Graphics Reference
In-Depth Information
Figure 8.3 shows that if we examine the relative distances between the two input
points (i.e.,
V
a
and
V
c
), we see that the distance in the
x
-direction is
dx
i
:
dx
i
=2
dy
i
=3
dx
o
=2
dy
o
=3
=
|
−
|
=
|
−
|
=
|−
|
=
,
dx
i
x
c
x
a
1
3
2
2
Output
Points
whereas the distance in the
y
-direction is
dy
i
:
Input
Poin
t
s
=
|
−
|
=
|
−
|
=
|−
|
=
.
dy
i
y
c
y
a
4
7
3
3
Figure 8.3.
Relative dis-
tances between input and
output points.
When we compare these distances to that from the output points (i.e.,
V
at
and
V
ct
)
dx
o
and
dy
o
:
dx
o
=
|
x
ct
−
x
at
|
=
|
6
−
8
|
=
|−
2
|
=
2
,
dy
o
=
|
y
ct
−
y
at
|
=
|
1
−
4
|
=
|−
3
|
=
3
.
We observe that
dx
o
=
dx
i
,
dy
o
=
dy
i
.
This observation should not be surprising because the translate operator offsets
all
input points by the same displacements and thus should not alter the relative
distances between the input and output positions.
In general, if input vertices
Translation operator.
Main-
tains the shape, size, and ori-
entation of an object.
to a translation operator are from a geometric shape (e.g., a triangle), then the
results of the translate operation would be a displaced triangle with the exact
same dimensions (edge lengths) and orientation.
Figure 8.4 shows the effect of
applying the
T
(
5
,−
3
)
translation operator on four vertices that form a rectangle:
Edges.
Describe the
adja-
cency
between vertices. For
example, in Figure 8.4, vertex
V
a
is adjacent to
V
b
and
V
c
.In
this case, we say
V
a
V
b
is an
edge and
V
a
V
c
is another edge.
Direction of edges.
If we treat
an edge as a vector connecting
two vertices, then each edge
has a unique direction.
Angles between edges.
These
are angles between connect-
ing edges. For example, in
Figure 8.4,
∠
V
a
V
b
V
c
is a 90-
degree angle.
⎧
⎨
V
a
=(
3
,
7
)
,
V
b
=(
1
,
7
)
,
Input points :
⎩
V
c
=(
1
,
4
)
,
V
d
=(
,
)
.
3
4
(
,−
)
We see that applying the
T
5
3
operator on the input points
=
(
,−
)
,
V
at
V
a
T
5
3
V
bt
=
(
,−
)
,
V
b
T
5
3
V
ct
=
V
c
T
(
5
,−
3
)
,
V
dt
=
V
d
T
(
5
,−
3
)
results in
⎧
⎨
V
at
=(
8
,
4
)
,
V
bt
=(
,
)
,
6
1
Output points :
⎩
V
ct
=(
6
,
4
)
,
V
dt
=(
8
,
1
)
.
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