Bug in RegionMember?

This bug still exists in version 12.2.

There is a workaround, which uses an alternative method of creating the RegionMemberFunction. I've shown that below, but first let's look at a couple of other possibilities that don't pan out.

The first idea one might consider is to define an alternate region member function, as follows.

polyQ = RegionMember[polygon, #] & 

This works, but it performs a couple orders of magnitude slower (presumably the reason RegionMemberFunction was invented), so it's not really an option.

Trying to fix the RegionMemberFunction directly doesn't work either, even though a workable repair seems apparent. Specifically, a RegionMemberFunction has three parts, the region specified in RegionMember, the dimensionality of the region (expressed as an integer), and a predicate (expressed as a pure Function). Given a point's coordinates, the predicate (part 3) returns True or False, according to whether or not the point lies in the region. In the cited case, evaluating and reducing the predicate makes the error clear (y<=(400+x)/2 should be y<=200).

polyQ = RegionMember[polygon];
Reduce@polyQ[[3]][{x, y}]

(0≤x<100 && 100≤y≤(400+x)/2) || (100≤x<200 && 100≤y≤300) || (200≤x≤250 && 0≤y≤400)

A correct predicate can be obtained as follows, where the expression here could be used as the body of a Function.

Element[(x|y),Reals] && Reduce@RegionMember[polygon,{x,y}]

(x|y)∈ℝ && ((0≤x<100 && 100≤y≤200) || (100≤x<200 && 100≤y≤300) || (200≤x≤250 && 0≤y≤400))

However, replacing the predicate in a RegionMemberFunction with a corrected version makes no difference, because the predicate seems not to be used when RegionMemberFunction is applied. As constructed, this predicate apparently only describes what will happen, so changing it amounts to assigning the wrong description, and nothing more.

A corrected predicate is of use though in defining an ImplicitRegion for which RegionMember produces a valid (and fast) RegionMemberFunction.

polyQ=RegionMember@ImplicitRegion[Reduce@RegionMember[polygon,{x,y}],{x,y}]

RegionMemberFunction[…]

The resulting RegionMemberFunction applies the correct predicate.

Moreover, the third part of the RegionMemberFunction describes correctly what the RegionMemberFunction will do.

polyQ[[3]][{x, y}]

(x|y)∈ℝ && ((0≤x<100 && 100≤y≤200) || (100≤x<200 && 100≤y≤300) || (200≤x≤250 && 0≤y≤400))

This can be generalized in a fixed RegionMember function, as follows:

ClearAll@fixedRegionMember;

fixedRegionMember[reg_?RegionQ] := 
  RegionMember@ImplicitRegion[Reduce@RegionMember[reg, {x, y}], {x, y}];

Here is the result.

polyQ = fixedRegionMember[polygon]

RegionMemberFunction[…]

With[{nPts = 10000},
 
 testPoints = Transpose@{RandomReal[{0, 300}, nPts], RandomReal[{0, 450}, nPts]};
 
 ListPlot[{testPoints, Select[testPoints, polyQ], polygonPoints}, 
    PlotStyle->{Red, Blue, {PointSize->Large, Green}}, 
    Prolog->{Gray, polygon}, ImageSize->Full] // AbsoluteTiming // Column
 ]