Nonuniform Rational B-Splines (NURBS)

NURBS

Bézier curve


NURBS (Nonuniform Rational Bézier-Splines) are industry standard tools for the representation and design of geometry. Some reasons for the use of NURBS are, that they:

    * offer one common mathematical form for both, standard analytical shapes (e.g. conics) and free form shapes;

    * provide the flexibility to design a large variety of shapes;

    * can be evaluated reasonably fast by numerically stable and accurate algorithms;

    * are invariant under affine as well as perspective transformations;

    * are generalizations of non-rational B-splines and non-rational and rational Bezier curves and surfaces.

However, one of the drawbacks NURBS have, is the need for extra storage to define traditional shapes (e.g. circles). This results from parameters in addition to the control points, but finally allow the desired flexibility for defining parametric shapes. NURBS-shapes are not only defined by control points; weights, associated with each control point are also necessary




In the mathematical field of numerical analysis, a Bézier curve is a parametric curve important in computer graphics. Generalizations of Bézier curves to higher dimensions are called Bézier surfaces, of which the Bézier triangle is a special case.

Bézier curves were widely publicised in 1962 by the French engineer Pierre Bézier, who used them to design automobile bodies. The curves were first developed in 1959 by Paul de Casteljau using de Casteljau's algorithm, a numerically stable method to evaluate Bézier curves.