Background Information and Techniques - Computer Animation: Algorithms and Techniques - page 424

Area of a polygon
The area of a polygon can be computed as a sum of the signed areas of simple elements. In the two-
dimensional case, the signed area under each edge of the polygon can be summed to form the area
( Figure B.15 ). The area under an edge is the average height of the edge times its width ( Eq. B.31 , where
subscripts are computed modulo n þ

1).

X n

ð y i þ y iþ 1 Þ

Area ¼ ð x iþ 1 x i Þ

2

i¼ 1

(B.31)

2 X n
i¼ 1

1

¼ ð y i x iþ 1 y iþ 1 x i Þ

The area of a polygon can also be computed by using each edge of the polygon to construct a tri-
angle with the origin ( Figure B.16 ). The signed area of the triangle must be used so that edges directed

Positive area

Negative area

Q

FIGURE B.15
Computing the area of a polygon.

B

A

E

C

D

Q

Area of polygon ( A , B , C , D , E ) Area of Triangle ( Q , A )

Area of Triangle( Q , B )
Area of Triangle( Q , C )

Area of Triangle( Q , D )
Area of Triangle( Q , E )

FIGURE B.16
The area of a two-dimensional polygon: the edges are labeledwith letters, triangles are constructed fromeach edge to