[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:
>
> ($) existence of a quadrilateral Q = ABCD (convex or not,
> but embedded in the plane) such that p runs from A to C
> within in Q and q runs from B to D within Q
I add today that the implicit condition (from my figure) that
the 4 sides AB, BC, CD, DA of the quadrilateral Q be linear
segments can be relaxed; for example, it is enough that they be
piecewise linear or piecewise bezier. That is sometimes helpful
generality.
Q is then still a 'Jordan curve' and the truth of the criterion is a
consequence of the 'Jordan curve theorem'.
Me on Fri, 8 Apr 2011 01:16:32 -0400 (EDT) > One byproduct
> [of ($)] 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.
Cheers
Laurent S.
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