Graphics Reference
In-Depth Information
Let
L
i
be the line determined by the segment [
Q
i
,
Q
i+1
]. Define intervals
I
i
= [a
i
,b
i
]
as follows:
Case 1:
L
is parallel to
L
i
.
(a) If
L
lies entirely in
H
i
, then let
I
i
= (-•, +•).
(b) If
L
lies entirely outside of
H
i
, then let
I
i
= f.
Case 2:
L
is not parallel to
L
i
.
In this case
L
will intersect
L
i
in some point
P
=
P
1
+ t
i
P
1
P
2
. We distinguish
between the case where the line
L
“enters” the halfplane
H
i
and where the
line “exits”
H
i
.
(a) (Line enters) If
P
1
P
2
•
N
i
≥ 0, then let
I
i
= [t
i
, +•).
(b) (Line exits) If
P
1
P
2
•
N
i
< 0, then let
I
i
= (-•, t
i
].
See Figure 3.4, where a segment
S
= [
P
1
,
P
2
] is being clipped against a triangle
Q
1
Q
2
Q
3
.
Note that finding the intersection point
P
in Case 2 is easy. All we have to do is solve
the equation
(
)
=
NP PPQ
i
•
+
t
-
0
1
1
2
i
for t.
Now let
I
0
= [a
0
,b
0
] = [0,1]. The interval
I
0
is the set of parameters of points which
lie is
S
. It is easy to see that the interval
k
max
I
=
I
i
i
=
0
=
[
]
a
, min
b
i
i
0
££
ik
0
££
ik
=
[
]
ab
,
Figure 3.4.
Cyrus-Beck line clipping.
