SOLID MODELING 249vertices. The polyhedral representation has advantages of object coherence and compact storage
which outweighs its disadvantage of spatial addressing (or point membership classification discussed
in Chapter 9) wherein an involved algorithm is used to determine whether a point is inside the object.
The polyhedral representation is still quite broad to encompass generic (freeform) definitions of
solids and boundary determinism is a strong property suggesting that one may not consider points
interior to a solid, rather only the simple closed connected surface b(V) would seen sufficient to
represent a solid in E^3. In polyhedral representation, the surface information pertaining to faces,
edges and vertices is stored in two parts. The first is geometry wherein physical dimensions and
locations in space of individual components are specified. The other is topologydescribing the
connectivity between components. It is the topology that renders the object coherence property to this
representation, and that the geometry alone is inadequate. Topology regards two points as vertices
that bounds a line to define an edge. Likewise, a closed ring of edges bounds a surface to define a
face. Both geometry and topology are essential for a complete shape description.
The study of topology ignores the dimensions (lengths and angles) from the geometry and studies
any bridge back to which the answer was in the negative. Topology, as a subject by itself, is very
broad though from the viewpoint of solid/volumetric modeling, we can restrict ourselves to the
understanding of topological properties of surfaces as suggested by boundary determinism.
8.2 Topology and Homeomorphism
The aim in topology is to identify a set of rules or procedures to recognize geometrical figures. Two
figures would belong to the same topogical class if they have the same basic, overall structure even
though differing much in details. Consider a cube, for instance, in Figure 8.4(a). So long as the
internal angles are all 90° and the edge length ais the same, the form remains a cube irrespective of
the edge size specified. So is true for a sphere whose form is independent of the radius specified. A
cube is a special case of a block, wherein although all internal angles are 90° each, dimensions of
three mutually orthogonal edges a,b and c (Figure 8.4b) can be different. If we let the internal angles
to have values other than 90°, and also the edge lengths to be different, we get a form shown in Figure
8.4(c). What is common in these figures, though they are of different shapes, is that they are all
hexahedrons (of six sides). Thus, from geometry of a solid, if we ignore the intricacies of size (lengths
and angles), we address the topology of that solid. The illustrations in Figure 8.4 (a-c) are topologically
identical. So is the illustration in Figure 8.4(d) wherein some of the internal angles are zero.
the latter for the notations of continuity and
closeness. Topology studies the patterns in
geometric figures for relative positions without
regard to size. Topology is sometimes referred to
as the rubber sheet geometry since a figure can
be changed into an equivalent figure by bending,
twisting, stretching and other such transformations,
but not by cutting, tearing and gluing. Previously
known as analysis situs, topology is thought to
be initiated by Euler when the solution to the
Königsberg bridge problem (Figure 8.3) was
provided in 1736. The problem was to determine
if the seven bridges (edges) in the city of
Königsberg across four land patches (nodes) can
all be traversed in a single trip without doubling
Figure 8.3 Königsberg bridge circuit (nodes are
the land patches and edges are bridges)1
3 27(^54)
6