3D graphics eBook - Course Materials Repository

Non-uniform rational B-spline 77
Non-uniform rational B-spline
Non-uniform rational basis spline
(NURBS) is a mathematical model
commonly used in computer graphics
for generating and representing curves
and surfaces which offers great
flexibility and precision for handling
both analytic and freeform shapes.
History
Development of NURBS began in the
1950s by engineers who were in need
of a mathematically precise
representation of freeform surfaces like
those used for ship hulls, aerospace
exterior surfaces, and car bodies,
which could be exactly reproduced
whenever technically needed. Prior
representations of this kind of surface
only existed as a single physical model
created by a designer.
The pioneers of this development were
Pierre Bézier who worked as an
engineer at Renault, and Paul de
Casteljau who worked at Citroën, both
Three-dimensional NURBS surfaces can have complex, organic shapes. Control points
influence the directions the surface takes. The outermost square below delineates the X/Y
extents of the surface.
A NURBS curve.
Animated version
in France. Bézier worked nearly parallel to de Casteljau, neither knowing about the work of the other. But because
Bézier published the results of his work, the average computer graphics user today recognizes splines — which are
represented with control points lying off the curve itself — as Bézier splines, while de Casteljau’s name is only
known and used for the algorithms he developed to evaluate parametric surfaces. In the 1960s it became clear that
non-uniform, rational B-splines are a generalization of Bézier splines, which can be regarded as uniform,
non-rational B-splines.
At first NURBS were only used in the proprietary CAD packages of car companies. Later they became part of
standard computer graphics packages.
Real-time, interactive rendering of NURBS curves and surfaces was first made available on Silicon Graphics
workstations in 1989. In 1993, the first interactive NURBS modeller for PCs, called NöRBS, was developed by CAS
Berlin, a small startup company cooperating with the Technical University of Berlin. Today most professional
computer graphics applications available for desktop use offer NURBS technology, which is most often realized by
integrating a NURBS engine from a specialized company.