationsByPairs returns a collineation fixing axis pointwise and centre blockwise (an elation) such that for each pair p of pairs p[1] is mapped on p[2]. If no such elation exists, fail is returned.

    ElationsByPairsSmall uses data as returned by ElationPrecalcSmall

  • AllElationsCentAx( centre, axis, data[, "generators"] ) O
  • AllElationsCentAx( centre, axis, blocks[, "generators"] ) O
  • AllElationsCentAxSmall( centre, axis, data[, "generators"] ) O

    Let centre be a point and axis a block of a projective plane defined by blocks (or by data as returned by ElationPrecalc). AllElationsCentAx returns a list of all non-trivial elations with centre centre and axis axis. If ``generators'' is set, a list of generators of the translation group is returned.

  • AllElationsAx( axis, data[, "generators"] ) O
  • AllElationsAx( axis, blocks ) O
  • AllElationsAxSmall( axis, data[, "generators"] ) O

    Let axis be a block of a projective plane defined by blocks (or by data as returned by ElationPrecalc). AllElationsAx returns a list of all non-trivial elations with axis axis.

  • IsTranslationPlane( infline, planedata ) O
  • IsTranslationPlaneSmall( infline, planedata ) O

    If the group of elations with axis infline is (sharply) transitive on the affine points (the points outside infline), IsTranslationPlane returns true, otherwise it returns false. This is faster than calculating the full translation group if the projective plane is not a translation plane.

  • HomologyByPairSmall( centre, axis, pair, data ) O

    HomologyByPairSmall returns the homology defined by the pair pair fixing centre blockwise and axis pointwise. The returned permutation fixes axis pointwise and centre linewise and takes pair[1] to pair[2].

  • GroupOfHomologiesSmall( centre, axis, data ) O

    returns the group of homologies with centre centre and axis axis.

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    RDS manual
    November 2006