Computing Rotation Matrices Between Two Sets of Vectors
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To compute the rotation matrix between two sets of vectors, several factors must be considered. First, we require same-name vectors with unit magnitude (normalized to length 1) in two different coordinate systems. The core algorithm typically involves calculating the optimal rotation using Singular Value Decomposition (SVD) - specifically the Kabsch algorithm which minimizes the RMSD between vector sets. The rotation matrix calculation involves these key steps: 1) Compute the covariance matrix between vector sets, 2) Perform SVD decomposition, 3) Ensure proper rotation (handling reflection cases), 4) Construct the 3x3 rotation matrix. This resulting orthogonal matrix transforms vectors from one coordinate system to another while preserving lengths and angles. The implementation should validate input vector normalization and handle degenerate cases through proper error checking.
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