Graphics Reference
In-Depth Information
Y
ax
+
by
-
d
> 0
ax
+
by
-
d
= 0
ax
+
by
-
d
< 0
X
Fig. 10.20.
The Hessian normal form of the line equation partitions space into two
zones.
Conversely, a point (
x
2
,y
2
) which is in the partition opposite to the direction
of the normal vector creates the inequality
ax
2
+
by
2
−
d<
0
(10.35)
This space-partitioning feature of the Hessian normal form is useful in clipping
lines against polygonal windows.
10.2.4 The Hessian Normal Form from Two Points
Given two points (
x
1
,y
1
)and(
x
2
,y
2
), we can compute the values of
a
,
b
and
d
for the Hessian normal form as follows. To begin with, we observe:
y
−
y
1
y
2
−
y
1
∆
y
∆
x
=
=
(10.36)
x
−
x
1
x
2
−
x
1
therefore
(
y
−
y
1
)∆
x
=(
x
−
x
1
)∆
y
(10.37)
and
x
1
∆
y
+
y
1
∆
x
= 0 (10.38)
which is the general equation of a straight line. For the Hessian normal form,
∆
x
2
+∆
y
2
=1
x
∆
y
−
y
∆
x
−
Therefore, the Hessian normal form is given by
x
∆
y
−
y
∆
x
−
(
x
1
∆
y
−
y
1
∆
x
)
∆
x
2
+∆
y
2
= 0
(10.39)
Let's test this with an example. Given the following points: (
x
1
,y
1
)=(0
,
1)
and (
x
2
,y
2
)=(1
,
0); ∆
x
=1
,
∆
y
=
−
1.
