Computer Aided Engineering Design

158 COMPUTER AIDED ENGINEERING DESIGN

property in Section 5.7G1. Shape functions N4,7(t) and N4,10(t) are only C^0 (position) continuous at
respective knots t 7 and t 10.

5.10 Parameterization

For an approximating spline in Eq. (5.34), we require n+1 control points b 0 ,b 1 ,... , bn and the order
p of the curve as input. From the properties mentioned earlier, it is required that the number of knots,
m+1 must satisfy the equality m = n + p. However, these knots are not known a priori and can be
chosen in a number of ways. One way is to assign a parameter ui to each control point bi and then
compute the knot vector from these parameters. Parameter assignment may be accomplished in any
of the following ways.

Uniformly spaced method
Forn+1 parameters to be equally spaced in [a,b], we have

uaii = + ban– ,i = 0,... , n (5.36)

This assignment scheme, though simple, does not work well when the control points are placed
unevenly. In such cases, curves with unsatisfactory shapes might result.

Chord length method
To ensure that the curve shape closely follows the shape of the corresponding polyline, this method
of parameterization may be employed. Herein, parameters are placed proportional to the chord
lengths of control polyline, that is, if the first parameter corresponding to b 0 is u 0 , then the subsequent
parametersuicorresponding to bi may be written as

uui

k

i

= 0 + Σ=1 | bbkk – –1|,i = 1,... , n (5.37)

We may normalize the parameterization in Eq. (5.37) by setting u 0 to 0 and dividing ui by the total

chord length of the polyline L
k

n

= Σ=1 | bbkk – –1|. Otherwise, we may choose to set this parameterization

in a chosen domain [a,b] for which

uabai

k

i

kk

k

n

kk

= + ( – )

| – |

| – |

=1 –1

=1 –1

Σ

Σ

bb

bb

,i = 1,... , n (5.38)

The chord length method is widely used and it usually performs well. Sometimes, a longer chord may
cause its curve segment to have a bulge bigger than necessary, which is a common problem with the
chord length method.

Centripetal method
This is derived from a concept analogous to the centripetal acceleration when a point is traversing
along a curve. The notion is that the centripetal acceleration should not be too large at sharp turns
(smaller radii of curvature). For the parameters to lie in the domain [a,b], the centripetal method
gives the parameter values as