Wolfram Language & System Documentation Center

RegionMemberFunction

RegionMemberFunction [reg,…]

represents a function whose values give whether a point is in a region reg or not.

Details

Details and Options Details and Options

Examples

Basic Examples

Scope

Special Regions

Formula Regions

Mesh Regions

Derived Regions

CSG Regions

Subdivision Regions

Applications

Properties & Relations

Possible Issues

RegionMemberFunction

RegionMemberFunction [reg,…]

represents a function whose values give whether a point is in a region reg or not.

Details

RegionMemberFunction works like Function .
RegionMemberFunction is generated by RegionMember [reg].
RegionMemberFunction [reg,…][{x,y,…}] gives True if the numerical point {x,y,…} is a member of the constant region reg where ConstantRegionQ [reg] is True .

Examples

open all close all

Basic Examples (1)

RegionMember for a region generates a RegionMemberFunction :

Wolfram Language code: mf = RegionMember[Disk[]]

Apply function repeatedly for member tests:

Wolfram Language code: Map[mf, RandomReal[{-1, 1}, {10, 2}]]

Scope (20)

Special Regions (5)

Use RegionMember to generate a RegionMemberFunction :

Wolfram Language code:

ℛ = Polygon[{{0, 0}, {2, -1}, {1, 0}, {2, 1}}];
Region[ℛ]

Wolfram Language code: mf = RegionMember[ℛ]

Apply it to a list of points to test membership:

Wolfram Language code: mf[{{1 / 2, 0}, {3, 4}}]

Wolfram Language code:

pts = RandomVariate[UniformDistribution[RegionBounds[ℛ]], 10 ^ 3];
col = mf[pts] /. {False -> Gray, True -> Red};

Wolfram Language code: Graphics[{ℛ, AbsolutePointSize[1.6], {col, Point /@ pts}}]

Test for points within an Interval :

Wolfram Language code:

ℛ = Interval[{0, 1}];
gr = NumberLinePlot[ℛ]

Wolfram Language code: mf = RegionMember[ℛ]

Apply it to a list of points to test membership:

Wolfram Language code:

pts = RandomVariate[UniformDistribution[{{-0.5, 1.5}}], 100];
col = mf[pts] /. {False -> Gray, True -> Red};

Wolfram Language code: gp = Graphics[{AbsolutePointSize[1.6], Table[{col[[i]], Point[{pts[[i, 1]], 3}]}, {i, 100}]}];

Wolfram Language code: Show[gr, gp, PlotRange -> All]

Test for points within a Disk :

Wolfram Language code:

ℛ = Disk[{0, 0}, 2];
Region[ℛ]

Wolfram Language code: mf = RegionMember[ℛ]

Apply it to a list of points to test membership:

Wolfram Language code:

pts = RandomVariate[UniformDistribution[RegionBounds[ℛ]], 10 ^ 3];
col = mf[pts] /. {False -> Gray, True -> Red};

Wolfram Language code: Graphics[{ℛ, AbsolutePointSize[1.6], {col, Point /@ pts}}]

Test for points within a Cylinder :

Wolfram Language code:

ℛ = Cylinder[{{0, 0, 0}, {2, 2, 2}}, 2];
gr = Graphics3D[{Opacity[0.3], ℛ}, Boxed -> False, Lighting -> "Neutral"]

Wolfram Language code: mf = RegionMember[ℛ]

Apply it to a list of points to test membership:

Wolfram Language code:

pts = RandomVariate[UniformDistribution[RegionBounds[ℛ]], 10 ^ 3];
col = mf[pts] /. {False -> Gray, True -> Red};

Wolfram Language code: Show[gr, Graphics3D[{AbsolutePointSize[2], {col, Point /@ pts}}]]

Regions in :

Wolfram Language code:

ℛ = Ball[{0, 0, 0, 0, 0, 0, 0}, 1];
pl = RandomReal[{0, 0.7}, {7, 7}];

Wolfram Language code: mf = RegionMember[ℛ];

Wolfram Language code: mf /@ pl

Formula Regions (3)

A union of two disks as an ImplicitRegion :

Wolfram Language code:

ℛ = ImplicitRegion[x^2 + y^2 ≤ 4∨(x - 3)^2 + y^2 ≤ 3, {x, y}];
DiscretizeRegion[ℛ, {{-6, 6}, {-2, 2}}]

Wolfram Language code: mf = RegionMember[ℛ]

Apply it to a list of points to test membership:

Wolfram Language code: mf[{{1, 1}, {2, 2}}]

Wolfram Language code:

pts = RandomVariate[UniformDistribution[RegionBounds[ℛ]], 10 ^ 4];
col = mf[pts] /. {False -> Gray, True -> Red};

Wolfram Language code: Graphics[{AbsolutePointSize[1], {col, Point /@ pts}}]

A union of two cylinders as an ImplicitRegion :

Wolfram Language code:

ℛ = ImplicitRegion[x^2 + y^2 ≤ 2∨(x - 2)^2 + y^2 ≤ 1, {x, y, {z, 0, 2}}];
DiscretizeRegion[ℛ, {2{-2, 2}, {-2, 2}, {0, 2}}]

Wolfram Language code: mf = RegionMember[ℛ]

Apply it to a list of points to test membership:

Wolfram Language code:

pts = RandomVariate[UniformDistribution[RegionBounds[ℛ]], 10 ^ 4];
col = mf[pts] /. {False -> Gray, True -> Red};

Wolfram Language code: Graphics3D[{AbsolutePointSize[1.6], {col, Point /@ pts}}]

A disk represented as a ParametricRegion :

Wolfram Language code: ℛ = ParametricRegion[{r Cos[θ], r Sin[θ]}, {{r, 0, 1}, {θ, 0, 2π}}];

Wolfram Language code: mf = RegionMember[ℛ]

Apply it to a list of points to test membership:

Wolfram Language code:

pts = RandomVariate[UniformDistribution[RegionBounds[ℛ]], 10 ^ 4];
col = mf[pts] /. {False -> Gray, True -> Red};

Wolfram Language code: Graphics[{AbsolutePointSize[1], {col, Point /@ pts}}]

Mesh Regions (6)

MeshRegion in 1D:

Wolfram Language code: ℛ = DelaunayMesh[RandomReal[1, {10, 1}]]

Wolfram Language code: mf = RegionMember[ℛ]

Apply it to a list of points to test membership:

Wolfram Language code: mf[{{1 / 2}, {3}}]

2D:

Wolfram Language code: ℛ = DelaunayMesh[RandomReal[2, {50, 2}]]

Wolfram Language code: mf = RegionMember[ℛ]

Apply it to a list of points to test membership:

Wolfram Language code:

pts = RandomVariate[UniformDistribution[RegionBounds[ℛ]], 10 ^ 4];
col = mf[pts] /. {False -> Gray, True -> Red};

Wolfram Language code: Graphics[{AbsolutePointSize[1], {col, Point /@ pts}}]

3D:

Wolfram Language code: ℛ = DelaunayMesh[RandomReal[1, {100, 3}]]

Wolfram Language code: mf = RegionMember[ℛ]

Apply it to a list of points to test membership:

Wolfram Language code:

pts = RandomVariate[UniformDistribution[RegionBounds[ℛ]], 10 ^ 4];
col = mf[pts] /. {False -> Gray, True -> Red};

Wolfram Language code: Graphics3D[{AbsolutePointSize[1], {col, Point /@ pts}}]

BoundaryMeshRegion in 1D:

Wolfram Language code: ℛ = ConvexHullMesh[RandomReal[1, {10, 1}]]

Wolfram Language code: mf = RegionMember[ℛ]

Apply it to a list of points to test membership:

Wolfram Language code: mf[{{1 / 2}, {3}}]

2D:

Wolfram Language code: ℛ = ConvexHullMesh[RandomReal[1, {50, 2}]]

Wolfram Language code: mf = RegionMember[ℛ]

Apply it to a list of points to test membership:

Wolfram Language code:

pts = RandomVariate[UniformDistribution[RegionBounds[ℛ]], 10 ^ 4];
col = mf[pts] /. {False -> Gray, True -> Red};

Wolfram Language code: Graphics[{AbsolutePointSize[1], {col, Point /@ pts}}]

3D:

Wolfram Language code: ℛ = ConvexHullMesh[RandomReal[1, {100, 3}]]

Wolfram Language code: mf = RegionMember[ℛ]

MeshCoordinates are always members of the region:

Wolfram Language code: AllTrue[MeshCoordinates[ℛ], mf]

Derived Regions (2)

RegionIntersection of two regions:

Wolfram Language code: ℛ = RegionIntersection[Disk[{0, 0}, 1], Disk[{0, 1}, 1]];

Wolfram Language code: Region[ℛ]

Wolfram Language code: mf = RegionMember[ℛ]

Apply it to a list of points to test membership:

Wolfram Language code: mf[{{0, 1}, {3, 4}}]

Wolfram Language code:

pts = RandomVariate[UniformDistribution[RegionBounds[ℛ]], 10 ^ 4];
col = mf[pts] /. {False -> Gray, True -> Red};

Wolfram Language code: Graphics[{AbsolutePointSize[1], {col, Point /@ pts}}]

TransformedRegion :

Wolfram Language code: ℛ = TransformedRegion[Disk[{0, 0}, 1], ScalingTransform[{3, 2}]];

Wolfram Language code: DiscretizeRegion[ℛ, {{-3, 3}, {-2, 2}}]

Wolfram Language code: mf = RegionMember[ℛ]

Apply it to a list of points to test membership:

Wolfram Language code:

pts = RandomVariate[UniformDistribution[RegionBounds[ℛ]], 10 ^ 4];
col = mf[pts] /. {False -> Gray, True -> Red};

Wolfram Language code: Graphics[{AbsolutePointSize[1], {col, Point /@ pts}}]

CSG Regions (2)

CSGRegion in 2D:

Wolfram Language code: ℛ = CSGRegion["Difference", {Disk[], Disk[{1 / 2, 1 / 2}]}];

Wolfram Language code: mf = RegionMember[ℛ]

Apply it to a list of points to test membership:

Wolfram Language code:

pts = RandomVariate[UniformDistribution[RegionBounds[ℛ]], 10 ^ 4];
col = mf[pts] /. {False -> Gray, True -> Red};

Wolfram Language code: Graphics[{AbsolutePointSize[1], {col, Point /@ pts}}]

CSGRegion in 3D:

Wolfram Language code: ℛ = CSGRegion["Difference", {Cube[3], Cuboid[{-1, -3, -1}, {3, 3, 3}]}];

Wolfram Language code: mf = RegionMember[ℛ]

Apply it to a list of points to test membership:

Wolfram Language code:

pts = RandomVariate[UniformDistribution[RegionBounds[ℛ]], 10 ^ 4];
col = mf[pts] /. {False -> Gray, True -> Red};

Wolfram Language code: Graphics3D[{AbsolutePointSize[1], {col, Point /@ pts}}]

Subdivision Regions (2)

SubdivisionRegion in 2D:

Wolfram Language code: ℛ = SubdivisionRegion[Rectangle[]];

Wolfram Language code: mf = RegionMember[ℛ]

Apply it to a list of points to test membership:

Wolfram Language code:

pts = RandomVariate[UniformDistribution[RegionBounds[ℛ]], 10 ^ 4];
col = mf[pts] /. {False -> Gray, True -> Red};

Wolfram Language code: Graphics[{AbsolutePointSize[1], {col, Point /@ pts}}]

SubdivisionRegion in 3D:

Wolfram Language code: ℛ = SubdivisionRegion[Tetrahedron[]];

Wolfram Language code: mf = RegionMember[ℛ]

Apply it to a list of points to test membership:

Wolfram Language code:

pts = RandomVariate[UniformDistribution[RegionBounds[ℛ]], 10 ^ 4];
col = mf[pts] /. {False -> Gray, True -> Red};

Wolfram Language code: Graphics3D[{AbsolutePointSize[1], {col, Point /@ pts}}]

Applications (3)

Generate points on a region by filtering a uniform set of points:

Wolfram Language code: ℛ = BoundaryDiscretizeGraphics[CountryData["US", {"Polygon", "Equirectangular"}]];

Get the region bounds:

Wolfram Language code: bounds = RegionBounds[ℛ];

Uniformly sample over the bounding box of the region:

Wolfram Language code: pts = RandomVariate[UniformDistribution[bounds], 10 ^ 4];

Select member points:

Wolfram Language code: mpts = Select[pts, RegionMember[ℛ]];

Visualize member points:

Wolfram Language code: Graphics[{AbsolutePointSize[1], Point[mpts]}]

Color points based on which region they belong to:

Wolfram Language code:

Subscript[ℛ, 1] = BoundaryDiscretizeGraphics[CountryData["US", {"Polygon", "Equirectangular"}]];
Subscript[ℛ, 2] = BoundaryDiscretizeGraphics[CountryData["Canada", {"Polygon", "Equirectangular"}]];

Create membership functions for each:

Wolfram Language code:

mf1 = RegionMember[Subscript[ℛ, 1]];
mf2 = RegionMember[Subscript[ℛ, 2]];

Generate points over a bounding box of both regions:

Wolfram Language code: pts = RandomVariate[UniformDistribution[RegionBounds[RegionUnion[Subscript[ℛ, 1], Subscript[ℛ, 2]]]], 10 ^ 4];

Visualize:

Wolfram Language code:

Graphics[{AbsolutePointSize[1], {Red, Point@Select[pts, mf1]}, {Blue, Point@Select[pts, mf2]}, {Black, Point@Select[pts, !mf1[#] && !mf2[#]&]}}]

Use random points in a region to perform Monte Carlo integration:

Wolfram Language code:

ℛ = ImplicitRegion[2 ≤ x^2 + y^2 ≤ 4, {x, y}];
mf = RegionMember[ℛ];
pts = Select[RandomVariate[UniformDistribution[RegionBounds[ℛ]], 10 ^ 5], mf];

Evaluate a function at each sample point and take their average:

Wolfram Language code:

f[x_, y_] := x^3 + y^4;
intVal = Mean[f@@@pts] * RegionMeasure[ℛ]

Compare with the exact value:

Wolfram Language code: Integrate[f[x, y], {x, y}∈ℛ]

Wolfram Language code: %//N

Properties & Relations (1)

RegionMemberFunction is generated by RegionMember :

Wolfram Language code: mf = RegionMember[Disk[{0, 0}, 1]]

Wolfram Language code: mf@RandomReal[1, {7, 2}]

Possible Issues (1)

RegionMemberFunction can only be generated for ConstantRegionQ regions:

Wolfram Language code: ℛ = Disk[{x, y}, r]

Wolfram Language code: ConstantRegionQ[ℛ]

Wolfram Language code: RegionMember[ℛ]

Generate the member function for a specific instance:

Wolfram Language code: mf = RegionMember[ℛ /. {x -> 0, y -> 2, r -> 1}]

Wolfram Language code: mf[{{0, 1}, {10, 10}}]

Related Guides

▪

Region Properties and Measures

History

Introduced in 2014 (10.0)

Wolfram Research (2014), RegionMemberFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionMemberFunction.html.

Text

Wolfram Research (2014), RegionMemberFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionMemberFunction.html.

CMS

Wolfram Language. 2014. "RegionMemberFunction." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RegionMemberFunction.html.

APA

Wolfram Language. (2014). RegionMemberFunction. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RegionMemberFunction.html

BibTeX

@misc{reference.wolfram_2026_regionmemberfunction, author="Wolfram Research", title="{RegionMemberFunction}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/RegionMemberFunction.html}", note=[Accessed: 04-September-2026]}

BibLaTeX

@online{reference.wolfram_2026_regionmemberfunction, organization={Wolfram Research}, title={RegionMemberFunction}, year={2014}, url={https://reference.wolfram.com/language/ref/RegionMemberFunction.html}, note=[Accessed: 04-September-2026]}

Top [フレーム]

More Learning

Tech Support

Data Science, Finance & Business

Engineering

Science & Technology

Technology for Education

Education Initiatives

Get Started

Grow Your Skills

Work with Us

Educational Programs for Adults

Educational Programs for Youth

Read

RegionMemberFunction

Details

Examples

Basic Examples (1)

Scope (20)

Special Regions (5)

Formula Regions (3)

Mesh Regions (6)

Derived Regions (2)

CSG Regions (2)

Subdivision Regions (2)

Applications (3)

Properties & Relations (1)

Possible Issues (1)

See Also

Related Guides

History

Text

CMS

APA

BibTeX

BibLaTeX