Rational continuity: parametric, geometric, and Frenet frame continuity of rational curves
ACM Transactions on Graphics (TOG) - Special issue on computer-aided design
Geometric Continuity of Parametric Curves: Three Equivalent Characterizations
IEEE Computer Graphics and Applications
Rational Beta-Splines for Representing Curves and Surfaces
IEEE Computer Graphics and Applications
Graphical Models
Modeling with rational biquadratic splines
Computer-Aided Design
Free-form splines combining NURBS and basic shapes
Graphical Models
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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 order Beta-continuous if and only if it satisfies the Beta-constraints for a fixed value of Beta (Beta1, Beta2, . . . Beta). Sums, differences, products, quotients, and scalar multiples of 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.