Non-uniform Rational B-spline - Technical Specifications

Technical Specifications

A NURBS curve is defined by its order, a set of weighted control points, and a knot vector. NURBS curves and surfaces are generalizations of both B-splines and Bézier curves and surfaces, the primary difference being the weighting of the control points which makes NURBS curves rational (non-rational B-splines are a special case of rational B-splines). Whereas Bézier curves evolve into only one parametric direction, usually called s or u, NURBS surfaces evolve into two parametric directions, called s and t or u and v.

By evaluating a Bézier or a NURBS curve at various values of the parameter, the curve can be represented in Cartesian two- or three-dimensional space. Likewise, by evaluating a NURBS surface at various values of the two parameters, the surface can be represented in Cartesian space.

NURBS curves and surfaces are useful for a number of reasons:

• They are invariant under affine as well as perspective transformations: operations like rotations and translations can be applied to NURBS curves and surfaces by applying them to their control points.
• They offer one common mathematical form for both standard analytical shapes (e.g., conics) and free-form shapes.
• They provide the flexibility to design a large variety of shapes.
• They reduce the memory consumption when storing shapes (compared to simpler methods).
• They can be evaluated reasonably quickly by numerically stable and accurate algorithms.

In the next sections, NURBS is discussed in one dimension (curves). It should be noted that all of it can be generalized to two or even more dimensions.