Graphics Reference
In-Depth Information
If we now solve for
x
and
y
we get
x
=
10
×
2
−
20
×
1
=
0
0
= undefined
2
×
2
−
4
×
1
y
=
2
×
20
−
4
×
10
=
0
0
= undefined
2
×
2
−
4
×
1
which yields undefined results. The reason for this is that (7.119) is the same
as (7.120) - the second equation is nothing more than twice the first equation,
and therefore brings nothing new to the relationship. When this occurs, the
equations are said to be
linearly dependent
.
Having shown the algebraic origins of the determinant, we can now go on
to investigate its graphical significance. Consider the transform
x
y
=
ab
cd
x
y
·
(7.121)
The determinant of the transform is
ad
-
cb
. If we subject the vertices of a
unit-square to this transform, we create the situation shown in Figure 7.30.
The vertices of the unit-square are moved as follows:
(0
,
0)
(0
,
0)
(1
,
0)
(
a, c
)
(1
,
1)
(
a
+
b, c
+
d
)
(0
,
1)
(
b, d
)
(7.122)
From Figure 7.30 it can be seen that the area of the transformed unit-square
A
is given by
area
=(
a
+
b
)(
c
+
d
)
−
B
−
C
−
D
−
E
−
F
−
G
1
2
bd
1
2
ac
1
2
bd
1
2
ac
=
ac
+
ad
+
cb
+
bd
−
−
cb
−
−
−
cb
−
=
ad
−
cb
(7.123)
which is the determinant of the transform. But as the area of the original
unit-square was 1, the determinant of the transform controls the scaling factor
applied to the transformed shape.
Let's examine the determinants of two transforms. The first 2D transform
encodes a scaling of 2, and results in an overall area scaling of 4:
20
02
and the determinant is
20
02
=4
