Computer Aided Engineering Design

SPLINES 161

With 7 data points and 4 as the order of the piecewise curves, the number of knots required are 11.
We can use u 0 ,... , u 6 as 7 knots with 4 free choices. Let these choices be arbitrary, say –2 and –1
to the left and, 2 and 3 to the right. Then the knot vector is

[t 0 ,... , tm]≡ [–2, –1, 0, 0.058, 0.225, 0.404, 0.682, 0.814, 1, 2, 3]

We can use the above knot sequence to compute the B-spline functions Np,p+i(t) in Eq. (5.43). Further,
we can use the u values above to compute the coefficient matrix in Eq. (5.45). Substituting for the

Figure 5.22 Interpolation with B-spline curves.

–2 02468 10

x(t)

10

5

0

–5

–10

y(t)

data points P, the unknown control points can be
determined as

B

0.69 –5.90

• 0.20 1.74
  2.80 2.38
  1.92 8.32
  4.77 2.34
  6.14 – 0.10
  8.86 –8.46

≡

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟

The control points ‘∗’ and the interpolated curve
(thick line) with data points ‘°’ are shown in
Figure 5.22. The interpolating spline is plotted
between 0 and 1 which are the lower and upper
bounds of parameter u. Note, however, that t∈
[0, 1] does not correspond to the interval of full
support, which is [0.058, 0.814].

Bézier curves are special cases of B-spline curves clamped at both ends. If the order of a B-spline
curve is chosen as the number of control points (i.e., p=n+1), then m+1 = 2n+2 = 2p knots are
required of which pknots are clamped at each end and the B-spline curve reduces to a Bézier curve.

5.12 Non-Uniform Rational B-Splines (NURBS)

Rational Bézier curves are first introduced in section 4.6 wherein, in addition to specifying the data
points, a user is also required to specify respective weights to gain additional design freedom.
However, local shape control is still not possible with rational Bézier segments. Noting that B-spline
basis functions are locally barycentric that render local shape control to B-spline curves, analogous
to Eqs. (4.66) and (5.34), rational B-spline curves can be defined as

b

b

() =

()

()

=0 ,+

=0 ,+

t

wN t

wN t

i

n

ippi i

i

n

ippi

Σ

Σ

(5.46)

The term non-uniformsignifies that the knots are not placed at regular intervals in a general setting.
Here again, setting wi to zero implies that the location of bi does not affect the curve’s shape. For
larger values of wi, the curve gets pushed towards bi. Because they offer a great deal of flexibility in
design and also because they possess local shape control and all other properties of B-spline curves,
NURBS are widely employed in freeform modeling of curves. NURBS are also capable of accurately
modeling many analytic curves. Since NURBS are the generalization of B-spline curves (setting all