Graphics Reference
In-Depth Information
is the set of parameters for the points in
S
«
X
. In other words, if
I
is not empty, then
[
]
SX P PPP PP
«= +
a
,
+
b
.
1
1
2
1
1
2
We shall explain this process with the example in Figure 3.4. In this example,
I
=
[]
« •
[
)
«-•
(
]
« •
[
)
=
[
]
01
,
t
,
,
t
t
,
t
,
t
,
1
2
3
3
2
which clearly gives the right answer.
3.2.3
Liang-Barsky Line Clipping
The Liang-Barsky line-clipping algorithm ([LiaB84]) optimizes the Cyrus-Beck line-
clipping algorithm in the case where we are clipping against a rectangle. It starts by
treating a segment as a parameterized set. Let
P
1
= (x
1
,y
1
) and
P
2
= (x
2
,y
2
). A typical
point
P
= (x,y) on the oriented line
L
determined by
P
1
and
P
2
then has the form
P
1
+ t
P
1
P
2
. See Figure 3.5. If we let Dx = x
2
- x
1
and Dy = y
2
- y
1
, then
xx
=+
=+
D
D
xt
1
yy
yt
.
1
If the window
W
we are clipping against is the rectangle [xmin,xmax] ¥ [ymin,ymax],
then
P
belongs to
W
if and only if
x
min
£+
x
D
D
x t
£
x
max
1
y
min
£+
y
y t
£
y
max
1
that is,
-£-
£ -
-£-
£
D
D
D
D
xt
x
min
max
x
1
xt
x
x
1
yt
y
y
min
1
yt
y
max
-
y
.
1
Figure 3.5.
Liang-Barsky line clipping.
