Nearrotations

Nearrotations are informal term used in mathematics and applied fields to describe linear transformations that are close to rotations. They are not exact elements of the special orthogonal group SO(n) but lie in a small neighborhood of it within GL(n, R).

Formally, for a square matrix A in R^{n×n}, A is a near-rotation if there exists R in SO(n) with a small distance ∥A − R∥, under a chosen norm such as the Frobenius or spectral norm. Equivalently, A can be expressed as A = R P where P is symmetric positive definite and P is near the identity. This captures the idea that A behaves like a rotation up to a modest distortion.

Polar decomposition provides a useful perspective: any nonsingular A can be written A = Q H, with Q in O(n) and H symmetric positive definite. If det Q = 1 and H is close to I, then A is a near-rotation. In numerical practice, the nearest rotation to A can be computed via the orthogonal Procrustes problem: given A, perform its singular value decomposition A = U Σ V^T and set R = U V^T.

Examples include a 2D matrix that is a rotation by θ plus a small radial scaling, or a 3D matrix derived from noisy orientation measurements that slightly violates orthogonality.

Notes: the set of near-rotations forms a neighborhood of SO(n) in GL(n, R); it is not closed under multiplication and can drift under successive perturbations. It is a practical concept in numerical analysis and estimation when exact rotations cannot be guaranteed due to measurement error.

Applications include computer vision, robotics, graphics, pose estimation, calibration, and Lie-group–based optimization where estimates are constrained to remain close to proper rotations.