Computer Aided Engineering Design

116 COMPUTER AIDED ENGINEERING DESIGN

rQr R( ) = 1 ( ), ( ) = ( )

=1

3

3

(^122) =1
3
3

uBuuBuiΣΣi i i i i 2

Defining the parameters u 1 = 2u, u 2 = 2u – 1, they satisfy the requirement that u 1 ∈ [0, 1], when
u∈ [0, 1/2], and u 2 ∈ [0, 1], when u∈ [1/2, 1]. The new control points of the two curve segments
areQ 0 , Q 1 ,Q 2 ,Q 3 and R 0 ,R 1 ,R 2 ,R 3. These are yet to be determined. The original curve and
its segments can be written as:

(a) r(u) = (1 – u)^3 P 0 + 3u(1 – u)^2 P 1 + 3u^2 (1 – u)P 2 + u^3 P 3

(b) r 1 (u 1 ) = (1 – u 1 )^3 Q 0 + 3u 1 (1 – u 1 )^2 Q 1 + 3 uu u 12 (1 – ) 12 QQ+ 13 3

(c) r 2 (u 2 ) = (1 – u 1 )^3 R 0 + 3u 2 (1 – u 2 )^2 R 1 + 3 uu u 22 (1 – ) 22 RR+^323

(d) r ̇(u) = –3(1 – u)^2 P 0 + 3[(1 – u)^2 – 2u(1 – u)]P 1 + 3[2u(1 – u) – u^2 ]P 2 + 3u^2 P 3

(e) r ̇

r

1

1

1

() = u d 1

du

du

du

= 2[–3(1–u 1 )^2 Q 0 +3[(1 –u 1 )^2 –2u 1 (1 – u 1 )]Q 1 + 3[2u 1 (1 – u 1 )–uu 12 ]+ 3 ]QQ 2 12 3

(f) ̇ ̇r(u) = 6(1 – u)P 0 + 3[–4 + 6u]P 1 + 3[2 – 6u]P 2 + 6uP 3

(g) ̇ ̇ ̇r(u) = –6P 0 + 18P 1 – 18P 2 + 6P 3

(h) ̇ ̇r 11 ()u = 4[6(1 – u 1 )Q 0 + 3[–4 + 6u 1 ]Q 1 + 3[2 – 6u 1 ]Q 2 + 6u 1 Q 3 ]

(i) ̇ ̇ ̇r 1 (u 1 ) = 8[–6Q 0 + 18Q 1 – 18Q 2 + 6Q 3 ]

Evaluate at u = 0, where the parameter u 1 = 0. From (a) and (b) we can find that (i) Q 0 = P 0.
From (d) and (e), since the slopes are equal

r ̇(0) = ̇r 1 (0)⇒ 3(P 1 – P 0 ) = 2 ∗ 3(Q 1 – Q 0 )⇒ (ii) (P 1 – P 0 ) = 2(Q 1 – Q 0 )

From (f) and (h), since the second derivatives are the same

̇ ̇r(0) = (0)r ̇ ̇ 1 ⇒ 6(P 0 – 2P 1 + P 2 ) = 4 ∗ 6(Q 0 – 2Q 1 + Q 2 )⇒ (iii) (P 0 – 2P 1 + P 2 ) = 4(Q 0 – 2Q 1 + Q 2 )

From (g) and (i) one obtains

̇ ̇ ̇r(0) = (0) ̇ ̇ ̇r 1 ⇒ (iv) (P 3 – 3P 2 + 3P 1 – P 0 ) = 8(Q 3 – 3Q 2 + 3Q 1 – Q 0 )

From the above equations (i) to (iv)

Q 1 = P 0

Q 1 =(P 0 + P 1 )/2

Q 2 = (P 2 + 2P 1 + P 0 )/4

Q 3 = (P 0 + 3P 1 + 3P 2 + P 3 )/8

DetermineR 0 ,R 1 ,R 2 ,R 3 in a similar manner in terms of P 0 ,P 1 ,P 2 ,P 3. this is left as an exercise.

4.4.3 Degree-Elevation of a Bézier Segment

The flexibility in designing a Bézier segment may also be improved by increasing its degree which
results in an addition of a data point. The shape of the segment, however, remains unchanged. Thus,