Uniform hyperbolic polynomial B-spline curves

This paper presents a new kind of uniform splines, called hyperbolic polynomial B-splines, generated over the space Omega = span(sinh t, cosh t, t(k-3), t(k-4),..., t, 1) in which k is an arbitrary integer larger than or equal to 3. Hyperbolic polynomial B-splines share most of the properties as those of the B-splines in the polynomial space. We give the subdivision formulae for this new kind of curves and then prove that they have the variation dimishing properties and the control polygons of the subdivisions converge. Hyperbolic polynomial B-splines can take care of freeform curves as well as some remarkable curves such as the hyperbola and the catenary. The generation of tensor product surfaces by these new splines is straightforward. Examples of such tensor product surfaces: the saddle surface, the catenary cylinder, and a certain kind of ruled surface are given in this paper. (C) 2002 Elsevier Science B.V. All rights reserved.