[metapost] MetaFont: Unexpected behavior of intersections times

luecking luecking at uark.edu
Thu Apr 7 04:50:08 CEST 2011

Quoting laurent at math.toronto.edu:

> One helpful sufficient condition for one or more genuine
> intersections of the two paths p and q is this
> criterion:
>  ($) existence of a quadrilateral Q = ABCD (convex or not,
> but embedded in the plane) such that p runs from A to C
> within in Q and q runs from B to D within Q
>      In practice, I claim (without proof) that this
> criterion will let you count and locate all sufficiently
> transverse and nonsingular intersections of paths p' and q'
> that are not very near to a path endpoint. Indeed, for any
> such intersection X of p' and q' there exists,
> after passage to subpaths p and q, a convex quadrilateral Q
> as in ($) proving intersection of p and q at X.
>  Dan Luecking > I once had two semicircles that crossed
>  > (theoretically) at the midpoint of each, and MP said
>  > they did not cross.
> Loosely interpreted, this sounds to me like a bug, so I
> would like to see a concrete example and its explanation.
> It seems to contradict the claim I have just made.

There is no contradiction. The point of intersection in my problem
case was a node, the endpoint of a Bezier segment. MF and MP
examine pairs of Bezier segments only, not the paths as a whole.
In my problem curves, no quadrilateral such as you have depicted
existed for those pairs of Bezier segments that met at the point of
(theoretical) intersection.


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