Graphics Reference
In-Depth Information
10.4 Point Inside a Triangle
We often require to test whether a point is inside, outside or touching a
triangle. Let's examine two ways of performing this operation. The first is
related to finding the area of a triangle.
10.4.1 Area of a Triangle
Let's declare a triangle formed by the anti-clockwise points (
x
1
,y
1
)
,
(
x
2
,y
2
)
and (
x
3
,y
3
), as shown in Figure 10.23. The area of the triangle is given by:
1
2
(
x
2
−
1
2
(
x
2
−
1
2
(
x
3
−
A
=(
x
2
−
x
1
)(
y
3
−
y
1
)
−
x
1
)(
y
2
−
y
1
)
−
x
3
)(
y
3
−
y
2
)
−
x
1
)(
y
3
−
y
1
)
which simplifies to
A
=
1
2
[
x
1
(
y
2
−
y
3
)+
x
2
(
y
3
−
y
1
)+
x
3
(
y
1
−
y
2
)]
and this can be further simplified to
x
1
y
1
1
A
=
1
2
x
2
y
2
1
(10.49)
x
3
y
3
1
Figure 10.24 shows two triangles with opposing vertex sequences. If we calcu-
late the area of the top triangle with anti-clockwise vertices, we obtain
A
=
1
2
[1(2
−
4) + 3(4
−
2) + 2(2
−
2)] = 2
Y
P
3
P
2
P
1
X
Fig. 10.23.
The area of the triangle is computed by subtracting the smaller triangles
from the rectangular area.
