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Bug persisting in 11.3

I found RegionMember quite useful in retrieving the condition for which a point $(x,y)$ belongs to a region. Thus, this command works:

RegionMember[Disk[{0, 0}, 3, {-\[Pi]/2, \[Pi]/4}]][{x, y}]

$$(x|y)\in \mathbb{R}\land x^2+y^2\leq 9\land \frac{x}{3}\geq 0\land -\frac{y}{3 \sqrt{2}}\geq -\frac{x}{3 \sqrt{2}}$$

However, it does not for this self-intersecting polygon:

RegionMember[Polygon[{{0, 0}, {1, 1}, {0, 1}, {1, 0}}]][{x, y}]

MMA returns

{}[{x, y}]

or

RegionMember[Polygon[{{0, 0}, {1, 1}, {0, 1}, {1, 0}}],{x, y}]

MMA returns the same:

RegionMember[Polygon[{{0, 0}, {1, 1}, {0, 1}, {1, 0}}], {x, y}]

Anyone could explain why RegionMember fails in this case. The region seems to be not complicated.

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    $\begingroup$ Looks like a bug to me. The documentation doesn't list any sort of restrictions, like self-intersections. Because it's not documented, then I would say it is not working as described. Particularly this: RegionMember[Polygon[{{0, 0}, {1, 1}, {0, 1}, {1, 0}}]] $\endgroup$
    – ktm
    Jan 29, 2018 at 20:56
  • $\begingroup$ If we apply DiscretizeRegion to the polygon, then we get a region that works with RegionMember. $\endgroup$
    – LouisB
    Jan 30, 2018 at 6:24
  • $\begingroup$ @LouisB what you propose does not work for me (MMA ver. 11.2). Further, why the need of discretizing the polygon and not the disk? $\endgroup$
    – José Antonio Díaz Navas
    Jan 30, 2018 at 11:41
  • $\begingroup$ I have just contacted support for assistance $\endgroup$
    – José Antonio Díaz Navas
    Jan 30, 2018 at 13:17
  • $\begingroup$ Perhaps RegionMember[RegionUnion@@Rationalize[Graphics`PolygonUtils`SimplePolygonPartition[Polygon[{{0,0},{1,1},{0,1},{1,0}}]]],{x,y}]. $\endgroup$
    – chyanog
    Jun 17, 2018 at 7:33

1 Answer 1

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The fact that the one-argument form returns an empty list is simply a bug.

For sufficiently complicated algorithm, it may be hard/impossible to return a useful answer. This true even of self-intersecting polygons, though this particular one is admittedly not that complicated. But then it should return unevaluated, rather than an empty list.

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