Graphics Reference
In-Depth Information
W
P
2
V
P
1
A
B
P
1
sV
P
2
B
tW
A
FIGURE B.13
Two lines are closest to each other at points
P
1
and
P
2
.
B.2.7
Area calculations
Area of a triangle
The area of a triangle consisting of vertices
V
1
,
V
2
,
V
3
is one-half times the length of one edge times the
perpendicular distance from that edge to the other vertex. The perpendicular distance from the edge to a
vertex can be computed using the cross-product (see
Figure B.14
). For triangles in two dimensions, the
z
-coordinates are essentially considered zero and the cross-product computation is simplified accord-
ingly (only the
z
-coordinate of the cross-product is non-zero).
The signed area of the triangle is required for computing the area of a polygon. In the two-
dimensional case, this is done simply by not taking the absolute value of the
z
-coordinate of
the cross-product. In the three-dimensional case, a vector normal to the polygon can be used to indi-
cate the positive direction. The direction of the vector produced by the cross-product can be com-
pared to the normal (using the dot product) to determine whether it is in the positive or negative
direction. The length of the cross-product vector can then be computed and the appropriate sign
applied to it.
1
2
1
2
1
2
V
1
Area(
V
1
,
V
2
,
V
3
)
V
2
V
3
V
1
V
3
sin
V
2
V
3
(
V
1
V
3
)
(
V
2
V
3
)
V
2
V
3
(
V
1
V
3
)
(
V
2
V
3
)
V
2
V
3
V
1
V
2
V
3
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