Rational Beta-splines for representing curves and surfaces

The rational Beta-spline representation, which offers the features of the rational form as well as those of the Beta-spline, is discussed. The rational form provides a unified representation for conventional free-form curves and surfaces along with conic sections and quadratic surfaces, is invariant under projective transformation, and possesses weights, which can be used to control shape in a manner similar to shape parameters. Shape parameters are an inherent property of the Beta-spline and provide intuitive and natural control over shape. The Beta-spline is based on geometric continuity, which provides an appropriate measure of smoothness in computer-aided geometric design. The Beta-spline has local control with respect to vertex movement, is affine invariant, and satisfies the convex hull property. The rational Beta-spline enjoys the benefit of all these attributes. The result is a general, flexible representation, which is amenable to implementation in modern geometric modeling systems