Planar Affine Group
According to Artzy, "The linear part of each affinity can be brought into one of the following standard forms by a coordinate transformation followed by a dilation from the origin:
- where the coefficients a, b, c, and d are real numbers."
Case (1) corresponds to similarity transformations which generate a subgroup of similarities. Euclidean geometry corresponds to the subgroup of congruencies. It is characterized by Euclidean distance or angle, which are invariant under the subgroup of rotations.
Case (2) corresponds to shear mappings. An important application is absolute time and space where Galilean transformations relate frames of reference. They generate the Galilean group.
Case (3) corresponds to squeeze mapping. These transformations generate a subgroup, of the planar affine group, called the Lorentz group of the plane. The geometry associated with this group is characterized by hyperbolic angle, which is a measure that is invariant under the subgroup of squeeze mappings.
Using the above matrix representation of the affine group on the plane, the matrix M is a 2 × 2 real matrix. Accordingly, a non-singular M must have one of three forms that correspond to the tricotomy of Artzy.
Read more about this topic: Affine Group
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