In O’ the mapping is like this:
=
=
⇒
=
=
=
⇒
=
=
=
⇒
=
′
h
v
u
p
dy
dx
d
w
v
u
p
dy
dx
d
v
u
p
p
0
~
0
~
0
0
~
0
0
2
2
2
2
2
2
1
1
1
1
1
1
0
0
0
0
This is a linear mapping the can be described by a 2x2 matrix.
2
1
22
21
12
11
0
0
~
d
M
h
d
M
w
p
m
m
m
m
p
M
p
⋅
=
∧
⋅
=
⇒
′
⋅
=
′
⋅
=
If you expand this and sort the equations you get 2 equation systems each with 2
unkowns. These can be described more easily with a new matrix.
Let
=
2
2
1
1
dy
dx
dy
dx
A
then the equation system can be rewritten as:
⋅
=
∧
⋅
=
12
11
12
11
0
0
m
m
A
w
m
m
A
w
This can be easily solved with determinants
Let
1
2
2
1
1
det
1
dy
dx
dy
dx
A
c
⋅
−
⋅
=
=
Then the resulting constants are:
dy
dv
dx
h
c
m
dx
dv
dx
h
c
m
dy
du
dx
w
c
m
dx
du
dy
w
c
m
=
⋅
⋅
−
=
=
⋅
⋅
=
=
⋅
⋅
−
=
=
⋅
⋅
=
2
22
1
21
2
12
2
11
To calculate the start values for u and v at the top of the bounding box the
transformation into O from O’ has to be reversed. Let u
s
and v
s
be the start values
then:
(
)
(
)
(
)
⋅
−
⋅
⋅
+
⋅
−
⋅
⋅
−
⋅
=
−
⋅
=
1
0
1
0
2
0
2
0
0
dx
y
dy
x
h
dx
y
dy
x
w
c
p
M
v
u
s
s
Drawing Engine
Chapter 8
Preliminary User's Manual S19203EE1V3UM00
243
Содержание uPD72257
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