Computer Aided Engineering Design

268 COMPUTER AIDED ENGINEERING DESIGN

in Figure 8.23 (b), the way is to instantiate two cylinders, transform (scale and translate) them
appropriately and cut (represented using ‘–’ sign) them from the respective blocks before uniting the
latter. The graphic representation of the procedure then appears as shown in Figure 8.23(c). For a
cylindrical object with radius and length as defining features, and scale factors as r and l, respectively,
the CSG expression for a bracket with two holes may be written as

[translate(scale(Block 1, x 1 ,y 1 ,z 1 ),a 1 ,b 1 ,c 1 ) – translate (scale(Cylinder 1, r 1 ,l 1 ),a 3 ,b 3 ,c 3 )]

• [translate(scale(Block 2, x 2 ,y 2 ,z 2 ),a 2 ,b 2 ,c 2 ) – translate(scale(Cylinder 2, r2,l 2 ),a 4 ,b 4 ,c 4 )]
  (E2)

Figure 8.23 The CSG tree representations for a bracket without and with holes

Transform (Block 1) Transform (Block 2)

Block 1 Block 2

(a)

(b)

+

Transform (Block 2) transform (cylinder 2)

Transform (Block 1) Transform (Cylinder 1)

Block 2 Cylinder 2

Block 1 Cylinder 1

(c)

+

Every solid constructed using the CSG scheme has a corresponding design expression and thus a
CSG tree associated with it. We may note, however, that a CSG solid may not necessarily be
represented by a unique tree as the operations for constructing the solid may not be unique. For
instance, the bracket frame above may result by cutting a block from another block. Alternatively, we
may join the two blocks first and then cut holes at respective sites. A CSG tree is concise, unambiguous,
closed and easy to create and edit. Its domain depends on the available set of primitive objects, as
well as the set of transformational and combinational operators.

8.9.2 Regularized Boolean Operations

The interior and boundary of a solid V have been defined in Section 8.1. Intuitively, the interior I(V) of
a solid comprises all points within the solid and not those on its boundary. A point Q is exterior to the