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等距變換
An isometric transformation (or isometry) is a shape-preserving transformation. The isometric transformations are reflection, rotation, translation and combinations of them such as the glide, which is the combination of a translation and a reflection.
Define
\[O(2, \mathbb{R}) = \{A \in GL_2(\mathbb{R}) | AA^T = I_2\},\]which contains all \(2\times2\) matrices of isometry. If \(\text{det}(A) = 1\), \(A\) is a rotation matrix; if \(\text{det}(A) = -1\), \(A\) is a reflection matrix. This way, we can show that the combination of rotation and reflection is still reflection since
\[\text{det}(MR) = \text{det}(M)\text{det}(R) = -1\times 1 = -1.\]