RegionEmbeddingDimension
gives the dimension of the space in which the region reg is embedded.
Details
- RegionEmbeddingDimension is also known as space dimension.
- The embedding dimension gives the dimension of the ambient space in which reg is embedded.
- Example cases with rows corresponding to RegionEmbeddingDimension and columns to geometric dimension:
Examples
open all close allBasic Examples (1)
Find the dimension of the space in which a region is embedded:
RegionEmbeddingDimension[Interval[{1, 2}]]RegionEmbeddingDimension[Circle[]]RegionEmbeddingDimension[Sphere[]]RegionEmbeddingDimension[Ball[{0, 0, 0, 0}, 1]]Scope (16)
Special Regions (4)
Regions in 
including Point:
RegionEmbeddingDimension[Point[{{0}, {1}}]]ℛ = Interval[{1, 2}];
NumberLinePlot[ℛ]RegionEmbeddingDimension[ℛ]Regions in 
including Point:
ℛ = Point[Tuples[Range[5], 2]];
Region[ℛ]RegionEmbeddingDimension[ℛ]Line:
ℛ = Line[{{1, 2}, {4, 3}}];
Region[ℛ]RegionEmbeddingDimension[ℛ]ℛ = Circle[{1, 2}, 3];
Region[ℛ]RegionEmbeddingDimension[ℛ]Disk:
ℛ = Disk[{1, 2}, {4, 3}];
Region[ℛ]RegionEmbeddingDimension[ℛ]Regions in 
including Point:
ℛ = Point[Tuples[Range[5], 3]];
Region[ℛ, Boxed -> True]RegionEmbeddingDimension[ℛ]Line:
ℛ = Line[{{1, 2, 3}, {6, 5, 4}}];
Region[ℛ, Boxed -> True]RegionEmbeddingDimension[ℛ]ℛ = Polygon[{{0, 0, 0}, {(5/3), (2/3), -(4/3)}, {(2/3), (2/3), -(1/3)}, {1, 2, 0}}];
Region[ℛ, Boxed -> True]RegionEmbeddingDimension[ℛ]ℛ = Cylinder[{{0, 0, 0}, {1, 1, 1}}, 2];
Region[ℛ]RegionEmbeddingDimension[Cylinder[{{0, 0, 0}, {1, 1, 1}}, 2]]Regions in 
including Simplex in 
:
RegionDimension[Simplex[{{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}}]]Cuboid in 
:
RegionDimension[Cuboid[{1, 2, 3, 4, 5}, {6, 7, 8, 9, 10}]]Ball in 
:
RegionDimension[Ball[{1, 2, 3, 4, 5, 6, 7}, 8]]Formula Regions (2)
The embedding dimension of a disk represented as an ImplicitRegion:
RegionEmbeddingDimension[ImplicitRegion[x^2 + y^2 ≤ 1, {x, y}]]RegionEmbeddingDimension[ImplicitRegion[x^2 + y^2 ≤ 1, {x, y, {z, 0, 2}}]]RegionEmbeddingDimension[ImplicitRegion[x^2 + y^2 ≤ 1 && z == 0, {x, y, z}]]The embedding dimension of a disk represented as a ParametricRegion:
RegionEmbeddingDimension[ParametricRegion[{r Cos[θ], r Sin[θ]}, {{r, 0, 1}, {θ, 0, 2π}}]]Using a rational parametrization of the disk:
RegionEmbeddingDimension[ParametricRegion[{r(1 - t^2/1 + t^2), r(2t/1 + t^2)}, {t, {r, 0, 1}}]]RegionEmbeddingDimension[ParametricRegion[{r Cos[θ], r Sin[θ], z}, {{r, 0, 1}, {θ, 0, 2π}, {z, 0, 2}}]]RegionEmbeddingDimension[ParametricRegion[{r Cos[θ], r Sin[θ], 0}, {{r, 0, 1}, {θ, 0, 2π}}]]Mesh Regions (3)
The embedding dimension of a BoundaryMeshRegion:
ConvexHullMesh[RandomReal[1, {10, 2}]]RegionEmbeddingDimension[%]DelaunayMesh[RandomReal[2, {20, 3}]]RegionEmbeddingDimension[%]The embedding dimension of a MeshRegion in 1D:
DelaunayMesh[RandomReal[1, {5, 1}]]RegionEmbeddingDimension[%]DelaunayMesh[RandomReal[1, {10, 2}]]RegionEmbeddingDimension[%]DelaunayMesh[RandomReal[2, {20, 3}]]RegionEmbeddingDimension[%]MeshRegion[{{0, 0}, {1, 0}, {2, -1}, {2, 1}}, Line[{{1, 2, 3, 4, 2}}]]RegionEmbeddingDimension[%]Derived Regions (3)
The embedding dimension of a RegionIntersection:
ℛ = RegionIntersection[Disk[{0, 0}, 1], Disk[{0, 1}, 1]];Region[ℛ]RegionEmbeddingDimension[ℛ]The embedding dimension of a TransformedRegion:
ℛ = TransformedRegion[Disk[], ScalingTransform[{3, 2}]];DiscretizeRegion[ℛ, {{-3, 3}, {-2, 2}}]RegionEmbeddingDimension[ℛ]The embedding dimension of a RegionBoundary:
ℛ = RegionBoundary[Ball[]];Region[ℛ]RegionEmbeddingDimension[ℛ]Geographic Regions (2)
Polygons with GeoPosition:
ℛ = Polygon[GeoPosition[{{{40.083441, -88.235716}, {40.083607, -88.257488}, {40.082603, -88.257149},
{40.076136999999996, -88.25740499999999}, {40.076178, -88.270888}, {40.076516, -88.271558},
{40.083686, -88.271512}, {40.083659999999995, -88.267046}, ... 33323}, {40.098112, -88.228687},
{40.095216, -88.228627}, {40.095179, -88.238547}, {40.094480999999995, -88.238546},
{40.094508999999995, -88.23267}, {40.094106, -88.232556}, {40.090666999999996, -88.232477},
{40.090741, -88.235745}}}]];RegionEmbeddingDimension[ℛ]Polygons with GeoPositionXYZ:
ℛ = Polygon[GeoPositionXYZ[{{{150451.6968462432, -4.884430486484052*^6, 4.085078564164219*^6},
{148595.27532671497, -4.884475441490381*^6, 4.085092666620835*^6},
{148626.35829777512, -4.884546311005128*^6, 4.0850073717259285*^6},
{148618.5908634042 ... 7*^6, 4.0860187668081024*^6},
{150697.56410771207, -4.8836599487428395*^6, 4.085984535480795*^6},
{150711.88303095422, -4.883905546449982*^6, 4.0856924143435075*^6},
{150433.15479548014, -4.883908845676418*^6, 4.0856987003255524*^6}}}]];RegionEmbeddingDimension[ℛ]Polygons with GeoPositionENU:
ℛ = Polygon[GeoPositionENU[{{{3378.2547059731055, -3369.2234780923936, -0.7440009205072329},
{1521.3211635380246, -3351.391253626573, -0.022340134218666208},
{1550.2571145363192, -3462.8657556618973, -0.08899812728964207},
{1528.5672303494055, -418 ... 63383291193, -0.37494203351275246},
{3654.121991908476, -2566.7472331234085, -0.5214977847472255},
{3375.420726854886, -2558.6597093173914, -0.3648706331350695}}},
GeoPosition[{40.11379115639895, -88.2753251202516, -1.0415787873318691}]]];RegionEmbeddingDimension[ℛ]Polygons with GeoGridPosition:
ℛ = Polygon[GeoGridPosition[{{{-0.9950503945490105, 1.2366760550756015},
{-0.9952074890903578, 1.2369207053693891}, {-0.9952196732768064, 1.2369073327446167},
{-0.9953160063787643, 1.236848436956935}, {-0.9954141759436825, 1.2369993898475449},
{-0. ... 197645333103}, {-0.9949098578570917, 1.2368130881428654},
{-0.9948663952535768, 1.2367477711687371}, {-0.9948714472169538, 1.2367426500757825},
{-0.9949211061652593, 1.2367089232486177}, {-0.9949439717990124, 1.236746107097628}}}, "Bonne"]];RegionEmbeddingDimension[ℛ]CSG Regions (1)
The embedding dimension of a CSGRegion in 2D:
ℛ = CSGRegion["Difference", {Disk[], Disk[{1 / 2, 1 / 2}]}]RegionEmbeddingDimension[ℛ]ℛ = CSGRegion["Difference", {Cube[2], Cylinder[{{1, 1, 1}, {1, -1, 1}}]}]RegionEmbeddingDimension[ℛ]Subdivision Regions (1)
The embedding dimension of a SubdivisionRegion in 2D:
ℛ = SubdivisionRegion[Rectangle[]]RegionEmbeddingDimension[ℛ]ℛ = SubdivisionRegion[Tetrahedron[]]RegionEmbeddingDimension[ℛ]Applications (2)
Select only full-dimensional primitives:
prims = {Triangle[{{0, 0, 0}, {1, 0, 0}, {0, 1, 0}}], Rectangle[], Sphere[], Ball[]};Select[prims, RegionDimension[#] === RegionEmbeddingDimension[#]&]Generate random points with the same embedding dimension as a given region:
RandomPoints[reg_] := Point[RandomReal[{-1, 1}, {40, RegionEmbeddingDimension[reg]}]]regs = {Interval[{0, 1}], Circle[], Cone[]};Visualize regions with random points:
Table[Show[DiscretizeRegion /@ {RandomPoints[r], r}], {r, regs}]Properties & Relations (8)
RegionEmbeddingDimension is the dimension of the space in which a region is embedded:
ℛ = Circle[];RegionEmbeddingDimension[ℛ]RegionDimension is the dimension of the point set that the region represents:
RegionDimension[ℛ]RegionEmbeddingDimension is always greater than or equal to RegionDimension:
RegionEmbeddingDimension[ℛ] ≥ RegionDimension[ℛ]RegionEmbeddingDimension is the number of variables needed for a point in the region:
ℛ = Sphere[];RegionEmbeddingDimension[ℛ]Since the embedding dimension is 3, three variables are needed:
Integrate[{x, y, z}, {x, y, z}∈ℛ]Since the embedding dimension is 3, the point p is taken to have length 3:
Integrate[p, p∈ℛ]RegionEmbeddingDimension for MeshRegion corresponds to the length of points:
MeshRegion[{{0, 0}, {1, 0}, {0, 1}}, Polygon[{1, 2, 3}]]RegionEmbeddingDimension[%]Similarly for BoundaryMeshRegion:
BoundaryMeshRegion[{{0, 0}, {1, 0}, {0, 1}}, Line[{1, 2, 3, 1}]]RegionEmbeddingDimension[%]For an ImplicitRegion, RegionEmbeddingDimension is the number of variables:
RegionEmbeddingDimension[ImplicitRegion[x ^ 2 + y ^ 2 == 1, {x, y}]]RegionEmbeddingDimension[ImplicitRegion[x ^ 2 + y ^ 2 == 1, {x, y, z}]]For a ParametricRegion, RegionEmbeddingDimension is the number of equations:
RegionEmbeddingDimension[ParametricRegion[{t, 2t}, {t}]]RegionEmbeddingDimension[ParametricRegion[{t, 2t, 3t}, {t}]]Boolean operations are defined for regions of the same RegionEmbeddingDimension:
RegionUnion[Ball[], Disk[]]
RegionEmbeddingDimension /@ {Ball[], Disk[]}With the same embedding dimension, the operation is well defined:
RegionUnion[Ball[], Cuboid[]]RegionEmbeddingDimension /@ {Ball[], Cuboid[]}Boolean operations do not change the RegionEmbeddingDimension:
{ℛ1, ℛ2} = {Ball[], Cuboid[]};RegionEmbeddingDimension /@ {ℛ1, ℛ2}Various Boolean operations all have the same embedding dimension:
RegionEmbeddingDimension /@ {RegionUnion[ℛ1, ℛ2], RegionIntersection[ℛ1, ℛ2]}The RegionEmbeddingDimension of a RegionProduct is the sum of the input dimensions:
ℛ1 = DiscretizeRegion[Disk[]];
ℛ2 = MeshRegion[{{0}, {1}}, Line[{1, 2}]];ℛ = RegionProduct[ℛ1, ℛ2]The resulting embedding dimension is the sum:
RegionEmbeddingDimension[ℛ] == RegionEmbeddingDimension[ℛ1] + RegionEmbeddingDimension[ℛ2]Text
Wolfram Research (2014), RegionEmbeddingDimension, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionEmbeddingDimension.html.
CMS
Wolfram Language. 2014. "RegionEmbeddingDimension." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RegionEmbeddingDimension.html.
APA
Wolfram Language. (2014). RegionEmbeddingDimension. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RegionEmbeddingDimension.html