On beta-continuous functions and their application to the construction of geometrically continuous curves and surfaces

A function is said to be β -continuous if it satisfies the Beta-constraints for the fixed values β = ( β 1 ,…, β n ). Sums, differences, products, quotients, and scalar multiples of β -continuous functions are shown to be β -continuous. These basic results are applied to various standard constructions of parametrically continuous curves and surfaces — such as rational splines, Catmull-Rom splines, affine combinations of curves, as well as ruled, lofted, tensor product, and Boolean sum surfaces — to generate geometrically continuous analogues. This analysis for geometric continuity defined by the Beta-constraints (reparametrization) is contrasted with a corresponding investigation for Frenet frame continuity.