# Any ideas on how to use the Region context?

If you type

?Region*


you'll get:

which seems a bunch of interesting and not documented symbols.

Any idea (or experience) on how to use them?

Edit

By using our "collective spelunking" I was able to work out this answer - Great! :)

And Silvia used it to write another one!

• RegionRegionMeasure[Circle[]] -> 2 Pi :) Dec 20, 2013 at 18:34
• Christmas has arrived in your region? :) Dec 20, 2013 at 18:47
• Check this: RegionRegionProperty[Polygon[{{1, 0}, {0, 1}, {2, 3}}], {x, y}, "FastDescription"] Dec 20, 2013 at 22:10
• Great find by the way. I wish we could get documentation for some of the useful stuff in these hidden-away packages. Dec 20, 2013 at 22:24
• In Simon's footsteps, trying to find valid arguments for RegionProperty I ran Union@Cases[ ToExpression[#, InputForm, DownValues] & /@ Names["Region*"], HoldPattern[RegionRegionProperty[__, s_String]] :> s, Infinity] to find {"Distance", "FastDescription", "ImplicitDescription", "Nearest", "SpaceDimension"}. SpecialRegionProperty can take {"Assumptions", "BoundingBox", "Centroid", "ConvexQ", "Distance", "ImplicitDescription", "InjectiveParametricDescription", "Instance", "Measure", "Nearest", "ParametricDescription", "RegionDimension", "SignedDistance", "SpaceDimension"} Dec 21, 2013 at 0:12

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[...],GraphicsRegionRegionDumpFillingMethod->"OddEvenRule"]  An example can be found here. • Great! In a few days we'll be able to write a manual! :) Dec 21, 2013 at 3:31 • @belisarius The Uncomplete Unofficial Documentation for Undocumented Functions :P Dec 21, 2013 at 4:02 • take a look! mathematica.stackexchange.com/a/39206/193 Dec 21, 2013 at 6:49 • @belisarius Thanks for you and Simon's enlightenment, I found a similar solution :) Dec 22, 2013 at 0:25 • @belisarius Found a new one SimplePolygonPartition :) Jan 28, 2014 at 6:30 This is quite a find. I've only had time to play with it a little, but are some interesting results: RegionConvexRegionQ[Disk[{1., 0.}]]  True  RegionRegionCentroid[Disk[{1., 0.}]]  {1., 0.}  RegionRegionMeasure[Disk[{1., 0.}]]  π  RegionRegionIntersection[Disk[{0, 0}], Disk[{1, 0}]]  seems to do nothing, but RegionRegionMeasure @ RegionRegionIntersection[Disk[{0, 0}], Disk[{1, 0}]]  -(Sqrt[3]/2) + (2 π)/3  It appears one can create regions and operate on them: box = RegionBoxRegion[{0, 0}, {2, 3}]; RegionRegionMeasure @ box  6  RegionRegionCentroid @ box  {1, 3/2} • +1 And it seems most of them also work for 3D cases. Dec 21, 2013 at 0:11 Its interesting to note that the Region context is loaded when you evaluate GraphicsRegionRegionInit[]. Old favourite GraphicsMesh gets loaded too. There is some interesting looking stuff in GraphicsRegion, clearly incomplete, for example one of the definitions is this... BoundingRegion[___] := "Implement me..."  I've not done much spelunking yet, but did find this: GraphicsRegionRegionInit[]; RegionConvert[Disk[]] (* MeshRegion[{2, 2}, {951, 2289, 1339}, <>] *) Graphics[Line @ MeshCoordinates[%, 1]]  • "Implement me..." There's some sentient code! Dec 22, 2013 at 0:02 Four more: RegionNearest[] returns the nearest point inside a region to a given point: AppendTo[$ContextPath, "Region"]

RegionNearest[Disk[], {3, 4}]
(*
{3/5, 4/5}
*)


RegionDifference[] seems to return unevaluated ... but no:

RegionMeasure@RegionDifference[Rectangle[], Disk[]]
(*
1 - π/4
*)


TransformedRegion[] also seems to return unevaluated ... but again:

RegionMeasure@TransformedRegion[Rectangle[], ScalingTransform[{3, 2}]]
(*
6
*)


ParametricRegion[]:

RegionMeasure@ParametricRegion[{{x, y}, 0 < x < 1 && 0 < y < 2}]
(*
2
*)

• @Silvia RegionMeasure@ParametricRegion[{{x, y}, 0 < x < 1 && 0 < y < 2}] Dec 21, 2013 at 5:58
• Sorry I missed that.. But ParametricRegion will be my most interested one :) Dec 21, 2013 at 5:58
• Wow! Brilliant! Dec 21, 2013 at 5:58

So far as I know, the region properties can be:

In[1]:= Union[
Join[RegionSpecialBoundaryProperty[];
Cases[GeneralUtilitiesDefinitions[
"SpecialBoundariesDumpSRBdryRule"], _String, -1],
RegionSpecialRegionProperty[];
Cases[GeneralUtilitiesDefinitions[
"SpecialRegionsDumpSRPropRule"], _String, \[Infinity]]]]

Out[1]= {"Assumptions", "Boundary", "BoundingBox", "Centroid", \
"ConvexQ", "Distance", "ImplicitDescription", \
"InjectiveParametricDescription", "Instance", "LinearGraphics", \
"Measure", "Nearest", "ParametricDescription", "Primitive", "Region", \
"RegionDimension", "SignedDistance", "SimpleBoundary", \
"SimplicialDecomposition", "SpaceDimension", "Type"}


Unfortunately, this list is incomplete. According to

In[2]:= RegionRegionProperty[Pyramid[],{x,y,z},"FastDescription"]
Out[2]= {{{x,y,z},-4 (-2-2 y+2 z)>=0&&-4 (-2+2 x+2 z)>=0&&-4 (-2+2 y+2 z)>=0&&4 (2+2 x-2 z)>=0&&16 z>=0}}
In[3]:= RegionRegionProperty[Pyramid[],{x,y,z},"FastImplicitDescription"]
Out[3]= -4 (-2-2 y+2 z)>=0&&-4 (-2+2 x+2 z)>=0&&-4 (-2+2 y+2 z)>=0&&4 (2+2 x-2 z)>=0&&16 z>=0
In[4]:= RegionRegionProperty[Pyramid[],{x,y,z},True,"ImplicitDescription"]
Out[4]= {(x==-1&&((y==-1&&z==0)||(-1<y<0&&z==0)||(y==0&&z==0)||(0<y<1&&z==0)||(y==1&&z==0)))||(-1<x<0&&((y==-1&&z==0)||(-1<y<x&&0<=z<=1+y)||(y==x&&0<=z<=1+y)||(x<y<0&&0<=z<=1+x)||(y==0&&0<=z<=1+x)||(0<y<-x&&0<=z<=1+x)||(y==-x&&0<=z<=1+x)||(-x<y<1&&0<=z<=1-y)||(y==1&&z==0)))||(x==0&&((y==-1&&z==0)||(-1<y<0&&0<=z<=1+y)||(y==0&&0<=z<=1-y)||(0<y<1&&0<=z<=1-y)||(y==1&&z==0)))||(0<x<1&&((y==-1&&z==0)||(-1<y<-x&&0<=z<=1+y)||(y==-x&&0<=z<=1+y)||(-x<y<0&&0<=z<=1-x)||(y==0&&0<=z<=1-x)||(0<y<x&&0<=z<=1-x)||(y==x&&0<=z<=1-y)||(x<y<1&&0<=z<=1-y)||(y==1&&z==0)))||(x==1&&((y==-1&&z==0)||(-1<y<0&&z==0)||(y==0&&z==0)||(0<y<1&&z==0)||(y==1&&z==0))),True}


"FastImplicitDescription" is also a valid region property.
So, is there a complete list?