256 COMPUTER AIDED ENGINEERING DESIGN
topology is a line for which reason curves are termed one-manifold.A surface, however curved and
complicated so long as it does not intersect with itself, can be thought of as composed of small, two-
dimensional Euclidean patches glued together. For this reason, a surface is called two-manifold.
Mathematically, a surface is two-manifold if and only if at point P on the surface, there exists an open
ballB of a sufficiently small radius r with center P such that the intersection of the ball with the
surface is homeomorphic to an open disc. For instance, a sphere and a torus (Figure 8.12a) are both
two-manifolds throughout since their intersection anywhere with an open ball yields a closed curve
homeomorphic to an open disc. However, a closed surface of two cubes sharing an edge shown in
Figure 8.12 (b) is not a two-manifold at the site shown as the intersection with an open ball yields two
intersecting discs that cannot be morphed into a single disc without gluing.
8.6 Representation of Solids: Half Spaces
Boundary determinism in section 8.1 (c) is a strong property which suggests that a solid V can be
identified by a closed and orientable surface b(V) which may either be analytical (a cube or a sphere
for example) or may be composed of different generic patches (Coon’s, Bézier, B-spline and others)
discussed in Chapter 7. We can locally control the shape of such patches to design the desired solid
model. A marked advantage achieved for representation schemes is that they are only required to
store the boundary surface information and not the points enclosed within b(V). In this regard, half
spaces contribute elegantly in the representation scheme for bounded solids, in that by combining
half spaces using set operations (union, intersection and difference discussed in section 8.9), various
solids can be constructed.
Half-spaces are unbounded geometric entities such that they divide the representation space E^3
into two infinite portions, one filled with material while the other empty. A half-space H can be
defined as
H = {P | P∈E^3 and f(P) < 0}whereP is a point in E^3 and f(P) = 0 is the equation of the surface. Most widely used half-spaces
amongst analytical are planar, cylindrical, spherical, conical and toroidal defined below
Planar half-space: H = {(x,y,z)|z < 0}(a) (b)Figure 8.12 (a) Two-manifolds and (b) not a two-manifold