Approximate spline of [formula omitted]-continuity on a generalized hyperbolic paraboloid

Abstract

In this paper, a generalized hyperbolic paraboloid is represented by bi-parametrization equipped with a shape parameter, which can appropriately control approximate behavior of splines. A unified method is presented to construct G2-continuous approximate spline curves and surfaces composed of a kind of curve segments and surface patches respectively. Any segment of this kind lying on the generalized hyperbolic paraboloid, is of certain tangent directions and bounded curvatures at two endpoints, that can be done by constraining the two parameters with a functional relationship and selecting suitable weight functions. There is also an alternative form given to improve the approximating effect. Moreover, the kind of patches is produced by tensor product of the same basis functions as for the curves. These segments therefore are conveniently connected into a curve with G2-continuity, as are the patches into a G2-continuous surface, both of which can arbitrarily approach their corresponding control polygon.