Graphics Reference
In-Depth Information
a reparameterization that is C
k
with respect to an orientation-preserving change of
parameters.
Note.
G
k
continuity really corresponds to the composite curve being a C
k
manifold
in a neighborhood of the point where they join. Unfortunately, this observation came
later in the historical development of the concept of G
k
continuity. Recognition of this
fact from the beginning would have clarified the issue. G
k
continuity has to do with
continuity of the shape of a curve whereas C
k
continuity has to do with the continu-
ity of the parameterization p(u).
Assume that r(u) = p(j(u)). The chain rule implies that
2
¢
()
=¢
()
(
)
¢
()
¢¢
()
= ¢
()
(
)
¢
()
+¢
()
(
)
¢¢
()
r u
q
jj
u
u
and
r
u
p
jj
u
u
p
jj
u
u
.
Letting
=
()
=
()
bj
1
and
b j
1
,
(11.131)
1
2
it follows that
¢
()
=
()
=¢
()
q
0
r
1
b
p
1
,
(11.132a)
¢¢
()
=
()
=¢¢
()
+¢
()
2
q
0
r
1
b
p
1
b
p
1
,
(11.132b)
2
with b
1
> 0.
Definition.
The bs are called
shape parameters
and equations (11.132) are called the
beta constraints
. The numbers b
1
and b
2
are called the
bias
and
tension
of the curve,
respectively.
Figure 11.34 expresses the geometry of the situation.
11.9.1
Theorem
(1) Two parametric curves meet with G
1
continuity if and only if they have the
same unit tangent vector at their common point.
Figure 11.34.
The shape parameters and geo-
metric continuity.
