3.3 Properties

NURBS generalize the nonrational parametric form. Like nonrational B-splines, the rational basis functions of NURBS sum to unity, they are infinitely smooth in the interior of a knot span provided the denominator is not zero, and at a knot they are at least continuous with knot multiplicity r, which enables them to satisfy different smoothness requirements. They inherit many of the properties of nonrational B-splines, such as the strong convex hull property, variation diminishing property, local support, and invariance under standard geometric transformations (see [11] for more details). Moreover, they have some additional properties:

• NURBS offer a common mathematical framework for implicit and parametric polynomial forms. In principle, they can represent analytic functions such as conics and quadrics precisely, as well as free-form shapes.
• NURBS include weights as extra degrees of freedom which influence local shape. If a particular weight is zero, then the corresponding rational basis function is also zero and its control point does not effect the NURBS shape. The spline is attracted toward a control point more if the corresponding weight is increased and less if the weight is decreased.

The most frequently used NURBS design techniques are the specification of a control polygon, or interpolation or approximation of data points to generate the initial shape. For surfaces or solids, cross-sectional design including skinning, sweeping, and swinging operations is also popular. The initial shape is then refined into the final desired shape through interactive adjustment of control points and weights and possibly the addition or deletion of knots. The refinement process is ad hoc and often tedious. To ameliorate it, we propose dynamic NURBS.