ComputerAided_Design_Engineering_amp_Manufactur.pdf

etween t 0 and t 1, while d 1 is the maximum deviation between t 1 and t 2. These maximum deviations are
obtained by using the Golden Section Search 32 and compared with the given tolerance �.
Step 1: If both d 0 and d 1 � �, set t 2h � t 2, t 2 � 0.5 (t 2 � t 2l), t 1h � t 2 and t 1 � 0.5 (t 0 � t 2), as shown in
Figure 2.13(b). This procedure is repeated as long as both d 0 and d 1 � �.
Step 2: If both d 0 and d 1 � �, set t 2l � t 2, t 2 � 0.5 (t 2 � t 2h), t 1h � t 2 and t 1 � 0.5 (t 0 � t 2), as shown in
Figure 2.13(c). This procedure is repeated as long as both d 0 and d 1 � �.
Step 3: If d 0 � � and d 1 � �, set t 1h � t 1 and t 1 � 0.5 (t 1 � t 1l), as shown in Figure 2.13(d). This procedure
is repeated as long as d 1 � � and d 0 � �.
Step 4: If d 0 � � and d 1 � �, set t 1l � t 1 and t 1 � 0.5 (t 1 � t 1h), as shown in Figure 2.13(e). This procedure
is repeated as long as d 0 � � and d 1 � �.
Steps 1 to 4 are repeated until:
both d 0 and d 1 � � by the specified convergence criterion, or,
both d 0 and d 1 � � and t 2 is equal to 1.
Then, the break point is considered to be found. t 2 is then assigned to an array element t(k). The center
of the circular arc is stored in x c(k) and y c(k), and its radius in r(k). Increment k.
Step 5: Set t 0 � t 2, t 1l � t 0, t 2l � t 0, t 2 � 1, t 1h � 1 and t 2h � 1, as shown in Figure 2.13(f). The maximum
deviation d 0 and d 1 are again calculated and compared with �.
Steps 1 to 5 are repeated until t(k) � 1.
End.
Figures 2.14a and 2.14b show the circular approximation for the same two Bezier curves as in Figures
2.12a and 2.12b. With the same tolerances and convergence criteria, the number of segments needed are
much fewer.
A C Continuity Approach
1
In the second approach, biarcs are used as the interpolating curve. A biarc is defined by two consecutive
circular arcs joined together in a continuous manner and was first proposed by Bolton14 C as an
interpolating curve to overcome the discontinuity of using just simple arcs. By imposing suitable end
conditions, chains of biarcs can be built up that maintain the continuity throughout. Depending on
the topological relationship of the two circular arcs, biarcs can be classified into C-shaped and S-shaped
1
C 0
C 1
FIGURE 2.14(a) Circular approximation of Bezier curve A. Tolerance : 0.6, convergence criteria : 0.06, number of
circular arcs: 4, processor time: 0.5.
© 2001 by CRC Press LLC