250 COMPUTER AIDED ENGINEERING DESIGN
In this regard, a square, a parallelogram, and a rectangle are topologically identical as well. Since
we are ignorant about edge lengths, we might as well have a side of a quadrilateral of zero length
(Figure 8.5b). The result, a triangle, would still be identical in topology to a quadrilateral. We may
further re-shape or re-morph the straight edges of a quadrilateral to have bends. A polygon (thick
lines in Figure 8.5c), and in case the number of bends approaches infinity a simple closed curve with
no self-intersection (thin lines in the figure), both, would be of the same topology as the parent
quadrilateral. It thus implies that a closed polygon is topologically equivalent to a circle and this
equivalence is termed as homeomorphism.Likewise, the hexahedrals in Figure 8.4 (a-d) are
homeomorphic to themselves and to a sphere in Figure 8.4(e) since we can deform (bend, stretch or
twist) the faces and edges of a hexahedral to blend with the surface of the sphere. If a cylindrical void
is cut through the sphere (Figure 8.4f), the resultant topology is not homeomorphic to a sphere
(without void) since no amount of bending, stretching or twisting would transform it to a sphere
and vice versa. However, a sphere with a through void is homeomorphic to a torus or doughnut in
Figure 8.4 (g) which, in turn, is homeomorphic to a coffee mug.
An interesting aspect to realize is the direction of motion along the boundary when viewing from
a point P inside the shapes in Figure 8.5. Starting from any point Q on a closed curve, the traverse
along the curve with the aim of getting back to Q would be undirectional, either anticlockwise or
clockwise, and moreover, it would be continuous. Topologically, since lengths are of no importance,
a line would result by fusing any two vertices of the triangle in Figure 8.5 (b) in a manner similar to
how the triangle was obtained from the quadrilateral. The resulting line, however, would not be the
same in topology to any of the closed curves. This is because the sense of direction of motion from
a point on the line to the same point would not remain unidirectional anymore. This non-homeomorphism
between a line and a closed curve can be explained alternatively. There is a cutinvolved, anywhere
on the closed curve with its two ends stretched, to obtain a line.
a
aabc(a) (b) (c)(e) (f) (g)(d)Figure 8.4 Various shapes of a hexahedral topology (a-d), all homeoporphic to a sphere in (e). A sphere
with a through hole (f ) is homeomorphic to a torus (g)