What kind of object results from RegionIntersection function?

Question

I'm trying to use RegionIntersection in a couple of Polygons. The polygons are the following:

p1=Polygon[{{-102.251, 21.8628}, {-102.252, 21.8628}, {-102.252, 21.8631}, {-102.251, 21.8631}, {-102.251, 21.8628}}]

and

p2=Polygon[{{-102.253, 21.8671}, {-102.253, 21.8682}, {-102.253, 21.8682}, {-102.253, 21.8682}, {-102.253, 21.8681}, {-102.252, 21.868}, {-102.252, 21.8678}, {-102.252, 21.8673}, {-102.253, 21.8672}, {-102.253, 21.8671}}]

When I run r = RegionIntersection[p1, p2] I was expecting that the resulting object was a "Region", however it is something like RegionIntersection[Polygon{{...}},Polygon{{...}}].

When I try to compute the Area of the resulting region it gives me:

Area::reg: "RegionIntersection[Polygon[{{-102.253,21.8671},{-102.253,21.8682},{-102.253,21.8682},{-102.253,21.8682},<<3>>,{-102.252,21.8673},{-102.253,21.8672},{-102.253,21.8671}}],<<1>>] is not a correctly specified region.

And if I test for a region with RegionQ[r] I obtain `true

Also if I try to plot the region discretizing with DiscretizeRegion[r] I obtain:

DiscretizeRegion[RegionIntersection[Polygon[{{-102.251, 21.8628}, {-102.252, 21.8628}, {-102.252, 21.8631}, {-102.251, 21.8631}, {-102.251, 21.8628}}],Polygon[{{-102.253, 21.8671}, {-102.253, 21.8682}, {-102.253, 21.8682}, {-102.253, 21.8682}, {-102.253, 21.8681}, {-102.252, 21.868}, {-102.252, 21.8678}, {-102.252, 21.8673}, {-102.253, 21.8672}, {-102.253, 21.8671}}]]]

So it seems that the resulting object from the Intersection is a region but it is not possible to compute anything on it.

Following the examples on the Documentation I notice that using circles it behaves correctly and it is possible to compute on the intersection, however if the intersection is nothing (i.e. the two regions don't intersect) the object is still a region. I was wondering if there is a way of knowing that I have a null region. Of curse my first guess was computing the Area on it but for my polygons it is apparently impossible.

Any thoughts on why regions behave like that? and on how to know if the Intersection region is a null region?

Answer 1

I think the problem is in the ordering of points in p2. if we plot p2 with the nodes we will find the following graphic:

p2 = Polygon[{{-102.253, 21.8671}, {-102.253, 21.8682}, {-102.253, 
     21.8682}, {-102.253, 21.8682}, {-102.253, 21.8681}, {-102.252, 
     21.868}, {-102.252, 21.8678}, {-102.252, 21.8673}, {-102.253, 
     21.8672}, {-102.253, 21.8671}}];
Show[Graphics[{Gray, p2}], 
 ListPlot[Tooltip /@ {{-102.253, 21.8671}, {-102.253, 
     21.8682}, {-102.253, 21.8682}, {-102.253, 21.8682}, {-102.253, 
     21.8681}, {-102.252, 21.868}, {-102.252, 21.8678}, {-102.252, 
     21.8673}, {-102.253, 21.8672}, {-102.253, 21.8671}}, 
  PlotStyle -> Directive[Red, PointSize[0.03]]]]

it can be seen that not all nodes are vertices of the polygon. if we remover those extra nodes the polygon will be like this:

p2 = Polygon[{{-102.253, 21.8681}, {-102.252, 21.868}, {-102.252, 
     21.8673}, {-102.253, 21.8672}}];
Show[Graphics[{Gray, p2}], 
 ListPlot[Tooltip /@ {{-102.253, 21.8681}, {-102.252, 
     21.868}, {-102.252, 21.8673}, {-102.253, 21.8672}}, 
  PlotStyle -> Directive[Red, PointSize[0.03]]]]

the area can be found easily like this:

    r = RegionUnion[p1, p2];
    Area[r]
   (* 1.1*10^-6*)

if you want to include all points you many rearrange them like this:

p2 = Polygon[{{-102.253, 21.8671}, {-102.252, 21.8673}, {-102.252, 
     21.8678}, {-102.252, 21.868}, {-102.253, 21.8682}, {-102.253, 
     21.8682}, {-102.253, 21.8682}, {-102.253, 21.8681}, {-102.253, 
     21.8672}, {-102.253, 21.8671}}];
Show[Graphics[{Gray, p2}], 
 ListPlot[Tooltip /@ {{-102.253, 21.8671}, {-102.252, 
     21.8673}, {-102.252, 21.8678}, {-102.252, 21.868}, {-102.253, 
     21.8682}, {-102.253, 21.8682}, {-102.253, 21.8682}, {-102.253, 
     21.8681}, {-102.253, 21.8672}, {-102.253, 21.8671}}, 
  PlotStyle -> Directive[Red, PointSize[0.03]]]]

and now the area can be found:

r = RegionUnion[p1, p2];
Area[r]
(*1.2*10^-6*)