Computer Aided Engineering Design

SPLINES 159

uabai k

i

kk

k

n

kk

= + ( – )

| – |

| – |

=1 –1

=1 –1

1

2

1

2

Σ

Σ

bb

bb

, i = 1,... , n (5.39)

withu 0 as a. Note that Eqs. (5.36), (5.38) and (5.39) can be generalized to

ua bai k

i

kk

e

k

n

kk

e

= + ( – )

| – |

| – |

=1 –1

=1 –1

Σ

Σ

bb

bb

,i = 1,... , n (5.40)

for some chosen exponent e≥ 0. For e = 0, uniformly spaced parameterization is obtained while for
e = 1 and^12 , respectively, chord length and centripetal parameterizations are achieved.

5.10.1 Knot Vector Generation

Once a set of parameters is obtained, the knot vector may be generated. The placement of knots
would, however, depend on the end conditions. For unclamped splines, all m+1 knots are simple. Of
those,n+1 knots (tp,... , tn+p) may be chosen as the parameters, ui,i = 0,... , n, respectively from
Eq. (5.40) while the remaining first p knots (t 0 ,... , tp– 1 ) may be chosen freely, that is, ti+p = ui,i =
0,... , n while t 0 ,t 1 ,... , tp– 1 are free choices of simple knots. In case the B-spline curve is clamped
at one end, the knot corresponding to that end must be repeated at least p–1 times. If the spline is
clamped at the first control point, then t 1 =... = tp– 1 and ti+p = ui,i = 0,... , n. We may still have
a free choice for t 0 that can be taken as equal to t 1. Likewise, to clamp the spline at the last control
pointti = ui,i = 0,... , n while tn+1 =... = tn+pis the free choice.
For a B-spline curve to be clamped at both ends, both knots tp–1 and tm–p+1, the two limits of the
full support range, may each be repeated p times, that is, t 0 =... = tp–1 and tn+1 =... = tn+p. With 2p
knots determined, the remaining n–p+1 internal knots tp,... , tn may be as follows:

Internal knots may be evenly spaced. The n–p+1 internal knots divide the chosen interval [a,b]
inton–p+ 2 spans. For their even spacing

t 0 = t 1 =... = tp–1 = a

taba

j

jp+–1 np

= + ( – )

• 
  
  
  
  • 2
    , j = 1, 2,... , n– p+1
  
  
  
  

tn+1 = tn+2 =... = tn+p=b (5.41)

The uniformly spaced knot vector does not require the knowledge of the position of control points,
and is simple to generate.

Internal knots may be averaged with respect to the parameters. As suggested by de Boor

t 0 = t 1 =... = tp–1 = a

t

p

jp uj n p

ij

jp

+–1 = i

+–2

=^1

• 1

Σ , = 1, 2,... , – + 1