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P 2 (0) = β 1 P 1 (1)
(6.21)
and
P 2 (0) = β 1 P 1 (1) + β 2 P 1 (1) .
(6.22)
We can observe that β 1 = 1 provides the continuity of the parametric
first derivative vector and, β 1 =1and β 2 = 0 provides the continuity of
the parametric first and second derivative vectors. For β 1 > 0and β 2
0,
they form a basis, i.e., they are linearly independent. With the coecients
so determined, one can compute the four w n values in equation (6.2). These
values after simplification can be written as
w 2 ( β 1 2 ,t )=2 β 1 (1
t ) 3
(6.23)
w 1 ( β 1 2 ,t )=[2 β 1 t [ t 2
3 t +3]+2 β 1 [ t 3
3 t 2 +2]
(6.24)
+2 β 1 [ t 3
3 t +2]+ β 2 [2 t 3
3 t 2 + 1]]
w 0 ( β 1 2 ,t )=[2 β 1 t 2 [
t +3]+2 β 1 t [
t 2 +3]
(6.25)
+ β 2 t 2 [
2 t +3]+2[
t 3 + 1]]
w 1 ( β 1 2 ,t )=2 t 3 /δ.
(6.26)
6.3 Design Criteria for a Curve
In order to design a curve with two pieces of curve segments, say P 1 ( t )and
P 2 ( t ), we need to maintain position continuity, first order continuity, and
curvature continuity. The i th curve segment in terms of β 1 and β 2 can be
written as
P i ( t )=(2 β 1 (1
t ) 3 ) V i− 2
+ ([2 β 1 t [ t 2
3 t +3]+2 β 1 [ t 3
3 t 2 +2]
+2 β 1 [ t 3
3 t +2]+ β 2 [2 t 3
3 t 2 + 1]] ) V i− 1
+ ([2 β 1 t [ t 2
3 t +3]+2 β 1 [ t 3
3 t 2 +2]
(6.27)
+2 β 1 [ t 3
3 t +2]+ β 2 [2 t 3
3 t 2 + 1]] ) V i− 0
+ ([2 β 1 t 2 [
t 2 +3]
t +3]+2 β 1 t [
+ β 2 t 2 [
t 3 + 1]] ) V i +1 .
2 t +3]+2[
The first derivative of the curve P i ( t ) can be computed through
w
6 β 1 (1
t 2 )
2 ( β 1 2 ,t )=
(6.28)
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