Uniform Hyperbolic Polynomial B-Splines and Space Curve Interpolation Through Isometric Mapping

Xunnian Yang
Department of Mathematics, Zhejiang University
Hangzhou, China


Abstract. This talk covers two topics on freeform curve and surface modeling.

The first part presents a new kind of uniform splines, named hyperbolic polynomial B-splines, over the space of which k is an arbitrary integer number larger than or equal to 3. Hyperbolic polynomial B-splines share the same properties as the B-splines in polynomial space. We present the subdivision formula for this new kind of curves and then prove that it has properties of V.D. and convergence of control polygons by subdivision. Hyperbolic polynomial B-splines can include both freeform curves and some remarkable curves, such as the hyperbola and the catenary. The generation of tensor product surfaces by this new spline is straightforward and the corresponding tensor product surfaces also contain many special surfaces, including saddle surface, catenary cylinder and a certain kind of ruled surface.

In the second part we present new methods for geometric Hermite interpolation by space curves. We show that a space curve is a surface curve lying on a cylinder and can be deformed into a plane curve by isometric mapping of the surface. Then we can obtain an interpolation space curve by constructing the plane directrix of the cylinder and computing the rest variable on the developed cylinder based on the technique of plane curves interpolation. Explicit algorithms are given to construct a quintic Bezier curve which interpolates the end curvatures and end torsions, a quartic Bezier curve that interpolates the end curvatures, both of which also match the unit tangents and unit normal vectors at the either ends. Even more, the additional freedoms for the interpolation can be used to control the shape of the final space curve. Several numerical examples are also presented to show the efficiency of the new method.