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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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