Computer Aided Engineering Design

COMPUTATIONS FOR GEOMETRIC DESIGN 283

C (x 3 ,y 3 ,z 3 ) such that the normal to the plane is given by n= AB×BC. The volume of the tetrahedron
ABCDis computed by calculating the determinant Δ as

Δ =

1

1

1

1

111

222

333

444

xyz

xyz

xyz

xyz

(9.3)

PointD can be placed in three possible ways with respect to the plane ABC.

(a) Point D lies on the same side of the plane as its normal if Δ is negative.
(b) Point D lies on the opposite side of the normal if Δ is positive.
(c) Point D lies in the plane if Δ is zero.

The normal n = AB×BC is given by

n

rjk

= – – –

• – –

212121

323232

xxyyzz

xxyyzz

IfD is placed on the same side of the normal, then

nBD > 0 or =

• – –
  
  
  • – –
  
  
  
  


• – –

1 > 0

424242

212121

323232

⋅Δ

xxyyzz

xxyyzz

xxyyzz

Performing a few row operations in Δ in Eq. (9.3) results in

Δ=

1

• – – 0


• – – 0


• – – 0

=

1

• – – 0


• – –

111

212121

323232

424242

111

424242

212121

xyz

xxyyzz

xxyyzz

xxyyzz

xyz

xxyyzz

xxyyzz 00

• – – 0

= –

323232

1

xxyyzz

Δ

Thus, for D on the side of the normal, Δ is negative and vice-versa.

Example 9.4. Three points A (1, 0, 0), B (0, 1, 0) and C (0, 0, 1) define a triangular lamina (Figure
9.10a). Find how the points: (a) D (0, 0, 0), (b) D (1, 1, 1), (c) D (1/3, 1/3, 1/3) and (d) D (1, 1, –1)
are placed with respect to this lamina.

(a)D (0, 0, 0). Find ΔABCD

Δ =

1001

0101

0011

0001

= 1 (> 0)