Page 286 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

APPENDIX 261

b

a

Figure 5.40 Closed cuts a and b do not divide the surface of the torus into separate
areas but leave it as one area. However, addition of any other closed curve would cut it
into two separate areas.

Separation Theorems on the Common Torus. (See Section 5.8.4). The
purpose of this subsection is to prove Theorem 5.8.4. The proof for its pla-
nar counterpart can be found in Ref. 110, VI.16.2, which uses the concept of
regular grating as the fundamental tool. We will use an analogous strategy to
prove Theorem 5.8.4. Since the topology of a torus is different from that of a
plane, we start with the modified definitions of regular grating, k-chains,and k-
1
cycles in torus T , and proceed with the corresponding operations and properties
(Theorems 5.9.1 to 5.9.7).
Several intermediate results are needed in order to prove Theorem 5.8.4. The
proofs for some of these are the same as their planar counterparts, in which case
we simply restate the statements and cite the source [110]. Proofs will be given
1
to statements that are valid only for T .
Regular grating is a convenient tool for studying the connectivity of a subset
1
of T . We show in Theorem 5.9.8 that a 1-cycle (a simple closed curve) does
not necessarily separate a torus into two halves as it would in a plane. This
major fact makes the proof of Theorem 5.8.4 different from its planar counter-
part.
Finally, to prove Theorem 5.8.4 we need to show that if a region (a connected
1
subset) D in T is uniformly connected, then the connectivity of any of its
boundary components is destroyed by the removal of two single points. This is
done by drawing a cross-cut L connecting the same boundary component of D
and showing that D-L has exactly two components (Theorem 5.9.12). The proof
of Theorem 5.9.12 in turn requires the intermediate results of Theorems 5.9.9 to
5.9.11.