Computer Aided Engineering Design

284 COMPUTER AIDED ENGINEERING DESIGN

Also, the normal n to the lamina is given by AB×BC=i + j + k.
This implies that D is positioned opposite to that of the normal of lamina ABC.

(b)D (1, 1, 1), ΔABCD = –2, thus D is positioned on the same side as of the normal of lamina ABC.

(c)D (1/3, 1/3, 1/3), ΔABCD = 0. Thus, D is coplanar with lamina ABC. Further we can check if
D lies within or outside the lamina by doing a point in polygon test (Section 9.3.1) on one of
the projections. If the xy plane is chosen, the projected points are A′ (1, 0), B′ (0, 1), C′(0, 0) and
D′ (1/3, 1/3). Since a triangular lamina is a convex polygon, we can use the special property and
confirm a point to be inside if it lies to the left of all three edges of the lamina.

DetermineΔABD′′ ′ =

101

011

001

= 1 (> 0) , implying D′ is to the left of A′B′.

Figure 9.10 Proximity of a point and a plane (a) point on the same side of the normal, (b) point to the
opposite side of the normal, (c) point within the triangular lamina and (d) point coplanar
with the given plane but not within the lamina.

C D

z

y B

x

O

A

C

D

n

B

z

y

O

x

A

C

D

n

B

z

y

O

x

A

C

D

z n

y B

x

O

A

n

(a) (b)

(c)

(d)