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This paper provides a rigorous mathematical foundation for geometric continuity of rational Beta-splines of arbitrary order. A function is said to be n^th order Beta-continuous if and only if it satisfies the Beta-constraints for a fixed value of Beta = (Beta1, Beta2, ... Betan). Sums, differences, products, quotients, and scalar multiples of Beta-continuous scalar-valued functions are shown to also be Beta-continuous scalar-valued functions (for the same value of Beta). Using these results, it is shown that the rational Beta-spline basis functions are Beta-continuous for the same value of Beta as the corresponding integral basis functions. It follows that the rational Beta-spline curve and tensor product surface are geometrically continuous.

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