[metapost] MetaFont: Unexpected behavior of intersections times

laurent at math.toronto.edu laurent at math.toronto.edu
Mon Apr 11 21:49:30 CEST 2011

Hi all,

Me on Tue, 5 Apr 2011 21:11:44 -0400 (EDT)
 > One helpful sufficient condition
 > for one or more genuine intersections of the two
 >  paths p and q is this criterion:
 > ($) existence of a quadrilateral Q = ABCD (convex or not,
 > but embedded in the plane) such that p runs from A to C
 > within in Q and q runs from B to D within Q

I add today that the implicit condition (from my figure) that
the 4 sides AB, BC, CD, DA of the quadrilateral Q be linear
segments can be relaxed; for example, it is enough that they be
piecewise linear or piecewise bezier. That is sometimes helpful

Q is then still a 'Jordan curve' and the truth of the criterion is a
consequence of the 'Jordan curve theorem'.

Me on Fri, 8 Apr 2011 01:16:32 -0400 (EDT) > One byproduct
 > [of ($)] should be a very sharp determination of the
 > transverse intersection point. MP is somewhat sloppy.

Dan on Sun, 10 Apr 2011 16:09:48 -0500 > ... the method
 > would be subject to pretty much the same
 > level of imprecision as the current one.

Possibly so.  Only an implementation can prove the contrary.
Dan is certainly right that there are difficulties in
practical application of ($). The devil is in the details.

It remains that there is a huge difference between
Knuth's intersection algorithms and my ($).
Knuth's NEVER assure an intersection point of the
mathematical paths p and  q , while criterion ($)
ALWAYS does. I think of the two as complementary.


Laurent S.

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