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P
3

w
3
= 4

w
3
= 2

P
1

w
3
= 1

w
3
= 1/2

w
3
= 1/4

P
0

P
4

P
2

Figure 11.18.
A nonuniform rational B-spline. The different curves show what happens

as the weight corresponding to the control point
P
changes.

All of the curves described in this chapter are invariant with respect to any

translation, rotation, or scaling transformation. That is, transforming the geomet-

rical constraints (e.g., the control points) and generating the curve produces the

same results as generating the curve using the untransformed geometrical con-

straints and then transforming the result. NURBS are also invariant with respect

to a homogeneous projection transformation. The curve generated by the homo-

geneous control points after projection is the same as the projection of the curve

generated using the unprojected control points. This property can be gained by a

nonuniform B-spline by promoting it to a NURBS curve in which every weight

has been assigned a value of 1.

NURBS have been widely adopted by computer modeling systems because

of their generality. NURBS can represent any of the other types of curves dis-

cussed in this chapter, and unlike nonrational curves, can represent conic sections

exactly.
1

1
See David F. Rogers and J. Alan Adams,
Mathematical Elements for Computer

Graphics
, Second Edition, McGraw-Hill, 1990.

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