Computer Aided Engineering Design

SOLID MODELING 269

solid if there exists an open ball B of radius r centered at Q such that the ball does not intersect with the
solid. That is, if any point p∈ B (|p–Q | < r), then p∉V.A set of all exterior points is termed the exterior
of the solid represented as E(V). Points that neither belong to the interior or exterior constitute the
boundaryb(V) of the solid. The closure of a solid C(V) is then defined as the union of its interior and
the boundary, that is, C(V) = I(V)∪b(V) or I(V) + b(V). In other words, the closure of a solid is the
complementE(V) of its exterior and contains all
points that do not belong to the exterior of the
solid. In a manner, V and C(V) are the same with
C(V) as the formal definition of the solid. The above
discussion seems necessary to circumvent certain
pitfals of the Boolean operations as given by an
example in Figure 8.24. For a block and cylinder
shown adjacent to each other, their intersection
yields a common disc (a one-manifold) that is not
a valid solid and the Boolean operation, as is,
violates the closure property in Section 8.1(f).
To eliminate the lower dimensional results of
set operations, we need to regularize the Boolean operations as follows:
We first compute the result as usual wherein the lower dimensional features (like the disc above)
may be generated. Then, the interior of the result is computed that eliminates all lower dimensional
components. In this step, we achieve only the interior of the solid which is united with its boundary
in the subsequent step by computing the closure. The regularized Boolean operations for solids A and
B can be summarized as

Regularized union: C[I (A∪B)]

Regularized intersection: C[I(A∩B)]

Regularized difference: C[I (A – B)]

Based on the above, the regularized intersection between the block and the cylinder shown in Figure
8.24 is an empty set. The two examples that illustrate the modeling procedure using constructive solid
geometry are: (i) a hexagonal bolt, different parts of which are shown in Figure 8.25 as components
of the history tree and (ii) a more complex one is of a Robosloth, the CSG model of which is shown
in Figure 8.26(a) with the realized prototype in Figure 8.26(b).

8.10 Other Modeling Methods

Many engineering components are such that the cross-section is uniform in the depth direction. Also,
many are axisymmetric. To model such components, solid modelers employ different sweep methods.
A planar wireframe cross-section composed of a simple (nonself-intersecting) closed contour of
edges (linear or curved) can be extrudedalong the vector perpendicular to the plane containing the
contour. This is called translational sweep an example of which is shown in Figure 8.27 (a). A simple
closed contour may also be revolved by a known angle about an axis to result in a solid of revolution
(Figure 8.27b). This is called rotational sweep. In many instances, the wireframe cross-section need
not follow a linear path and the sweep path may be represented by a curve. An example of a solid
obtained using nonlinear sweep is depicted in Figure 8.27 (c). A sweep path is often termed as the
directrix. In a hybrid sweep, we can combine two or more sweep solids using the regularized set
operations discussed above.

A

B

A∩ B

Figure 8.24 Boolean intersection operation with

a block and a cylinder