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RegionMember[reg,{x,y,…}]

给出 True 如果数值点 {x,y,…} 是恒定区域 reg 的成员，否则将给出 False.

RegionMember[reg,{x,y,…}]

给出点 {x,y,…} 作为 reg 的成员的条件.

RegionMember[reg]

返回一个可以被重复应用到不同点的 RegionMemberFunction[…].

更多信息
Details and Options Details and Options
范例  
基本范例  
范围  
基本用途  
特定区域  
公式区域  
Show More Show More
网格区域  
导出区域  
CSG 区域  
细分区域  
应用  
基本应用  
区域中的随机点  
Monte Carlo 积分  
属性和关系  
参见
相关指南
历史
按以下格式引用：

RegionMember

RegionMember[reg,{x,y,…}]

给出 True 如果数值点 {x,y,…} 是恒定区域 reg 的成员，否则将给出 False.

RegionMember[reg,{x,y,…}]

给出点 {x,y,…} 作为 reg 的成员的条件.

RegionMember[reg]

返回一个可以被重复应用到不同点的 RegionMemberFunction[…].

更多信息

范例

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基本范例  (3)

检验一个特定的点是否在某个区域内：

Wolfram Language code: RegionMember[Polygon[{{0, 0}, {1, 0}, {0, 1}}], {1 / 3, 1 / 3}]

获得点的隶属条件：

Wolfram Language code: RegionMember[Disk[], {x, y}]

生成 RegionMemberFunction 应用到不同的点：

Wolfram Language code: mf = RegionMember[Disk[{0, 0}, 3]]
Wolfram Language code: mf[RandomReal[4, {7, 2}]]

范围  (23)

基本用途  (4)

直接检验一个点是否在某个区域内：

Wolfram Language code: ℛ = Rectangle[{0, 0}, {2, 2}];
Wolfram Language code: {RegionMember[ℛ, {1, 1}], RegionMember[ℛ, {3, 3}]}

直接检验一个点的列表是否在某个区域内：

Wolfram Language code: ℛ = Ball[{0, 0, 0}, 2];
Wolfram Language code: RegionMember[ℛ, RandomReal[2, {7, 3}]]

用变量 {x,y} 获取点的隶属条件：

Wolfram Language code: ℛ = Rectangle[{0, 0}, {2, 2}];
Wolfram Language code: RegionMember[ℛ, {x, y}]

生成 RegionMemberFunction 应用到不同的点：

Wolfram Language code: ℛ = Disk[{0, 0}, 2];
Wolfram Language code: mf = RegionMember[ℛ]
Wolfram Language code: mf[RandomReal[3, {7, 2}]]

特定区域  (6)

在 的区域：

Wolfram Language code: RegionMember[Point[{1}], {x}]
Wolfram Language code: RegionMember[Interval[{1, 5}], {x}]
Wolfram Language code: RegionMember[Interval[{1, 5}], {6}]

在 的区域：

Wolfram Language code: RegionMember[Point[{1, 2}], {x, y}]
Wolfram Language code: RegionMember[Circle[{0, 0}, 2], {x, y}]
Wolfram Language code: RegionMember[Disk[{1, 2}, {3, 4}], {x, y}]
Wolfram Language code: RegionMember[Circle[{0, 0}, 2], {3, 3}]

可视化在 的区域隶属：

Wolfram Language code: ℛ = Triangle[{{0, 0}, {1, 2}, {2, 1}}];

获得区域的边界：

Wolfram Language code: bounds = RegionBounds[ℛ]

在边界框以上对区域进行非均匀采采样：

Wolfram Language code: pts = RandomVariate[UniformDistribution[bounds], 10 ^ 4];
Wolfram Language code: col = RegionMember[ℛ, pts] /. {False -> Gray, True -> Red}; Graphics[{AbsolutePointSize[1], {col, Point /@ pts}}]

在 的区域：

Wolfram Language code: RegionMember[Point[{1, 2, 3}], {x, y, z}]
Wolfram Language code: RegionMember[Line[{{1, 2, 3}, {4, 5, 6}}], {x, y, z}]
Wolfram Language code: RegionMember[Polygon[{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}], {x, y, z}]
Wolfram Language code: RegionMember[Cylinder[{{0, 0, 0}, {1, 1, 1}}, 2], {x, y, z}]
Wolfram Language code: RegionMember[Ball[{0, 0, 0}, 2], {3, 3, 3}]

可视化在 的区域隶属：

Wolfram Language code: ℛ = Cuboid[{-1, -1, -1}, {1, 1, 1}]; in = RegionCentroid[ℛ]; out = {2, 0, 0};
Wolfram Language code: {RegionMember[ℛ, in], RegionMember[ℛ, out]}
Wolfram Language code: Graphics3D[{{Opacity[0.2], ℛ}, PointSize[Large], {Green, Point[in]}, {Red, Point[out]}}, Boxed -> False]

在 的区域：

Wolfram Language code: RegionMember[Simplex[{{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}}], {x, y, z, t}]
Wolfram Language code: RegionMember[Cuboid[{1, 2, 3, 4, 5}, {6, 7, 8, 9, 10}], {x, y, z, t, u}]
Wolfram Language code: RegionMember[Ball[{0, 0, 0, 0, 0, 0, 0}, 8], {1, 2, 3, 4, 5, 6, 7}]

公式区域  (3)

隐区域：

Wolfram Language code: RegionMember[ImplicitRegion[x^2 + y^2 ≤ 1, {x, y}], {x, y}]
Wolfram Language code: RegionMember[ImplicitRegion[x^2 + y^2 ≤ 1, {x, y, {z, 0, 2}}], {x, y, z}]

可视化在 的区域隶属：

Wolfram Language code: ℛ = ImplicitRegion[x ^ 2 - 2y ^ 2 <= 1, {{x, -3, 3}, {y, -4, 4}}];

获得区域的边界：

Wolfram Language code: bounds = RegionBounds[ℛ]

在边界框以上对区域进行非均匀采采样：

Wolfram Language code: pts = RandomVariate[UniformDistribution[bounds], 10 ^ 4];
Wolfram Language code: col = RegionMember[ℛ, pts] /. {False -> Gray, True -> Red}; Graphics[{AbsolutePointSize[1], {col, Point /@ pts}}]

参数区域：

Wolfram Language code: RegionMember[ParametricRegion[{(1 - t ^ 2) / (1 + t ^ 2), 2t / (1 + t ^ 2)}, {t}], {x, y}]
Wolfram Language code: RegionMember[ParametricRegion[{{u + v + w, u + v - w, u - v - w, -u - v - w}, -1 ≤ u ≤ 1 && -1 ≤ v ≤ 1 && -1 ≤ w ≤ 1}, {u, v, w}], {x, y, z, t}]

网格区域  (4)

在二维的 MeshRegion：

Wolfram Language code: ℛ = DelaunayMesh[RandomReal[8, {100, 2}]]; in = RegionCentroid[ℛ]; out = {9, 9};
Wolfram Language code: {RegionMember[ℛ, in], RegionMember[ℛ, out]}
Wolfram Language code: Show[ℛ, Graphics[{PointSize[Large], {Green, Point[in]}, {Red, Point[out]}}]]

在三维：

Wolfram Language code: ℛ = DelaunayMesh[RandomReal[8, {100, 3}]]; in = RegionCentroid[ℛ]; out = {9, 9, 9};
Wolfram Language code: {RegionMember[ℛ, in], RegionMember[ℛ, out]}
Wolfram Language code: Show[HighlightMesh[RegionBoundary[ℛ], Style[2, Opacity[0.1]]], Graphics3D[{PointSize[Large], {Green, Point[in]}, {Red, Point[out]}}]]

在二维的 BoundaryMeshRegion：

Wolfram Language code: ℛ = ConvexHullMesh[RandomReal[8, {100, 2}]]; in = RegionCentroid[ℛ]; out = {9, 9};
Wolfram Language code: {RegionMember[ℛ, in], RegionMember[ℛ, out]}
Wolfram Language code: Show[ℛ, Graphics[{PointSize[Large], {Green, Point[in]}, {Red, Point[out]}}]]

在三维的 BoundaryMeshRegion：

Wolfram Language code: ℛ = ConvexHullMesh[RandomReal[8, {1000, 3}]]; in = RegionCentroid[ℛ]; out = {9, 9, 9};
Wolfram Language code: {RegionMember[ℛ, in], RegionMember[ℛ, out]}
Wolfram Language code: Show[HighlightMesh[ℛ, Style[2, Opacity[0.2]]], Graphics3D[{PointSize[Large], {Green, Point[in]}, {Red, Point[out]}}]]

导出区域  (4)

两个区域的 RegionIntersection：

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

获得区域的边界：

Wolfram Language code: bounds = RegionBounds[ℛ]

在边界框以上对区域进行非均匀采采样：

Wolfram Language code: pts = RandomVariate[UniformDistribution[bounds], 10 ^ 4];
Wolfram Language code: col = RegionMember[ℛ, pts] /. {False -> Gray, True -> Red}; Graphics[{AbsolutePointSize[1], {col, Point /@ pts}}]

混合维度区域的 RegionUnion：

Wolfram Language code: ℛ = RegionUnion[Circle[{1, 0}, 1], Disk[{0, 0}, 1]]; Region[ℛ]
Wolfram Language code: RegionMember[ℛ, {x, y}]

TransformedRegion：

Wolfram Language code: ℛ = TransformedRegion[Disk[{0, 0}, 1], ShearingTransform[θ, {1, 0}, {0, 1}]];
Wolfram Language code: RegionMember[ℛ, {x, y}]

RegionBoundary：

Wolfram Language code: ℛ = RegionBoundary[Disk[{0, 0}, 1]]; Region[ℛ]
Wolfram Language code: RegionMember[ℛ, {x, y}]

CSG 区域  (1)

二维空间中的 CSGRegion：

Wolfram Language code: ℛ = CSGRegion["Difference", {Disk[], Disk[{1 / 2, 1 / 2}]}]; in = {-1 / 2, -1 / 2}; out = {0, 0};
Wolfram Language code: {RegionMember[ℛ, in], RegionMember[ℛ, out]}
Wolfram Language code: Show[ℛ, Graphics[{PointSize[Large], {StandardGreen, Point[in]}, {StandardRed, Point[out]}}]]

三维：

Wolfram Language code: ℛ = CSGRegion["Difference", {Cube[2], Cylinder[{{1, 1, 1}, {1, -1, 1}}]}]; in = {0, 0, 0}; out = {1, 1, 1};
Wolfram Language code: {RegionMember[ℛ, in], RegionMember[ℛ, out]}
Wolfram Language code: Show[Region[BoundaryDiscretizeRegion[ℛ], BaseStyle -> Opacity[0.2]], Graphics3D[{Opacity[1], PointSize[Large], {StandardGreen, Point[in]}, {StandardRed, Point[out]}}]]

细分区域  (1)

二维空间中的 SubdivisionRegion：

Wolfram Language code: ℛ = SubdivisionRegion[Rectangle[]]; in = {1 / 2, 1 / 2}; out = {1, 1};
Wolfram Language code: {RegionMember[ℛ, in], RegionMember[ℛ, out]}
Wolfram Language code: Show[ℛ, Graphics[{PointSize[Large], {StandardGreen, Point[in]}, {StandardRed, Point[out]}}]]

三维：

Wolfram Language code: ℛ = SubdivisionRegion[Tetrahedron[]]; in = {0, 0, 0}; out = {1 / 10, 1 / 10, 1 / 10};
Wolfram Language code: {RegionMember[ℛ, in], RegionMember[ℛ, out]}
Wolfram Language code: Show[Region[BoundaryDiscretizeRegion[ℛ], BaseStyle -> Opacity[0.2]], Graphics3D[{Opacity[1], PointSize[Large], {StandardGreen, Point[in]}, {StandardRed, Point[out]}}]]

应用  (6)

基本应用  (2)

将给定国家的多边形数据转换成一个 MeshRegion：

Wolfram Language code: ℛ = DiscretizeGraphics[CountryData["Austria", {"FullPolygon", "Mercator"}]]

算出一个城市的隶属：

Wolfram Language code: city = GeoGridPosition[GeoPosition[Entity["City", {"Vienna", "Vienna", "Austria"}]], "Mercator"]["Data"]
Wolfram Language code: RegionMember[ℛ, city]
Wolfram Language code: Show[ℛ, Epilog -> {Red, Point@city}]

对一个参数化的区域，RegionMember 可以给出关于决定给定点是成员的参数的条件：

Wolfram Language code: ℛ = Disk[{0, 0}, {Subscript[r, x], Subscript[r, y]}]; RegionMember[ℛ, {3, 4}]

找出一个区域包含点的实例：

Wolfram Language code: FindInstance[%, {Subscript[r, x], Subscript[r, y]}]

可视化：

Wolfram Language code: Graphics[{LightGray, ℛ /. %, Red, Point[{3, 4}]}]

区域中的随机点  (2)

通过过滤一个点的均匀集合，生成在区域上的点：

Wolfram Language code: ℛ = ImplicitRegion[1 < x ^ 2 + y ^ 2 < 3, {x, y}];

获得区域的边界：

Wolfram Language code: bounds = RegionBounds[ℛ];

在边界框以上对区域进行非均匀采采样：

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

选择成员点：

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

可视化成员点：

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

将给定国家的多边形数据转换成一个 MeshRegion：

Wolfram Language code: ℛ = [image];

获得区域的边界：

Wolfram Language code: bounds = RegionBounds[ℛ]

在边界框以上对区域进行非均匀采采样：

Wolfram Language code: pts = RandomVariate[UniformDistribution[bounds], 10 ^ 4];
Wolfram Language code: col = RegionMember[ℛ, pts] /. {False -> Gray, True -> Red}; Graphics[{AbsolutePointSize[1], {col, Point /@ pts}}]

Monte Carlo 积分  (2)

做蒙特卡罗积分，估算单位圆盘的面积：

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

获得区域的边界：

Wolfram Language code: bounds = RegionBounds[ℛ]

在边界框以上对区域进行非均匀采采样：

Wolfram Language code: pts = RandomVariate[UniformDistribution[bounds], 10 ^ 4];
Wolfram Language code: col = RegionMember[ℛ, pts] /. {False -> Gray, True -> Red}; Graphics[{ℛ, AbsolutePointSize[1], {col, Point /@ pts}}]

计算区域以内的样本的数目：

Wolfram Language code: inside = Count[RegionMember[ℛ, pts], True]

获取区域内样本与样本点总数的比例：

Wolfram Language code: ratio = N[(inside/Length[pts])]

获得边界内的面积：

Wolfram Language code: area = Area[Cuboid@@(bounds)]

获得区域的近似面积：

Wolfram Language code: area * ratio

用区域内的随机点做蒙特卡罗积分：

Wolfram Language code: ℛ = ImplicitRegion[2 ≤ x^2 + y^2 ≤ 4, {x, y}]; pts = Select[RandomVariate[UniformDistribution[RegionBounds[ℛ]], 10 ^ 5], RegionMember[ℛ, #]&];

在每个样本点对函数求值并取它们的平均值：

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

与精确值比较：

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

属性和关系  (5)

Element 可以用来检验恒定区域的区域隶属：

Wolfram Language code: ℛ = Disk[{0, 0}, 2];
Wolfram Language code: {RegionMember[ℛ, {0, 0}], {0, 0}∈ℛ}

对成员 RegionDistance 为零：

Wolfram Language code: ℛ = Disk[{0, 0}, 2];
Wolfram Language code: {RegionMember[ℛ, {0, 0}], RegionDistance[ℛ, {0, 0}]}

对成员 SignedRegionDistance 为非正：

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

对非成员 SignedRegionDistance 为正：

Wolfram Language code: {RegionMember[ℛ, {-2, 0}], SignedRegionDistance[ℛ, {-2, 0}]}

使用 RegionNearest 找到最近的成员：

Wolfram Language code: ℛ = Disk[{0, 0}, 2];
Wolfram Language code: {RegionMember[ℛ, RegionNearest[ℛ, {3, 3}]], RegionMember[ℛ, {3, 3}]}

可视化：

Wolfram Language code: Graphics[{{LightBlue, ℛ}, {Green, Point[RegionNearest[ℛ, {3, 3}]]}, {Red, Point[{3, 3}]}}]

使用 FindInstance 找出特殊和公式区域的多个实例：

Wolfram Language code: ℛ = ImplicitRegion[x ^ 2 - 2y ^ 2 <= 1, {{x, -3, 3}, {y, -4, 4}}];
Wolfram Language code: Region[ℛ]

找出在两个区域的点：

Wolfram Language code: pts = FindInstance[{x, y}∈ℛ, {x, y}, 100];
Wolfram Language code: RegionPlot[ℛ, Epilog -> {Red, Point[{x, y} /. pts]}]
Wolfram Research (2014)，RegionMember，Wolfram 语言函数，https://reference.wolfram.com/language/ref/RegionMember.html.

文本

Wolfram Research (2014)，RegionMember，Wolfram 语言函数，https://reference.wolfram.com/language/ref/RegionMember.html.

CMS

Wolfram 语言. 2014. "RegionMember." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/RegionMember.html.

APA

Wolfram 语言. (2014). RegionMember. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/RegionMember.html 年

BibTeX

@misc{reference.wolfram_2026_regionmember, author="Wolfram Research", title="{RegionMember}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/RegionMember.html}", note=[Accessed: 30-October-2026]}

BibLaTeX

@online{reference.wolfram_2026_regionmember, organization={Wolfram Research}, title={RegionMember}, year={2014}, url={https://reference.wolfram.com/language/ref/RegionMember.html}, note=[Accessed: 30-October-2026]}