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We present a new algorithm for linear spectral mixture analysis, which is capable of supervised unmixing of hyperspectral data while respecting the constraints on the abundance coefficients. This simplex-projection unmixing algorithm is based upon the equivalence of the fully constrained least squares problem and the problem of projecting a point onto a simplex. We introduce several geometrical properties of high-dimensional simplices and combine them to yield a recursive algorithm for solving the simplex-projection problem. A concrete implementation of the algorithm for large data sets is provided, and the algorithm is benchmarked against well-known fully constrained least squares unmixing (FCLSU) techniques, on both artificial data sets and real hyperspectral data collected over the Cuprite mining region. Unlike previous algorithms for FCLSU, the presented algorithm possesses no optimization steps and is completely analytical, severely reducing the required processing power.