Any ideas on how to use the Region` context?

For a more clear view, here is a table of some of the Region functions.

AppendTo[$ContextPath, "Region`"]

Clear[testfunc]
testfunc[reg_] := {ToString /@ #, Through[#[reg]]} &[{
                    ConvexRegionQ,
                    BoundedRegionQ,
                    RegionDimension,
                    Module[{dim = RegionEmbeddingDimension[#]},
                           var = Symbol["x" <> ToString[#]] & /@ Range[dim];
                           dim] &,
                    RegionMeasure,
                    RegionCentroid,
                    RegionProperty[#, var, "FastDescription"] &,
                    RegionProperty[#, var, "ImplicitDescription"] &,
                    RegionElement,
                    LevelFunction[RegionProperty[#, var, "FastDescription"][[1, 2]]] &
                  }] // 
          Grid[Insert[#, {ConvexRegionQ, BoundedRegionQ, RegionDimension, 
               RegionEmbeddingDimension, RegionMeasure, RegionCentroid, 
               FastDescription, ImplicitDescription, RegionElement, 
               LevelFunction}, 2]\[Transpose], Dividers -> All, 
            FrameStyle -> GrayLevel[.8], Alignment -> Left] & // Quiet

In addition of BoxRegion, other *Regions also seems to be used to declare regions:

Names["Region`*Region"]

{"BallRegion", "BooleanRegion", "BoxRegion", "EllipsoidRegion", "EmptyRegion", "FullRegion", "InverseTransformedRegion", "ParametricRegion", "SimplexRegion", "TransformedRegion"}

For example, a 2D triangle embeded in 7D space:

tri3d = RandomInteger[{-10, 10}, {3, 3}];
tri7d = ArrayFlatten[{{tri3d, ConstantArray[0, {3, 4}]}}];
(* a random rotate in 7D space: *)
rt7d = RotationTransform[{{0, 0, 1, 0, 0, 0, 0}, RandomInteger[{-1, 1}, 7]},
                         ConstantArray[0, 7]];
tri7d = rt7d /@ tri7d;
testfunc@SimplexRegion[tri7d]

Maybe some of them (LevelFunction) work only on "full-rank" regions?

simplex = Function[dim, SimplexRegion[RandomInteger[{-10, 10}, {dim + 1, dim}]]] @ 4
testfunc @ simplex

Some regions look like special cases:

RegionDimension@EmptyRegion[2]

$-\infty$

RegionMeasure@FullRegion[3]

$\infty$

Edit:

SimplePolygonPartition can be used to divide self-intersecting Polygon to simple pieces. The usage is like

SimplePolygonPartition[Polygon[...]]
SimplePolygonPartition[Polygon[...],Graphics`Region`RegionDump`FillingMethod->"OddEvenRule"]

An example can be found here.

This is quite a find. I've only had time to play with it a little, but are some interesting results:

Region`ConvexRegionQ[Disk[{1., 0.}]]

True

Region`RegionCentroid[Disk[{1., 0.}]]

{1., 0.}

Region`RegionMeasure[Disk[{1., 0.}]]

π

Region`RegionIntersection[Disk[{0, 0}], Disk[{1, 0}]]

seems to do nothing, but

Region`RegionMeasure @ Region`RegionIntersection[Disk[{0, 0}], Disk[{1, 0}]]

-(Sqrt[3]/2) + (2 π)/3

It appears one can create regions and operate on them:

box = Region`BoxRegion[{0, 0}, {2, 3}];
Region`RegionMeasure @ box

6

Region`RegionCentroid @ box

{1, 3/2}

Its interesting to note that the Region context is loaded when you evaluate Graphics`Region`RegionInit[]. Old favourite Graphics`Mesh gets loaded too. There is some interesting looking stuff in Graphics`Region, clearly incomplete, for example one of the definitions is this...

BoundingRegion[___] := "Implement me..."

I've not done much spelunking yet, but did find this:

Graphics`Region`RegionInit[];

RegionConvert[Disk[]]
(* MeshRegion[{2, 2}, {951, 2289, 1339}, <>] *)

Graphics[Line @ MeshCoordinates[%, 1]]