Basic Geometric Modeling Tools - Computer Graphics and Geometric Modeling - page 250

Figure 6.18. Computing the area of a polygon.

n

pp
pp

() Ê
Ë

ˆ
¯

 12
2

01 i -

A = det . (6.30)

0 i

i =

Proof. Consider Figure 6.18 and use Formula 6.7.3. The argument also works for
nonconvex polygons.

6.7.5 Formula. The signed volume V of a tetrahedron with vertices p 1 , p 2 , p 3 , and

p 4 is given by

ppp
ppp
ppp
ppp

1
1
1
1

Ê ˆ

11 12 13

pp
pp
pp

-
-
-

Ê ˆ

1 2

Á
Á
Á

˜
˜
˜

21 22 23

Á
Á

˜
˜

= () = () V 16 det 16 det , (6.31)

1 3

31 32 33

Ë ¯

1 4

Ë ¯

41 42 43

where p i = (p i1 ,p i2 ,p i3 ). The value for V will be positive if we order the points so that
the ordered basis ( p 1 - p 2 , p 1 - p 3 , p 1 - p 4 ) induces the standard orientation of R 3 .

Proof. The first equality follows from Formula 6.7.2 and the fact that the paral-
lelolopiped defined by p 1 - p 2 , p 1 - p 3 , and p 1 - p 4 can be decomposed into six tetra-
hedra of equal volumes. The second equality in formula (6.31) follows from basic
properties of the determinant, in particular, the fact that the determinant of a matrix
is unchanged if a row of the matrix is subtracted from another row.

Let g: [a,b] Æ R 2 be a differentiable curve in the plane. Let p Œ R 2 . The set Definition.

[ ] () tab p g t

Π[ ]

,

is called the region subtended by the curve g from the point p . See Figure 6.19(a).

6.7.6 Formula. Let g: [a,b] Æ R 2 , g(t) = (g 1 (t),g 2 (t)), be a curve in the plane. The
signed area A of the region subtended by the curve g from the origin is given by