Developing Images of Affine Tori

Affine transformation on ℝ²

An affine transformation is a function (x', y') = f(x, y) givne by
     x' = A₁₁ x + A₁₂ y + a₁
     y' = A₂₁ x + A₂₂ y + a₂
or equivalently
     f(v) = A v + a
in the matrix notation where v = (x, y), A = (A_ij) and a = (a₁, a₂).

The set of all affine transformation on ℝ² is called the Affine group and denoted by Aff(2) = ℝ × GL(2, ℝ). The orientation preserving subgroup is denoted by Aff₊(2) = ℝ × GL₊(2, ℝ).

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