CG Modeling 2: NURBS - An Introduction to Computer Graphics for Artists - page 252

Fig. 14.11 A Möbius strip is non-orientable. This can be seen by tracing a line on a Möbius strip,
which will cover both sides of the paper without lifting the pencil from the paper to switch sides
Fig. 14.12 The shape of a sphere, cone, and square may be different, but their topology is the
same. Each is made of a four-sided surface, or in the case of a cube, six four-sided surfaces
What is important about this problem is that it showed that the exact shape and
dimensions, as well as the location of the bridges, were totally unimportant to his
solution. In the same way, topology, at least the branch of topology relating to
surfaces, studies surfaces that are, or are not, homotopic , or topologically identical.
A NURBS sphere, cylinder, and plane are all homotopic surfaces because they are
topologically identical (Fig. 14.12 ). They could each be made into the other simply
by moving their included control vertices to the right position, just as a square sheet
can be draped around a cone or a sphere to take on those shapes.
All NURBS primitives are four-sided. This is a topological limitation. To use
them effectively, you will have to be able to see this four-sided topology in every-
thing you intend to build. The easiest way to accomplish it is to remember the
unfolded carton from the fi rst exercise. When you look at objects, analyze your
target to discern how it will unfold in two-dimensional space. By doing this, you
will understand its topology and know how to build the object using four-sided
NURBS surfaces.