Graphics Reference
In-Depth Information
And the solution guaranteed by the theorem is
t
=
M
†
b
=
0.4
0.2
4
3
=
2.2.
(10.51)
Example 2: An underdetermined system
The system
13
x
y
=
4
(10.52)
+
=
y
=
3/4
x
3
y
4
x
=
4
has a great many solutions; any point
(
x
,
y
)
on the line
x
+
3
y
=
4 is a solution
(see Figure 10.10). The solution that's
closest to the origin
is the point on the line
x
+
3
y
=
4 that's as near to
(
0, 0
)
as possible, which turns out to be
x
=
0.4
;
y
=
Figure 10.10: Any point of the
blue line is a solution; the red
point is closest to the origin.
1.2. In this case, the matrix
M
is
13
; its SVD and pseudoinverse are simply
M
=
UDV
T
=
1
√
10
1
√
10
√
10
and
(10.53)
/
3
/
M
†
=
VD
†
U
=
1
/
√
10
1
√
10
1
=
1
.
/
10
/
√
10
(10.54)
/
3
/
10
3
And the solution guaranteed by the theorem is
M
†
b
=
1
4
=
0.4
1.2
.
/
10
(10.55)
3
/
10
Of course, this kind of computation is much more interesting in the case where
the matrices are much larger, but all the essential characteristics are present even
in these simple examples.
A particularly interesting example arises when we have, for instance, two
polyhedral models (consisting of perhaps hundreds of vertices joined by trian-
gular faces) that might be “essentially identical”: One might be just a translated,
rotated, and scaled version of the other. In Section 10.4, we'll see how to represent
translation along with rotation and scaling in terms of matrix multiplication. We
can determine whether the two models are in fact essentially identical by listing
the coordinates of the first in the columns of a matrix
V
and the coordinates of the
second in a matrix
W
, and then seeking a matrix
A
with
AV
=
W
.
(10.56)
This amounts to solving the “overconstrained system” problem; we find that
A
=
V
†
W
is the best possible solution. If, having computed
A
, we find that
AV
=
W
,
(10.57)
then the models are essentially identical; if the left and right sides differ, then the
models are not essentially identical. (This entire approach depends, of course, on
corresponding vertices of the two models being listed in the corresponding order;
the more general problem is a lot more difficult.)
We now describe a way to apply linear transformations to generate
translations,
and at the same time give a nice model for the points-versus-vectors ideas we've
espoused so far.
