Geometry can have transformation groups that have invariants {Erlanger program} {Erlanger Programme}. Possible invariants are linearity, collinearity, cross ratio, harmonic point set, and conic section. According to the Erlanger Programme [1872] (Felix Klein), geometry is point spaces and their transformation groups (mappings). For a geometry, specific invariant space structures do not change under its transformations. For Euclidean geometry, rigid motions (displacement and rotation transformations) do not change invariant separation. Euclidean geometry has between-ness and separation (length). For similarity or extended Euclidean geometry, rigid motions, translations, rotations, and uniform scalings (similarity transformations) do not change invariant length ratios. Extended Euclidean geometry has between-ness, length ratios, and angles. For affine geometry, translations, rotations, skewings, non-isotropic scalings, and nonsingular linear mappings (affine transformations) do not change invariant line at infinity or three-collinear-point cross ratio. Affine geometry has between-ness but no length ratios. For projective geometry, all linear mappings (projective collineations) do not change invariant four-collinear-point cross ratio. Projective geometry has no separation, length ratios, or between-ness. Euclidean geometry has the most invariants, and projective geometry has the least.
Mathematical Sciences>Geometry
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Date Modified: 2022.0224