Graphics Reference
In-Depth Information
1.8
1.6
1.4
1.2
1
y
0.8
[0
5]
−
0.6
0.4
0.2
[
−
5 0]
0
−
1
−
0.5
0
0.5
1
1.5
x
Fig. 8.11.
A Hermite curve between the points (0, 0) and (1, 1) with tangent vectors
(
−
5
,
0) and (0
, −
5).
which become
⎡
⎣
⎤
⎦
−
7
13
−
x
=[
t
3
t
2
t
1
1]
7
t
3
+13
t
2
·
=
−
−
5
t
5
0
⎡
⎣
⎤
⎦
−
7
8
0
0
y
=[
t
3
t
2
t
1
1]
7
t
3
+8
t
2
·
=
−
When these polynomials are plotted over the range 0
1, we obtain the
curve shown in Figure 8.11. We have now reached a point where are starting
to discover how parametric polynomials can be used to generate space curves,
which is the subject of the next chapter. So, to conclude this chapter on
interpolants, we will take a look at interpolating vectors.
≤
t
≤
8.3 Interpolating Vectors
So far we have been interpolating between a pair of numbers. Now the question
arises: can we use the same interpolants for vectors? Perhaps not, because
a vector contains both magnitude and direction, and when we interpolate
between two vectors we must ensure that both quantities are preserved. For
example, if we interpolated the
x
-and
y
-components of the vectors [2 3]
T
and
[4 7]
T
, the in-between vectors would preserve the change of orientation but
ignore the change in magnitude. To preserve both, we must understand how
the interpolation should operate.
Figure 8.12 shows two unit vectors
V
1
and
V
2
separated by an angle
θ
.The
interpolated vector V can be defined as a proportion of
V
1
and a proportion
