Computer Aided Engineering Design

SOLID MODELING 267

Block 2, respectively which may be identified by the three dimensions (length, width and height) and
the location of their respective reference points or local coordinate origins. Initially of standard size,
we may scale the three dimensions of the blocks by factors, say, x,y and z using the scale command.
Thus, for Block 1, the first CSG operation would be scale (Block 1, x 1 ,y 1 ,z 1 ). This object would then
require to be translated in that the reference point of the block would be shifted by, say, (a,b,c) with
respect to a global coordinate frame as shown. This may be accomplished using the translate command,
translate (scale (Block 1, x 1 ,y 1 ,z 1 ),a 1 ,b 1 ,c 1 ). Similar operations for Block 2 would be translate
(scale (Block 2, x 2 ,y 2 ,z 2 ),a 2 ,b 2 ,c 2 ). At this stage, the origins (and local axes) of the two blocks
would be positioned appropriately and the blocks would be united using the Boolean union or JOIN
command that would appear as

JOIN (translate (scale(Block 1, x,y 1 ,z 1 ),a 1 ,b 1 ,c 1 ), translate (scale (Block 2, x,y 2 ,z 2 ),a 2 ,b 2 ,c 2 ))
or if the union is expressed using the ‘+’ sign then

translate (scale (Block 1, x 1 ,y 1 ,z 1 ),a 1 ,b 1 ,c 1 ) + translate (scale (Block 2, x 2 ,y 2 ,z 2 ),a 2 ,b 2 ,c 2 )

(E1)

8.9.1 Boolean Operations

Given two sets (solids) A and B, their union (A ∪ B or A + B) consists of all points belonging to A
and B. Their intersection (A ∩ B) consists of points common to both A and B and the difference
A – B consists of points in A but not in B. Similarly, B–A would consist of points only in B and not
in A. Consider, for instance, the Boolean interactions between a sphere A and a block B (Figure 8.22
a) which is a cube of side length the same as the diameter of the sphere. The sphere is placed over
the cube such that the center of the sphere coincides with that of the top face of the cube. Figures 8.22
(b-e) show the union, intersection and difference operations A–B and B–A, respectively.

(a)

(b) Union (c) Intersection (d) Cube-sphere (e) Sphere-cube

Figure 8.22 Boolean operations using a cube and a sphere

Note that expression (E1) for the bracket design can be expressed graphically in the form of a
history tree or the CSG tree shown in Figure 8.23 (a). In addition, to cut holes in the bracket as shown