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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