Computer Aided Engineering Design

SOLID MODELING 255

(Figure 8.9c) that is closed but not orientable. For closed polyhedral surfaces, Möbius suggested a
way to determine whether they are orientable. The edges enclosing a face may be traversed clockwise
such that the normal to the face points into the solid and the face is to the right of the direction. For
a closed surface, each edge will receive two arrows, one for each face that it bounds. The surface is
orientable, if and only if, the directions of the two arrows are opposite to each other. An example is
shown in Figure 8.10 (c) for a hexahedral topology. Perimeter edges bounding a face may also be
traversed anticlockwise so long as this sense of traverse is maintained for all faces.

(c)Genus.It is an integer g that counts the number of handles (or voids) for closed orientable
surfaces (ε = 1) or crosscaps for closed non-orientable surfaces (ε = 0). For instance, g for a sphere
is zero and for a torus is 1. For surfaces with boundary components, one sets the genus to be equal
to that corresponding to a closed-up surface. A disc and a sphere have the same genus as zero, and
the genus of a Möbius band is the same as that of the projective plane, which is 1.

(d)Euler characteristic. In addition to the above, another invariant for polyhedrons based on Euler’s
law is called the Euler characteristic χ given as

χ = v – e + f (8.1)

wherev is the number of vertices, e the number of edges and f the number of faces. The rule holds
for any polydedron that is homeomorphic to a sphere. Thus, for hexahedrons in Figure 8.4 (a-d), χ
= 8 – 12 + 6 = 2. For a tetrahedron without holes, χ = 4 – 6 + 4 = 2. For surfaces, the Euler
characteristic can be expressed in terms of the above three invariants

χ = 2 – 2g – c ifε = 1

and χ = 2 – g – c ifε = 0 (8.2)

Since a hexahedral is homeomorphic to a sphere, χ for a sphere with genus 0 is expected to be 2

Figure 8.11 A soccer ball

which is confirmed by Eq. (8.2). Alternatively,
we may represent a sphere in discrete form as
a soccer ball, for example Figure 8.11 that has
60 vertices, 90 edges and 32 faces. The Euler
characteristic is 60 – 90 + 32 = 2.

(e)Connectivity number of a surface. This
number is equal to the smallest number of closed
cuts, or cuts connnecting points on different
boundaries or on previous cuts that can be made
to separate a surface into two or more parts. For closed surfaces, the connectivity number is 3 – χ
while for a surface with boundaries, it is 2 – χ. A surface with connectivity number 1, 2 or 3 is termed
simply, doubly ortriplyconnected respectively. A sphere is simply connected as it needs a single
closed cut to be separated into two parts while a torus is triply connected. The first closed cut will
render an open cylinder, the second cut joining the two boundaries of this cylinder will result in a
plane while the third cut across the plane boundary will separate it into two parts. For the surface in
Figure 8.6, c = 2 and g = 0. Thus, χ = 0 implying that the surface is doubly connected. One can make
two cuts, each joining the outer and inner boundaries to separate the surface into two parts.

8.5 Surfaces as Manifolds

Manifolds are local shapes describing the local topology of geometric entities. For a curve, its local