[metapost] (no subject)

[laurent at math.toronto.edu]

Hi all,

Dan raised some points worth debating
<Date: Tue, 28 Jun 2011 12:42:36 -0500>

 > [Also, for the concept of winding number,
 > imprecision in the location of a point and the
 > calculation of a path can turn a point that is "on"
 > a path to one that is not "on" it, producing
 > similar problems (not to mention the already
 > discussed problem of intersection of paths).]

I always repeat to myself and to anyone listening
that when MF/MP affirm an intersection of two
objects, they merely mean (and do prove) than a
small neighborhoods of the objects do intersect.
There are times when we should know what small
means. And there are situations when several
different orders of smallness should come into
play. With 64 binary bits available to represent
any number in our calculations there will be more
room to exploit several orders of smallness. By
placing the orders of smallness into the hands of
the user, MP could become less chancy and more
adaptable.

 > Read back in the archives when we disussed path
 > intersection. One of the examples included was path
 > (call it P), which was a subpath of another (call
 > it Q). Clearly P and intersect, but the
 > intersetiontimes primitive said they did not,
 > presumably because of the inherent problems of
 > fixed precision computation.

Reference please! This sort of thing alarms me. We
should all understand when and why this can happen.
The one reason I can see offhand is a dire shortage
of available time instants on the paths. Does the
same (sort of) example recur with 64 bit precision?

 > Exactly such a situation would render the
 > winding of of Q about a point problematic. So
 > in this sense a puzzling case has already
 > surfaced. If one cannot determine reliably
 > intersection of paths, one cannot
 > reliably determine whether a point is on a path
 > and therefore whether a winding number exists.

I believe that in allowing cusps in the paths
accepted by a stable "turning number" function one
has to also accept some "near cusps" as cusps. This
will cause instability of turningnumber where those
"near cusps" that are really tiny loops become just
big enough to be abruptly recognized as loops
instead of "near cusps". This will be a residual
UNSTABLE region of "turning number" behavior. I
consider it to be a 'useless' region; the curves
involved can be considered garbage.

The pure mathematics of the curves and regions of
MP/MF is quite accessible but deserves respect; it
requires dilligence  -- sometimes more then I can
expend. The computer MF/MP computations involving
them add a very considerable layer of complexity
making a whole that can be truly daunting !-)

Cheers

Laurent S.