Graphics Programs Reference
In-Depth Information
(
x
*
,
y
*
)
(
x
*
,
y
*
)
S
R
(
x
,
y
)
(
x
,
y
)
Q
(
x
*
,
y
*
)
P
P
P
Q
(
x
,
y
)
(a)
(b)
(c)
Figure 1.14: Halfturns.
1.2.7 Glide Reflections
This transformation is a special combination of three reflections. Imagine the two
vertical parallel lines
x
=
L
and
x
=
M
and the horizontal line
y
=
N
(Figure 1.15a).
Reflecting a point
P
=(
x, y
) about the line
x
=
L
is done by translating the line to the
y
axis, reflecting about that axis, and translating back. The transformation matrix is
⎛
⎞
⎛
⎞
⎛
⎞
⎛
⎞
100
010
−
−
100
010
001
100
010
L
−
100
010
2
L
⎝
⎠
⎝
⎠
⎝
⎠
=
⎝
⎠
,
L
01
01
01
and the transformed point is
⎛
⎝
⎞
⎠
=(
−
100
010
2
L
(
x, y,
1)
−
x
+2
L, y,
1)
.
01
Reflecting this point about the line
x
=
M
results in
⎛
⎞
100
010
2
M
−
⎝
⎠
=(
x
(
−
x
+2
L, y,
1)
−
2
L
+2
M, y,
1)
01
(a translation), and reflecting this about the horizontal line
y
=
N
yields
⎛
⎞
100
0
−
10
02
N
⎝
⎠
=(
x −
2
L
+2
M,−y
+2
N,
1)
.
(
x −
2
L
+2
M, y,
1)
1
This particular glide reflection is therefore a translation in
x
andareflectionin
y
.A
general glide reflection is the product of three reflections, the first two about parallel
lines
L
and
M
and the third about a line
N
perpendicular to them (Figure 1.15b).
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