An orthogonal transformation is a linear transformation that preserves the inner product of vectors, which in turn means it also preserves lengths and angles between vectors. In practical terms, if you apply an orthogonal transformation to a set of vectors, the transformed vectors will maintain their geometric relationships. Mathematically, a transformation \( T: \mathbb{R}^n \to \mathbb{R}^n \) can be represented using a matrix \( A \).
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An orthogonal transformation of a finite-dimensional real inner product space is a linear map preserving its inner product: . In an orthonormal basis, its matrix satisfies ; in an arbitrary basis with Gram matrix , the condition is . It preserves inner products, lengths, angles, and therefore Gaussian curvature when applied to an embedded surface in Euclidean space.