Graphics Programs Reference
In-Depth Information
x
=L
x
=M
y
=N
(a)
(b)
Figure 1.15: Glide Reflection.
1.2.8 Improper Rotations
A rotation followed by a reflection about one of the coordinate axes is called an
improper
rotation
. The transformation matrices for the two possible improper rotations in two
dimensions (Figure 1.16) are
cos
θ
10
0
=
cos
θ
,
−
sin
θ
sin
θ
sin
θ
cos
θ
−
1
sin
θ
−
cos
θ
cos
θ
−
=
−
,
−
sin
θ
10
01
cos
θ
−
sin
θ
sin
θ
cos
θ
−
sin
θ
cos
θ
and the transformation rules therefore are
x
∗
=
x
cos
θ
+
y
sin
θ,
y
∗
=
x
sin
θ
−
y
cos
θ,
x
∗
=
y
∗
=
−
x
cos
θ
−
y
sin
θ,
−
x
sin
θ
+
y
cos
θ.
Notice that the determinant of an improper rotation matrix equals
−
1, like that of a
pure reflection.
(a)
(b)
Figure 1.16: Improper Rotations.
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