248 COMPUTER AIDED ENGINEERING DESIGN
(c) Boundary determinism: Jordan’s theorem^1 for a two-dimensional Euclidean plane E^2 states that
a simple (non self intersecting) closed (Jordan) curve divides E^2 into regions interior and exterior
to the curve. Formally, for C as a continuous simple closed curve in E^2 ,E^2 \C (complement of C
inE^2 ) has precisely two connected^2 components. Equivalently in E^3 , a simple and orientable
closed surface b (V) divides a solid V into the interior I (V) and exterior E^3 – V spaces. In other
words, if the closed surface b (V) of a solid V is known, the interior I (V) of the solid V is
unambiguously determined.
(d) Homogenous three-dimensionality: A solid set V must not have disconnected or dangling subsets
as shown in Figure 8.2 as such sets defy boundary determinism above.
(e) Rigidity:The relative positions between any two points p 1 and p 2 in V must be invariant to re-
positioning or re-orientation of the solid in E^3.
(f) Closure: Any set operation (union, intersection and subtraction) when applied to solids V 1 and V 2
must yield a solid V 3 satisfying all the aforementioned properties.
The discussion above suggests two ways in which a generic solid may be represented. The first, most
general representation is through a set of contiguous points in 3-space. A solid object may be
represented as a set of adjacent cells using a three-dimensional array. The cell size is usually the
maximum resolution of the display. Space arrays have two advantages as a representation; (a) spatial
addressing wherein it is easy to determine whether a point belongs to a solid or not and (b) spatial
uniquenessthat assures that two solids cannot occupy the same space. In contrast, it has two grave
disadvantages; (a) this representation lacks object coherence in that there is no explicit relation
between cells occupying the solid. Note that in most space arrays, only the occupancy state of a cell
is stored. (b) Also, the representation is very expensive in terms of storage space.
A cell in the interior of a solid in a space array representation has the same occupancy state as its
adjacent cells. However, at the object’s boundaries, this is not so. Thus, a more concise approach to
represent a solid is through the boundary points (or the bounding surface) that partitions the points
internal and external to the body, as suggested by the boundary determinism property. In this polyhedral
representation, usually, the bounding surface is subdivided into faces that may be planar or curved.
Each face may be identified by a perimeter ring of edges that again may be planar or curved. A face
may have one or more internal rings to define voids or holes. Lastly, adjacent edges intersect at
(^1) The theorem seems geometrically plausible though its proof requires concepts from topology.
(^2) For a pair of non-empty subsets U and V of E (^2) ,U and V are connected if U∩V = {} and U∪V = E (^2).
Figure 8.1 Interior and boundary points of a solid
r p
Ball
B
Interior
I(V)
B 1
Space
E^3 – V
Boundaryb(V)
Figure 8.2 Example of dangling plane and line with
a cube