Computer Aided Engineering Design

SPLINES 137

Φ 0 (t) = a 0 + a 1 t + a 2 t^2 + a 3 t^3

Noting that Φi(0) = ΦΦii′′(0) = (0) = 0,a 0 = a 1 = a 2 = 0 so that Φ 0 (t) = a 3 t^3 wherea 3 is an unknown.

Next, a cubic expression for φ 1 (t) may be written as

Φ 1 (t) = b 0 + b 1 t + b 2 t^2 + b 3 t^3

As the spline is continuous up to the second derivative, at t 1 = 1, we have Φ 0 (1) = Φ 1 (1),ΦΦ 01 ′′(1), (1)
andΦΦ 01 ′′(1) = ′′(1). These conditions, respectively, yield

b 0 + b 1 + b 2 + b 3 =a 3

b 1 + 2b 2 + 3b 3 = 3a 3

2 b 2 + 6b 3 = 6a 3

Solving the above in terms of b 3 , we get

Φ 1 (t) = (a 3 – b 3 )– 3(a 3 – b 3 )t + 3(a 3 – b 3 )t^2 + b 3 t^3

Also, since the spline is symmetric about t = 2, it is expected that φ 1 ′′(2) = 0 which gives

–3(a 3 – b 3 ) + 12(a 3 – b 3 ) + 12b 3 = 0 or b 3 = –3a 3

Thus

Φ 1 (t) = 4a 3 – 12a 3 t + 12a 3 t^2 – 3a 3 t^3

The unknown constant a 3 can be determined using the standardization integral in Eq. (5.12). Using
symmetry and noting that the order of the curve is 4,

0

4

0

2

0

1

0

1

2

∫∫∫ ∫ΦΦΦ Φ( ) = 2 tdt ( ) = 2 tdt ( ) + 2 tdt^1 ( )tdt

=

2

+

11

2

= 6 =^1 =^1

4

=^1

24

33

33

aa

a

m

⇒a

Thus,Φ 0 () =^13
24
tt and Φ 1 () =^1
24
t [4 – 12t + 12t^2 – 3t^3 ]. Since Φ(t) is symmetric about t= 2,

Φ(2 + δ) = Φ(2 –δ). For 2 –δ = t, 2 + δ = 4 –t and so Φ(t) = Φ(4 –t). More specifically, the splines

in knot spans 2≤t≤ 3 and 3≤t≤ 4 are Φ 2 (t) =^1
24
[4 – 12(4 –t) + 12(4 – t)^2 – 3(4 –t)^3 ] and

Φ 4 (t) =^1
24
(4 –t)^3. The plot of the computed spline is shown in Figure 5.5.
The general form of the standardized spline in the knot span ti–4≤t≤ti is shown in Figure 5.6.
Note that the spline is extended indefinitely from the end points ti− 4 to the left and ti to the right,
respectively, on the t axis. Thus, the spline has an indefinite number of spans and is non-zero over
precisely 4 spans. It is also termed as a fundamental spline, or the spline of minimal support the
support being the number of spans over which the spline is non-zero. Note that this spline is of the
lowest order that can be C^2 continuous for which reason, it is called the fundamental spline. We
would realize later in this chapter that such standardized splines have barycentric properties similar