# Affine Transformations and the Orthogonality of Cartesian Rotations

The derivation shows that a Cartesian coordinate transformation, which is an affine transformation ( $$x^{h^{\prime}}=R\_i^{i^{\prime}} x^i+A^{i^{\prime}}$$ ) that preserves the form of the metric tensor ( $$g\_{i j}=\delta\_{i j}$$ ), necessarily implies that the transformation matrix R is orthogonal. This is mathematically expressed as the orthogonality condition, $$R\_i^{i^{\prime}} R\_j^{i^{\prime}}=\delta\_{i j}$$. This requirement ensures that the transformation represents a rigid-body motion (rotation and/or reflection) in Euclidean space. Furthermore, using this orthogonality condition, the inverse relationship can be derived and shown to have the same affine form: $$x^i=R\_i^{i^{\prime}} x^{i^{\prime}}+B^i$$, where $$B^i$$ is a new constant translation vector related to $$A^{i^{\prime}}$$.

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