254 COMPUTER AIDED ENGINEERING DESIGN
With some basic surfaces aforementioned, the main classification theorem for surfaces states that
every closed surface is homeomorphic to a sphere with some handles^3 or crosscaps attached, that is,
every single surface is one of the following: (a) a sphere, (b) a connected sum of tori or (c) a
connected sum of projective planes or cross caps. The scope of this chapter shall be restricted to
closed connected non-intersecting surfaces that can be constructed in three-dimensions. Thus, we
would deal with surfaces homeomorphic to a sphere or a connected sum of tori.
8.4 Invariants of Surfaces
To better understand surfaces, we would require some characteristics that capture the essential qualitative
properties, and remain invariant under homeomorphic transformations. These are
(a)Number of boundary components. This number is represented by an integer c. For instance, for
a sphere or a torus, c= 0, for a disc or a hemisphere c= 1, for an open-ended cylinder c = 2 which
is the case as well for the surface shown in Figure 8.6.
(b)Orientability. Consider a sphere in Figure 8.10(a) with a circle of arbitrary radius drawn about
the center Q, a point on the sphere. Let an outward normal n be drawn at Q suggesting the orientation
of the circle to be anticlockwise, and let C be any arbitrary closed path on the sphere. If Q traverses
alongC, the orientation of the normal would be preserved if Q returns to its original position. Such
surfaces are termed orientable. Like a sphere, a hexahedral surface (a polyhedral surface in general)
and torus are both orientable. Consider the sketch of a Möbius surface in Figure 8.10(b). If the circle
at point Q commences to traverse towards the left along the closed path shown, the direction of the
normal is reversed (dotted line) when Q reaches its original position. Such surfaces are termed non-
orientable. Orientability is a Boolan value ε such that ε = 1 for all orientable surfaces while 0 for the
non-orientable ones.
(^3) A handle is analogous to that in a coffee mug. Note that a torus (doughnut) and a coffee mug are homeomorphic
to each other
Figure 8.10 (a) Orientable sphere, (b) non-orientable Möbius surface and (c) Möbius rule for orientability
of closed polyhedral surfaces
n
Q
C
n
Q
(a) (b) (c)
An orientable surface has two distinguishable sides which can be labeled as the inside and outside.
Being closed is not a sufficient condition for a surface to be orientable. An example is the Klein bottle