DESIGN OF CURVES 119

Eq. (4.58) implies that with the first segment given, the position continuity constraints the first
control point q 0 of the neighboring segment. For slope (C^1 ) continuity at the junction point pm = q 0
(Figure 4.20b),

αα 1 1

1

2

2

2

(1)

=

(0)

=

d

du

d

du

rr

t

or α 1 m(pm – pm–1) = α 2 n(q 1 – q 0 )

or q 1 = λ(pm – pm–1) + pm = (λ + 1)pm – λpm–1 (4.59)

whereα 1 and α 2 are normalizing scalars for the slope along the unit tangent vector t and λ
α
α
=^1.
2

m

n

p 0

(u 1 = 0)

p 1 p 2

pm–1

(u 2 = 1) qn

q 2

pm=q (^0) q 1
(m 1 = 1)
(u 2 = 0)
(a)C^0 continuity
p 0
p 1
p 2
pm–1
qn
pm=q 0
p 1
pn–1
(b)C^0 continuity q 1 = (λ + 1) pm – λpm–1
Figure 4.20 C^0 and C^1 continuous Composite Bézice curves
Thus, for two Bézier segments with position continuity at the junction point, slope continuity
further constraints the second control point q 1 of r 2 (u 2 ) to be collinear with the last leg pm–1pm of
the first polyline. For the segments to have the curvature (C^2 ) continuity at the junction point
κ 1 (1) = κ 2 (0), where κ 1 (u 1 ) and κ 2 (u 2 ) are curvature expressions for the two segments. Or
d
du
d
du
d
du
d
du
d
du
d
du
1
1
2
1
2 1
1
1
3
2
2
2
2
2 2
2
2
3
(1) (1)
(1)

(0) (0)
(0)
rr
r
rr
r
××
(4.60)