Computer Aided Engineering Design

252 COMPUTER AIDED ENGINEERING DESIGN

are all homeomorphic to a circle. If we attach a disc to each boundary component of a surface S, the

resulting surface Sˆ will be a closed one. This closing-up operation preserves homeomorphism types,

that is, S 1 and S 2 are homeomorphic to each other (S 1 ≈S 2 ) if and only if SSˆˆ 12 .≈ Thus, we can divide
surfaces into classes, where two surfaces are in the same class if they have the same homeomorphic

closed-up surfaces. Given SSˆˆ 12 and as two closed surfaces, we can cut out a disc from each one and
attach the resultants along their cut boundaries. The result is a closed surface SSˆˆ 12 # called the
connected sum of two surfaces. As an example, in Figure 8.8, discs are cut from two spheres and the
resulting surfaces are joined at the boundaries to get a double sphere which is a closed surface. The
connected sum of any two surfaces does not depend not the choice of discs cut from each surface and

that the connected sum operation respects homeomorphism. Thus, if S 1 ≈ SSS 122 ′ and ≈ ′, then S 1 #

S 2 ≈ SS 12 ′′ #.

Hemisphere

Disc

(a) (b) (c)

Figure 8.7 (a) Homeomorphism between a disc and a hemisphere (b) an open ended cylinder having two
boundaries (c) a cylindrical surface having one boundary

8.3.2 Some Basic Surfaces and Classification

A sphere and torus introduced above are some examples of basic closed surfaces in three-dimensions.
We can build a torus (doughnut) from a rectangular piece of paper (Figure 8.9a) by gluing together
the edges with corresponding arrows shown. Note that as an intermediate step, an open ended
cylinder is obtained with two boundaries. A Möbius strip is obtained by gluing two opposite ends of
a rectangular strip (Figure 8.9b) with a twist. We may find that it has only one boundary. The
construction of a torus and Möbius strip can be combined (Figure 8.9c) in a manner that we get an
open cylinder in an intermediate step to twist one end by 180° and then glue the two ends. The
resulting surface is a Klein bottle which cannot be built in a three-dimensional space without self
intersection. Figure 8.9 (c) shows two projections of the Klein bottle which is a closed surface (zero
boundaries). Finally, we can glue the opposite sides of a rectangular strip such that there is a twist
about both the horizontal and vertical axes as suggested in Figure 8.9(d). The resultant surface is a

U =

Figure 8.8 Connected sum of two spheres