Computer Aided Engineering Design

102 COMPUTER AIDED ENGINEERING DESIGN

j= 1, i = 0; bbb bb 01 = (1 – )uu 00 + 10 = (1 – )uu 01 +

j = 1, i = 1: bbb bb 11 = (1 – )uu 10 + 20 = (1 – ) + uu 12

j = 2, i = 0: bbb 02 = (1 – )uu 01 + 11

= (1 – u){(1 – u)b 0 + ub 1 } + u{(1 – u)b 1 + ub 2 }

= (1 – u)^2 b 0 + 2u(1 – u)b 1 + u^2 b 2

=^2 C 0 (1 – u)^2 u^0 b 0 +^2 C 1 (1 – u)^1 u^1 b 1 +^2 C 2 (1 – u)^0 u^2 b 2 (4.31)
Forn= 3

j = 1, i = 0: bbbbb 01 = (1 – )uu 00 + 10 = (1 – )uu 01 +

j = 1, i = 1: bbbbb 11 = (1 – )uu 10 + 20 = (1 – ) + uu 12

j = 1, i = 2: bbbbb 21 = (1 – )uu 20 + 30 = (1 – )uu 23 +

j = 2, i = 0: bbb b bb 02 = (1 – )uu 01 + 112 = (1 – )u uu)u 01 + 2 (1 – +^22

j= 2, i= 1: bbb b bb 12 = (1 – )uu 11 + 212 = (1 – )u 12 + 2 (1 – )uuu +^23

j = 3, i = 0: bbb b^30 = (1 – )uu 02 + 123 = (1 – )u uu uuu 0 + 3 (1 – )^2 b 1 + 3^2 (1 – )bb 2 +^33

=^3 Cuu Cuu Cuu Cuu 0 (1 – )^30 bbb b 0 +^31 (1 – )^211 +^32 (1 – )^122 +^33 (1 – )^033

(4.32)

withbbi^0 = .i Here bij, called de Casteljau points, represent intermediate points like e,f and rin
Figure 4.11, for the nth degree curve. For instance, b 32 represents the fourth point in the second level
of linear interpolation. After linear interpolation is exhausted, the final point b 0 n lies on the nth degree
curve for some parameter u limited in the range [0, 1]. The line segments b 0 b 1 ,b 1 b 2 ,... , bn–1bn,
calledlegs,when joined in this order form a polylinemostly referred to as the control polyline. The
working of the algorithm is illustrated for n= 2 and 3, respectively, and the schematic of geometric
construction is given in Figure 4.12 (a) and (b). For n= 2

Figure 4.12 Generalized de Casteljau’s algorithm for: (a) n = 2 and (b) n= 3.

All line segments are divided in the ratio u: (1 – u)

b 0

b 1

b 2

b 1

b 0

b 2

b 3

b 01

b 01

b 02

b 02

b 11

(a)

b 11

b 12

b 21

b 03

(b)

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