Graphics Reference
In-Depth Information
1.2
1
0.8
0.6
n
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
−
0.2
−
0.4
t
Fig. 8.10.
The four Hermite interpolating curves.
and, unpacking the constants and polynomial terms, we obtain
⎡
⎤
⎡
⎤
2
−
211
n
1
n
2
s
1
s
2
⎣
⎦
·
⎣
⎦
−
33
1
0010
1000
−
2
−
n
=[
t
3
t
2
t
1
1]
·
(8.27)
This type of interpolation is called
Hermite interpolation
, after the French
mathematician Charles Hermite (1822-1901). Hermite also proved in 1873
that e is transcendental (see page 9).
This interpolant can be used as shown above to blend a pair of numerical
values and their tangent vectors, or it can be used to interpolate between
points in space. To demonstrate the latter we will explore a 2D example, and
it is very easy to implement the technique in 3D.
Figure 8.11 illustrates shows how two points (0, 0) and (1, 1) are to be
connected by a cubic curve that responds to the initial and final tangent
vectors. At the start point (0, 1) the tangent vector is [
−
50]
T
, and at the
final point (1, 1) the tangent vector is [0
5]
T
.The
x
and
y
interpolants are
−
⎡
⎤
⎡
⎤
2
−
211
0
1
−
5
0
⎣
⎦
·
⎣
⎦
−
33
1
0010
1000
−
2
−
x
=[
t
3
t
2
t
1
1]
·
⎡
⎤
⎡
⎤
2
−
211
0
1
0
−
⎣
⎦
·
⎣
⎦
−
33
1
0010
1000
−
2
−
y
=[
t
3
t
2
t
1
1]
·
5
