Graphics Reference
In-Depth Information
The rotation operator turns these four points to
⎧
⎨
V
ar
=(
2
.
82
,
8
.
49
)
,
V
br
=(
0
.
71
,
6
.
36
)
,
Output points :
⎩
V
cr
=(
2
.
12
,
4
.
95
)
,
V
dr
=(
4
.
24
,
7
.
07
)
.
R
o
Cen
ter of
Rot
ation
In this case, we verify the earlier observation that distances between input points
and output points are maintained. We notice that the output vertices define a rect-
angle that is the same size as the input rectangle, only at a rotated orientation. In
general, the rotate operator does not change the shape or the area (size) of objects,
just the orientation is altered. In the case of Figure 8.15, one worrying observation
45
o
Original
Rectangle
is that although the
R
operator has accomplished the desired/expected results
of rotating the input rectangle, the operator has also moved the resulting rectangle
in a not entirely intuitive manner. Similar to the scaling operator, the moving of
the resulting rectangle is due to the fact that rotation is defined with respect to
to the origin. Figure 8.16 shows an example of a perhaps more intuitive rotation
with the center of rotation being at point
(
45
)
Figure 8.16.
Example of
an “intuitive” rotation op-
eration.
(
,
)
5
4
. We will learn how to accomplish
this kind of rotation in the next chapter.
Summary of the Rotation Operator
Order of operations.
As in previous operators, with our vector representation,
RV
a
is an undefined operation.
Geometric properties.
In general, all corresponding input and output edges
will be of the same length. In addition, the angles formed by the input edges will
be exactly the same as the corresponding output angles.
R
−
1
(
θ
)=
(
−
θ
)
Reversibility.
R
.
Identity.
R
(
0
)
.
Extension to 3D.
Rotation in three dimensions is a much more complex oper-
ation. For example, instead of a center of rotation position (e.g., origin), in 3D,
rotation is defined with respect to an axis. We will examine rotations in 3D space
in Chapter 16.
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