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There have been numerous attempts to quantify the shape of the dental arch mathematically, with orthogonal polynomial curves providing a robust and versatile method for quantifying variation in both shape and asymmetry. Lu (1966) first presented the theoretical basis for fitting orthogonal polynomials to arch shape data. Whilst theoretically sound, Lu’s original paper contained several arithmetic errors and a number of incorrect assumptions. In this paper we present corrections for these errors and extrapolate the theory to unequally-spaced arch shape data using a simple recursive procedure first developed by Robson (1959).