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13.1. Parametric Cur ves 587
It is possible to achieve even better continuity than that, using in Equa-
tion 13.15 the c defined by Equation 13.16 [332]:
c =
t
2
− t
1
t
1
− t
0
. (13.16)
This is also shown in Figure 13.9. If we instead set t
2
=2.0, then
c =1.0, so when the time intervals on each curve segment are equal, then
the incoming and outgoing tangent vectors should be identical. However,
this does not work when t
2
=3.0. The curves will look identical, but the
speed at which p(t) moves on the composite curve will not be smooth. The
constant c in Equation 13.16 takes care of this.
Some advantages of using piecewise curves are that lower degree curves
can be used, and that the resulting curves will go through a set of points.
In the example above, a degree of three, i.e., a cubic, was used for each
of the two curve segments. Cubic curves are often used for this, as those
are the lowest-degree curves that can describe an S-shaped curve, called
an inflection. The resulting curve p(t) interpolates, i.e., goes through, the
points q
0
, q
3
= r
0
,andr
3
.
At this point, two important continuity measures have been introduced
by example. A slightly more mathematical presentation of the continuity
concept for curves follows. For curves in general, we use the C
n
notation
to differentiate between different kinds of continuity at the joints. This
means that all the n:th first derivatives should be continuous and nonzero
all over the curve. Continuity of C
0
means that the segment should join
at the same point, so linear interpolation fulfills this condition. This was
the case for the first example in this section. Continuity of C
1
means that
if we derive once at any point on the curve (including joints), the result
should also be continuous. This was the case for the third example in this
section, where Equation 13.16 was used.
There is also a measure that is denoted G
n
. Let us look at G
1
(ge-
ometrical) continuity as an example. For this, the tangent vectors from
the curve segments that meet at a joint should be parallel and have the
same direction, but nothing about the lengths is assumed. In other words,
G
1
is a weaker continuity than C
1
, and a curve that is C
1
is always G
1
except when the velocities of two curves go to zero at the point where the
curves join and they have different tangents just before the join [349]. The
concept of geometrical continuity can be extended to higher dimensions.
The middle illustration in Figure 13.9 shows G
1
-continuity.
13.1.4 Cubic Hermite Interpolation
B´ezier curves are very useful in describing the theory behind the construc-
tion of smooth curves, but their use is not always intuitive. In this section,