SOLID MODELING 2518.3 Topology of Surfaces
To understand the polyhedral representation of solids better, study of the topological properties of
surfaces becomes essential. Surfaces are compact and connected topological objects on which each
point has a neighborhood (a closed curve around a point on the surface) homeomorphic to either a
QPRFigure 8.6 Boundary and interior points on the
surfaceplaneR^2 or a half plane H^2. Points of the first
type are interior points (point P in Figure 8.6)
while of the second type are the boundary points
(Q and R in the figure). A set of all boundary
points constitutes the boundary of the surface.
The boundary can comprise one or more
components, each of which is homeomorphic to
a circle. In Figure 8.6, there are two boundary
components, the exterior and the interior, each of
which can be morphed into respective circles.
The bounding surface in polyhedral
representation of a solid must satisfy certain
properties so that the solid is well-defined. A
valid solid consists of a complete set of spatial
points occupied by an object. Solids may vary
in form with different applications. However, in
general, they are boundedandconnected. A boundedsolid is defined within a finite space. If
connected, there exists a (continuous) path, totally interior to the solid, that connects any pair of
points belonging to it. Note that this is true even when a solid may have multiple cavities. For
bounded and connected solids, the bounding surface must be (a) closed, (b) orientable, (c) connected
and (d) nonself-intersecting. Nonself-intersection is essential for otherwise, a bounding surface may
enclose two or more domains or volumes defying the Jordon’s curve theorem.
A closed surface is one having no boundary. For instance, a sphere and a torus in Figure 8.4 (e)
and (g). A sphere and a cube, both with cylindrical through holes, being homeomorphic to a torus are
also closed surfaces. A disc has one boundary curve, a circle, and is topologically the same as a
hemisphere (Figure 8.7a). A cylinder (Figure 8.7b), open at both ends (discs removed from both
ends), has two boundary curves. However, a cylindrical surface (Figure 8.7c) has only one boundary
curve.
8.3.1 Closed-up Surfaces
A generic surface (as in Figure 8.6) can be thought to be composed of boundary components, which
PP P(a) (b) (c)
Figure 8.5 Various simply closed planar shapes of identical topology