Civil Engineering Reference
In-Depth Information
Major axis
Preferred cut
Minor axis
Figure 7.35
Cutting along principal axes.
Preferred
Figure 7.36
Cutting a cylinder into two pieces.
the major principal axis, produces two open surfaces, whereas the cut along the transverse
direction, normal to the minor principal axis, produces two cylindrical surfaces.
7.5.4 Euler-Poincare characteristics
The topology of an open surface and that of a cylinder can be easily differentiated by con-
sidering the Euler-Poincare characteristics of the objects:
χ = v + f − e
where v = Number of vertices, f = Number of faces and e = Number of edges.
For closed surfaces without holes, χ = 2. An open surface is topologically equivalent to
a closed surface upon removing one of its faces; hence, χ = 1 for open surfaces. Take for
instance an open surface shown in Figure 7.37a; there are 9 vertices, 8 faces and 16 edges,
and χ = 9 + 8 − 16 = 1. As for the open cylindrical surface shown in Figure 7.37b, there are 8
vertices, 8 faces and 16 edges, and χ = 8 + 8 − 16 = 0. χ = 2 for closed surfaces can be restored
by adding two faces to the open cylinder, one at the top and one at the bottom. In fact, the
cylindrical surface is topologically equivalent to an open surface with an interior opening,
as shown in Figure 7.37c. If only one face is added to the bottom of the cylinder, effectively
we are filling up the interior opening, and an open surface will be resulted.
(a)
(b)
(c)
Figure 7.37
Topology of some simple objects: (a) open surface; (b) cylinder with open ends; (c) open surface
with a hole.
