Graphics Reference
In-Depth Information
Thus, the matrix for the transformation sending the
u
's to the
v
's is just
KM
−
1
.
Let's make this concrete with an example. We'll find a matrix sending
u
1
=
2
u
2
=
1
and
(10.27)
3
−
1
to
v
1
=
1
1
v
2
=
2
−
,
and
(10.28)
1
respectively. Following the pattern above, the matrices
M
and
K
are
M
=
21
3
(10.29)
−
1
K
=
12
1
.
(10.30)
−
1
Using the matrix inversion formula (Equation 10.17), we find
−
M
−
1
=
−
1
5
1
−
1
(10.31)
−
32
so that the matrix for the overall transformation is
J
=
KM
−
1
=
12
1
−
·
−
1
5
1
−
1
(10.32)
−
1
−
32
=
7
.
/
−
/
5
3
5
(10.33)
−
2
/
53
/
5
As you may have guessed, the kinds of transformations we used in WPF in
Chapter 2 are internally represented as matrix transformations, and transformation
groups are represented by sets of matrices that are multiplied together to generate
the effect of the group.
Inline Exercise 10.12:
Verify that the transformation associated to the matrix
J
in Equation 10.32 really does send
u
1
to
v
1
and
u
2
to
v
2
.
Inline Exercise 10.13:
Let
u
1
=
1
3
; pick any two nonzero
vectors you like as
v
1
and
v
2
, and find the matrix transformation that sends
each
u
i
to the corresponding
v
i
.
and
u
2
=
1
4
The recipe above for building matrix transformations shows the following:
Every linear transformation from
R
2
to
R
2
is determined by its values on two
independent vectors. In fact, this is a far more general property: Any linear trans-
formation from
R
2
to
R
k
is determined by its values on two independent vectors,
and indeed, any linear transformation from
R
n
to
R
k
is determined by its values
on
n
independent vectors (where to make sense of these, we need to extend our
definition of “independence” to more than two vectors, which we'll do presently).
