Graphics Reference
In-Depth Information
and another transformation T
2
that transforms T
1
:
x
=
Ax
+
By
y
=
Cx
+
Dy
T
2
×
T
1
(7.5)
If we substitute the full definition of T
1
we get
x
=
A
(
ax
+
by
)+
B
(
cx
+
dy
)
y
=
C
(
ax
+
by
)+
D
(
cx
+
dy
)
T
2
×
T
1
(7.6)
which simplifies to
x
=(
Aa
+
Bc
)
x
+(
Ab
+
Bd
)
y
y
=(
Ca
+
Dc
)
x
+(
Cb
+
Dd
)
y
T
2
×
T
1
(7.7)
Caley proposed separating the constants from the variables, as follows:
x
y
=
ab
cd
x
y
T
1
·
(7.8)
where the square matrix of constants in the middle determines the trans-
formation. The algebraic form is recreated by taking the top variable
x
,
introducing the = sign, and multiplying the top row of constants [
ab
]in-
dividually by the last column vector containing
x
and
y
. We then examine
the second variable
y
, introduce the = sign, and multiply the bottom row of
constants [
cd
] individually by the last
column
vector containing
x
and
y
,to
create
x
=
ax
+
by
y
=
cx
+
dy
(7.9)
Using Caley's notation, the product T
2
×
T
1
is
x
y
=
AB
CD
x
y
·
(7.10)
But the notation also intimated that
x
y
=
AB
CD
ab
cd
x
y
·
·
(7.11)
and when we multiply the two
inner matrices
together they must produce
x
=(
Aa
+
Bc
)
x
+(
Ab
+
Bd
)
y
y
=(
Ca
+
Dc
)
x
+(
Cb
+
Dd
)
y
(7.12)
or in matrix form
x
y
=
Aa
+
Bc
x
y
Ab
+
Bd
·
(7.13)
Ca
+
Dc
Cb
+
Dd
