Graphics Programs Reference
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( 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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