Graphics Reference
In-Depth Information
called the
blending function
approach to interpolation and approximation. We shall
see more examples of this in the future. We want to make one last important obser-
vation in this context.
Let
A
Õ
R
k
and consider a function p :
A
Æ
R
m
of the form
()
=
()
++
()
()
++
()
pu
f u
p
...
f u
p
+
g u
v
...
g u
v
,
(11.27)
1
1
s
s
1
1
t
t
where
p
i
,
v
j
Œ
R
m
, the
p
i
are “points,” the
v
j
are “vectors,” and the f
i
(u) and g
i
(u) are
real-valued functions. The difference between a “point” and a “vector” here has to do
with how they transform. We assume that an affine map T of
R
m
sends a “vector”
v
(thought of as a directed segment from the origin to the
point v
) to the “vector”
T(
0
)T(
v
). Given an arbitrary affine map T of
R
m
, express T in the form
()
=
()
+
TM
q
q
q
0
,
where M is a linear transformation (see Chapter 2 in [AgoM05], in particular, Theorem
2.5.8). Let
X
= p(
A
) Õ
R
m
. The question we want to ask is how one can compute the
tranformed set
Y
= T(
X
). An arbitrary point p(u) of
X
gets mapped by T to
() ( )
++
() ( )
+
() ( )
++
() ( )
+
f uM
p
...
f uM
p
g uM
v
...
g uM
v
q
.
(11.28)
1
1
s
s
1
1
t
t
0
On the other hand, if we simply replaced the points
p
i
and vectors
v
i
in equation
(11.27) by their transformed values, we would get
() ( )
+
(
)
++
() ( )
+
(
)
+
() ( )
++
() ( )
=
fuM
pq
...
fuM
pq
guM
v
...
guM
v
1
1
0
s
s
0
1
1
t
t
s
Ê
Á
ˆ
˜
Â
() ( )
++
() ( )
+
()
() ( )
+++
() ( )
fuM
p
...
fuM
p
fu
q
+
guM
v
...
guM
v
.
(11.29)
1
1
s
s
j
01
1
t
t
j
=
1
Definition.
The function p(u) defined by equation (11.27) is said to be
affinely invari-
ant
if expressions (11.28) and (11.29) define the same point.
11.2.2.3 Theorem.
The function p(u) defined by an equation of the form (11.27)
is affinely invariant if and only if
s
Â
()
=
fu
1
.
j
j
=
1
Proof.
This is easy. One simply equates expressions (11.28) and (11.29).
11.2.2.4
Corollary.
The Lagrange and Hermite interpolating curves are affinely
invariant.
Proof.
This follows from Theorem 11.2.2.3 and equations (11.5) and (11.24).
The importance of being affinely invariant lies in the fact that in order to move a
curve (or the set traced out by an arbitrary parameterization defined by equation
(11.27)) we do not have to move every point on it (which would not be very feasible
even in the case of computer graphics where the screen consists of only a finite
