[metapost] turningnumber revisited

[Dan Luecking]

At 06:03 AM 6/27/2011, Boguslaw Jackowski wrote:

>So, I support Larry's point -- turningnumber is a discontinuous operation
>on the path -- an infinitesimal (i.e., arbitrarily small) change of 
>a path (nodes position) may cause the change of a turningnumber.

Actually, if there is no ambiguity in the turningnumber (no cusps),
then an infinitesimal change cannot change the turning number.

What one can say is that when a path has an ambiguous turningnumber
(cusps exist) then, no matter what disambiguation rule is chosen,
an infinitessimal change in the path can change the turning number.

But the same can be said of many other useful operations that
can sometimes be ambiguous or undefined (angle of a vector,
quotient of two numbers). This doesn't make them useless, it
merely limits their usefulness.

>The windingnumber for a given point yields the constant value. More 
>precisely, the countinuous change of a path does not change the 
>winding number as long as the path does not touch the given
>point. I this case the result is undefined.

In complex analysis, a curve can usefully be defined to have a
winding number around any point through which it passes
smoothly: the average of the values on either side. Of course,
if this definition is chosen, then winding number is also
discontinuous (and now undefined at a cusp--or at least
ambiguous).

In an ideal world where exact calculations were possible, one
would like turningnumber to trigger an error when a path has a
cusp, (and winding number to trigger an error when a point is
on a path). Unfortunately, in a computer, where calculations have
to be digitized, a cusp can seem to disappear after something so
simple as a rotation, and the same is true of the concept of a
point being "on" a path.

Regards,

Dan


Daniel H. Luecking
Department of Mathematical Sciences
Fayetteville, Arkansas
http://www-cs-faculty.stanford.edu/~knuth/iaq.html