SOLID MODELING 259Due to ambiguous representation, wireframes are limited in use in solid modeling though they are
popular in applications like preview oranimationssince one does not need to render a complex model
or an animated sequence which could be very time consuming.
8.8 Boundary Representation Scheme
B-rep for short, this solid modeling scheme can be regarded as an extension of wireframe modeling
to include the face information. The faces, individually, can either be analytical surfaces or design
patches discussed in Chapter 7. B-rep directly employs the Jordan’s curve theorem on boundary
determinism stating that a closed, connected, orientable and nonself-intersecting surface determines
the interior of a solid. As in wireframe models, both topological and geometric information is stored
in B-rep as well wherein relationships among vertices, edges, faces and orientations form a part of the
topological data while design equations of edges and faces are stored as geometric input. Face
orientations may be recorded such that a normal to the face points into the solid. This can be ensured
by the clockwise ordering of vertices (right-handed rule) associated with the face. Once done for all
faces, we can then inspect the normal vectors to distinguish the interior of the solid from its exterior.
Thus, for a tetrahedron in Figure 8.14, the vertices of the front face may be ordered as 2, 1 and 4. For
the face on the right, the order should be 3, 2 and 4. Likewise, for the back and bottom faces, the order
should be 1, 3, 4 and 1, 2, 3, respectively.
8.8.1 Winged-Edge Data Structure
A data structure in wide use for a B-rep model is the Baumgart’s winged-edge data structure for
polyhedrons, which is also applicable to homeomorphic solids that can be achieved by stretching the
straight edges to curved ones to have curved faces. An advantage is that the winged-edge structure
employs only edges to document the connectivity. First, the data structure is described for polyhedrons
with no voids. Consider a tetrahedron (Figure 8.17a) which shows the edges numbered within
Figure 8.16 A wireframe representing a block void within a block and its solid
model interpretations (below)