Curves in Computer Graphics - Computer Graphics and Geometric Modeling - page 430

b i (u) are B-splines of order k. The curve p(u) is called a n onuniform r ational B - s pline
(NURBS) curve of order k with domain [a,b] if b i (u) = N i,k (u) with respect to a knot
vector

= () = U u ( ,..., , a auu , ,..., ub , ,..., ). b

i

144

kk + 1 n

144

k k

(The N i,k (u) are the B-splines defined by equations (11.69).) In any case, the points p i
are called the control points of the curve p(u) and the numbers w i are called its weights .

The ordinary Bézier and B-spline curves are clearly a special case of the rational
ones since we get them by using weights that are all equal to 1. Note further that if
we define the function R i (u) by

()

wb u

ii

() = Ru , (11.104a)

i

n

 0

()

wb u

jj

j =

then

n

Â

() = ()

pu R u p , (11.104b)

i i

i = 0

so that p(u) is again a curve of a form (like that of equation (11.67)) that we have seen
many times before.

Definition. The functions R i (u) in equations (11.104) are called the rational basis
functions for the curve p(u).

NURBS curves (and surfaces) have become very popular in recent years and a
number of modeling programs are based on them. Some general references for these
and rational Bézier curves are [PieT95], [Pieg91], [Fari95], [Roge01], or [RogA90]. In
the rest of this section we shall look at some examples and properties of NURBS
curves, leaving a discussion of how to compute them efficiently to the next section.

11.5.3.1 Example. Suppose that we want to find a NURBS representation for the

unit circle.

Solution. Consider the first quadrant of the unit circle. By equation (11.100) we have
the rational parameterization

2

Ê
Á

1
1

-
+

u
u

2 u

u

ˆ
˜

() =

Π[] pu , , for u 01 , .

2 2

1 +

In homogeneous coordinates this can be written as

( )

() =-

2 2

Pu 1 u , 2 u , 0 1 , + u

= ( ) + ( ) +-

2

( )

(11.105)

1001 ,,, u 0200 ,,, u 1001 ,,, .