CG Modeling 2: NURBS - An Introduction to Computer Graphics for Artists - page 251

Fig. 14.10 Intersecting curves may not have control vertices at the point of intersection, or if they
do, nearby points may not be aligned to allow intersection

If you must intersect curves to fi nd an intersection point or to cut curves and then
build a surface from the remainder, the operation you intend to perform should be
performed immediately after they have been made to intersect. The reason is that if
you move any CVs on either curve after they intersect, you are likely to move them
out of alignment so that they no longer intersect (Fig. 14.10 ).

14.3 Topology

If you take a strip of paper, twist it, then tape the ends together, you will have a Möbius
strip (Fig. 14.11 ). Because the ends of this object have been twisted before connecting
them, one may trace a line across the surface of both sides of the strip without ever
breaking the line or changing sides. This property is sometimes called one-sidedness ,
although at any given point on the object's surface, it does have another side.
Topology, or the part of it that is most closely related to CG, is the study of sur-
face properties that remain the same regardless of deformations made to an object.
The mathematician Leonhard Euler wrote the fi rst major proof on topology in 1736.
The problem he described in the paper is known as the Seven bridges of Königsberg
(Horak 2006 ) . The problem he was asked to solve was to fi nd a way to cross each of
seven bridges exactly once. By abstracting the problem into nodes that represented
each endpoint and lines that connected them, he was able to prove that it could not
be done because of the way the bridges were connected. He went on to show that it
could only be done if either none of the endpoints or two of them (and only two) had
an odd number of connections.