Computer Aided Engineering Design

270 COMPUTER AIDED ENGINEERING DESIGN

The cross-section contour is often chosen to lie on a plane. The contour should be closed and
nonself-intersecting in that it should bind a single domain with finite area. However, the wireframe
may not be connected. That is, it can have an outer ring of edges and a few inner ones to depict the
voids in sweep solids, as shown in Figures 8.27 (a) and (b). Further, the directrix should be such that
self-intersection does not occur at any instant when a contour is swept along it.
The other modeling approach is an extension of the tensor product method for surface patches to
three-dimensional parametric space. The resulting solid is called a hyperpatchsince it is bounded by
surface patches. For three normalized parameters u,v and w, points on or inside the hyperpatch are
expressed using an ordered Cartesian triple. That is

P( , , ) = [ ( , , ) ( , , ) ( , , )] = =0 =0

=0

uw xuwyuwzuwi uw

m

j

n

k

p

ijk

vvvvvΣΣΣC ij k (8.4)

whereCijk are the data points for Bézier or B-spline hyperpatches in three-space. A tricubic Bézier
hyperpatch for example is obtained for m = n = p = 3. The face surfaces, edge curves, and corner
vertices can be obtained by substituting appropriate values of the parameters into the above equation.
The face surfaces are given by P(0, v,w), P(1, v,w), P(u, 0, w), P(u, 1, w), P(u,v, 0), and P(u,v, 1).
Similarly, the equation of any curve is obtained by fixing two parametric variables. P(u, 0, 0) and
P(u, 1, 0) are two of the 12 boundary curves. The eight corner vertices correspond to the values of
u,v and w as 0 and 1 only.

Join 55

Figure 8.25 The CAD of a bolt with its history tree (arrows relate between the nodes of the history tree
and the corresponding features)