Graphics Reference
In-Depth Information
The scaling action is relative to the origin, i.e. the point (0,0) remains (0,0)
All other points move away from the origin. To scale relative to another point
(
p
x
,p
y
) we first subtract (
p
x
,p
y
)from(
x
,
y
) respectively. This effectively trans-
lates the reference point (
p
x
,p
y
) back to the origin. Second, we perform the
scaling operation, and third, add (
p
x
,p
y
)backto(
x
,
y
) respectively, to com-
pensate for the original subtraction. Algebraically this is
x
=
s
x
(
x
p
x
)+
p
x
y
=
s
y
(
y − p
y
)+
p
y
−
(7.25)
which simplifies to
x
=
s
x
x
+
p
x
(1
−
s
x
)
y
=
s
y
y
+
p
y
(1
−
s
y
)
(7.26)
or in a homogeneous matrix form
⎡
⎤
⎡
⎤
⎡
⎤
x
y
1
s
x
0
p
x
(1
−
s
x
)
x
y
1
⎣
⎦
=
⎣
⎦
.
⎣
⎦
0
s
y
p
y
(1
−
s
y
)
(7.27)
00
1
For example, to scale a shape by 2 relative to the point (1, 1) the
matrix is
⎡
⎤
⎡
⎤
⎡
⎤
x
y
1
20
−
1
x
y
1
⎣
⎦
=
⎣
⎦
·
⎣
⎦
02
1
00 1
−
7.3.3 2D Reflections
The matrix notation for reflecting about the
y
-axis is:
⎡
⎤
⎡
⎤
⎡
⎤
x
y
1
−
100
010
001
x
y
1
⎣
⎦
=
⎣
⎦
·
⎣
⎦
(7.28)
or about the
x
-axis
⎡
⎤
⎡
⎤
⎡
⎤
x
y
1
100
0
x
y
1
⎣
⎦
=
⎣
⎦
.
⎣
⎦
10
001
−
(7.29)
However, to make a reflection about an arbitrary vertical or horizontal
axis we need to introduce some more algebraic deception. For example, to
make a reflection about the vertical axis
x
= 1, we first subtract 1 from the
x
-coordinate. This effectively makes the
x
= 1 axis coincident with the major
y
-axis. Next we perform the reflection by reversing the sign of the modified
