Computer Aided Engineering Design

SPLINES 133

be known. For the second span, t 1 ≤t≤t 2 ,Φ 1 (t 1 ) = y 1 and Φ 1 (t 2 ) = y 2 are known. The remaining two
conditions may be obtained by incorporating C^1 and C^2 continuity at t = t 1. That is, ΦΦ 10 ′′() = tt 11 ()and
ΦΦ 11 ′′() = tt0 1′′().
Proceeding likewise, cubic segments Φi(t),i= 0,... , n–1 over all the knot spans can be
determined. In practice however, polynomial splines are not computed in this manner. First, it is not
recommended to specify second or higher order derivatives as input since they are usually not very
accurate. Second, there may be a possibility for accumulation of errors especially when the number
of knot spans is large.
Alternatively, a polynomial spline may be computed as follows. Since Φi(t) is cubic, let

Φi(t) = a 0 + a 1 t + a 2 t^2 + a 3 t^3 (5.5)

inti≤t≤ti+1. Also, let si and si+1 be the unknown slopes at t = ti and t = ti+1 respectively. The
unknownsa 1 ,i = 0,.. ., 3 can be determined using the following conditions.

1

1

01 2 3

012 3

= or

23

+1^2 +1^3 +1

2

+1 +1

2

0

1

2

3

+1

+1

0

1

2

3

tt t

tt t

tt

tt

a

a

a

a

y

y

s

s

a

a

a

a

i ii

i ii

i i

i i

i

i

i

i

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎡

⎣⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

=

1

1

01 2 3

012 3

23

+1^2 +1^3 +1

2

+1 +1

2

–1

+1

+1

tt t

tt t

tt

tt

y

y

s

s

i ii

i ii

i i

i i

i

i

i

i

(5.6)

for which Eq. (5.5) becomes

Φi

i ii

i ii

i i

i i

i

i

i

i

tttt

tt t

tt t

tt

tt

y

y

s

s

() = [1 ]

1

1

01 2 3

012 3

23

23

+1^2 +1^3 +1

2

+1 +1

2

–1

+1

+1

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

= [1 ]

(– 3) (3 – ) – –

6–6(2+ )(+ 2)

23

+1 +1

2

3

+1

2

3

+1

2

2

2

+1

2

+1

3

+1

3

+1 +1

2

+1

tt t

ttt

h

ttt

h

tt

h

tt

h

tt

h

tt

h

ttt

h

ttt

h

i i i

i

i i i

i

ii

i

i i

i

ii

i

ii

i

i ii

i

i i i

i

22

+1

3

+1

3

+1

2

+1

2

3322

+1

+1

–3(+ ) 3(+ ) – (+2 ) –(2+ )

2 –2 11

tt

h

tt

h

tt

h

tt

h

hhhh

y

y

s

s

i i

i

i i

i

i i

i

i i

i

iiii

i

i

i

i

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥⎥

⎥

⎥

(5.7)
Eq. (5.7) ensures that the curve Φ(t) = {Φi(t),i = 0, ...,n– 1} is position and slope continuous for

t 0 ≤t≤tn. For continuity of the second derivative, one must impose ΦΦi′′–1() = ttiii′′(). Differentiating

Eq. (5.7) twice gives