CG Modeling 2: NURBS - An Introduction to Computer Graphics for Artists - page 243

fi xed with a ratchet or by other means. This has the effect of modifying the curvature
of the wood running through each spline.

14.1.1.2 Auto Design Splines

Splines used in the auto design industry are based on boat splines, but designed to
be used indoors. An auto design spline is made of a thin strip of wood or metal with
heavy weights connected to it by fl exible cords. The weights could be exchanged
with lighter or heavier weights to decrease or increase their effect on the strip.
Designers then traced the curves with a pencil or pen.

14.1.1.3 Bezier

The word Bezier refers to Pierre Bezier, a French automobile designer who published
the fi rst paper on NURBS curves, as used in the auto design industry. Bezier and Paul
de Casteljau created the method at about the same time (Schmitt et al. 1986 ), but
because de Casteljau didn't publish the results, they are now known as Bezier curves.

14.1.1.4 Rational

The word Rational refers to the fact that the curve can be rational, depending on
weighted values at each control point. Most curves are non-rational. They are ratio-
nal only if the weight of each point is the same.

14.1.1.5 Uniform

The term uniform refers to a curve that is evenly divided based on knot position. Knots
control curvature at control points. The more knots there are, the tighter the curve
becomes at that point until it can form a sharp edge, instead of a smooth tangent . In a
uniform curve, it starts and ends with full multiplicity knots , with evenly spaced simple
knots between them. For example, the following group of knot values in a three-degree
seven control point curve is uniform: 0,0,0,2,4,6,8,8,8. In the previous example, the
knots at values 0 and 8 are full-multiplicity knots. Full-multiplicity knots are hard
knots that can have a tangency break. Simple knots have no tangency breaks. A non-
uniform curve does not have evenly distributed knots, as in a uniform curve.

14.1.2 Limitations

NURBS surfaces are always four-sided. This limitation allows surfaces to be defi ned
parametrically because it is a fi xed topological feature of all NURBS surfaces.