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RegionIntersection[reg1,reg2,…]

给出区域 reg1、reg2、… 相交的部分.

更多信息和选项
Details and Options Details and Options
范例  
基本范例  
范围  
特殊区域  
公式区域  
网格区域  
导出区域  
CSG 区域  
细分区域  
应用  
属性和关系  
可能存在的问题  
巧妙范例  
参见
相关指南
历史
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RegionIntersection

RegionIntersection[reg1,reg2,…]

给出区域 reg1、reg2、… 相交的部分.

更多信息和选项

范例

打开所有单元 关闭所有单元

基本范例  (2)

两个圆盘的交集：

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

可视化：

Wolfram Language code: Region[ℛ]

两个 MeshRegion 对象的交集：

Wolfram Language code: RegionIntersection[[image], [image]]

范围  (16)

特殊区域  (7)

对于某些区域，交集是显式计算：

Wolfram Language code: Subscript[ℛ, 1] = Triangle[{{0, 0}, {2, 3}, {-2, 3}}]; Subscript[ℛ, 2] = Triangle[{{0, 2}, {2, -1}, {-2, -1}}];
Wolfram Language code: Subscript[ℛ, 3] = RegionIntersection[Subscript[ℛ, 1], Subscript[ℛ, 2]]

可视化交集：

Wolfram Language code: Graphics[{Opacity[0.5], {Red, Subscript[ℛ, 1]}, {Green, Subscript[ℛ, 2]}, {Blue, Subscript[ℛ, 3]}}]

无穷直线和球的交集：

Wolfram Language code: Subscript[ℛ, 1] = InfiniteLine[{{-1, -2, -3}, {1, 2, 3}}]; Subscript[ℛ, 2] = Ball[{0, 0, 0}, 3];
Wolfram Language code: Subscript[ℛ, 3] = RegionIntersection[Subscript[ℛ, 1], Subscript[ℛ, 2]]

可视化交集：

Wolfram Language code: Graphics3D[{Opacity[0.5], {Red, Subscript[ℛ, 1]}, {Green, Subscript[ℛ, 2]}, {Blue, Subscript[ℛ, 3]}}]

Line 区域的交集：

Wolfram Language code: ℛ = RegionIntersection[Line[{{1}, {3}}], Line[{{2}, {4}}]]

可视化：

Wolfram Language code: Region[ℛ]

Polygon 区域的交集：

Wolfram Language code: ℛ1 = Polygon[{{0, 0}, {-1, 3}, {0, 2}, {1, 3}}]; ℛ2 = Polygon[{{0, 5}, {1, 2}, {0, 3}, {-1, 2}}];
Wolfram Language code: ℛ = RegionIntersection[ℛ1, ℛ2];

可视化:

Wolfram Language code: Region[ℛ]

两个 Disk 区域的交集：

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

可视化：

Wolfram Language code: Region[ℛ]

长方体和圆锥相交的部分：

Wolfram Language code: ℛ = RegionIntersection[Cuboid[], Cone[]];

可视化:

Wolfram Language code: Region[ℛ]

有不同 RegionDimension 的区域的交集：

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

可视化：

Wolfram Language code: Region[ℛ]

公式区域  (2)

ImplicitRegion 对象相交的部分是一个 ImplicitRegion：

Wolfram Language code: Subscript[ℛ, 1] = ImplicitRegion[x ≤ 1, {x}]; Subscript[ℛ, 2] = ImplicitRegion[x ≥ -1, {x}];
Wolfram Language code: RegionIntersection[Subscript[ℛ, 1], Subscript[ℛ, 2]]

2D:

Wolfram Language code: Subscript[ℛ, 1] = ImplicitRegion[x^2 + y^2 ≤ 1, {x, y}]; Subscript[ℛ, 2] = ImplicitRegion[x^2 + (y - 1)^2 ≤ 1, {x, y}];
Wolfram Language code: RegionIntersection[Subscript[ℛ, 1], Subscript[ℛ, 2]]

3D:

Wolfram Language code: Subscript[ℛ, 1] = ImplicitRegion[x^2 + y^2 + z^2 ≤ 1, {x, y, z}]; Subscript[ℛ, 2] = ImplicitRegion[(x - 1)^2 + y^2 + z^2 ≤ 1, {x, y, z}];
Wolfram Language code: RegionIntersection[Subscript[ℛ, 1], Subscript[ℛ, 2]]

nD:

Wolfram Language code: Subscript[ℛ, 1] = ImplicitRegion[x^2 + y^2 + z^2 + u^2 + v^2 ≤ 1, {x, y, z, u, v}]; Subscript[ℛ, 2] = ImplicitRegion[(x - 1)^2 + y^2 + z^2 + u^2 + v^2 ≤ 1, {x, y, z, u, v}];
Wolfram Language code: RegionIntersection[Subscript[ℛ, 1], Subscript[ℛ, 2]]

ParametricRegion 对象相交的部分：

Wolfram Language code: Subscript[ℛ, 1] = ParametricRegion[{u, v}, {{u, 0, 2}, {v, 0, 2}}]; Subscript[ℛ, 2] = ParametricRegion[{u + 1, v + 1}, {{u, 0, 2}, {v, 0, 2}}];
Wolfram Language code: ℛ = RegionIntersection[Subscript[ℛ, 1], Subscript[ℛ, 2]];

可视化：

Wolfram Language code: Region[ℛ]

网格区域  (2)

BoundaryMeshRegion 对象相交的部分是一个 BoundaryMeshRegion：

Wolfram Language code: RegionIntersection[[image], [image]]
Wolfram Language code: BoundedRegionQ[%]

2D:

Wolfram Language code: RegionIntersection[[image], [image]]
Wolfram Language code: BoundedRegionQ[%]

3D:

Wolfram Language code: RegionIntersection[[image], [image]]
Wolfram Language code: BoundedRegionQ[%]

全维 (full-dimensional) MeshRegion 对象相交的部分是一个 MeshRegion：

Wolfram Language code: RegionIntersection[[image], [image]]
Wolfram Language code: MeshRegionQ[%]

2D:

Wolfram Language code: RegionIntersection[[image], [image]]
Wolfram Language code: MeshRegionQ[%]

3D:

Wolfram Language code: RegionIntersection[[image], [image]]
Wolfram Language code: MeshRegionQ[%]

导出区域  (2)

BooleanRegion 对象相交的部分：

Wolfram Language code: Subscript[ℛ, 1] = BooleanRegion[Or, {Triangle[{{0, 0}, {2, 3}, {-2, 3}}], Triangle[{{0, 2}, {2, -1}, {-2, -1}}]}]; Subscript[ℛ, 2] = BooleanRegion[And, {Triangle[{{0, 0}, {2, 3}, {-2, 3}}], Triangle[{{0, 2}, {2, -1}, {-2, 2}}]}];
Wolfram Language code: ℛ = RegionIntersection[Subscript[ℛ, 1], Subscript[ℛ, 2]];

可视化：

Wolfram Language code: Region[ℛ]

TransformedRegion 对象相交的部分：

Wolfram Language code: Subscript[ℛ, 1] = TransformedRegion[Cuboid[], RotationTransform[Pi / 8, {1, 0, 0}]]; Subscript[ℛ, 2] = TransformedRegion[Cuboid[], RotationTransform[Pi / 8, {0, 1, 0}]];
Wolfram Language code: ℛ = RegionIntersection[Subscript[ℛ, 1], Subscript[ℛ, 2]];

可视化：

Wolfram Language code: Region[ℛ]

CSG 区域  (2)

二维空间中 CSGRegion 对象的交集：

Wolfram Language code: Subscript[ℛ, 1] = CSGRegion[{Rectangle[{-1, -8}, {1, 8}], Rectangle[{-3, -8}, {-5, 8}], Rectangle[{-7, -8}, {-9, 8}], Rectangle[{3, -8}, {5, 8}], Rectangle[{7, -8}, {9, 8}]}]; Subscript[ℛ, 2] = CSGRegion["Difference", {Disk[{0, 0}, 8], Disk[{0, 0}, 4]}];
Wolfram Language code: RegionIntersection[Subscript[ℛ, 1], Subscript[ℛ, 2]]

三维空间中 CSGRegion 对象的交集：

Wolfram Language code: Subscript[ℛ, 1] = CSGRegion[{Cube[1.5], Cube[{0, 0, 1}, 0.8]}]; Subscript[ℛ, 2] = CSGRegion[{Ball[], Ball[{0, 0, 1}, 0.5]}];
Wolfram Language code: RegionIntersection[Subscript[ℛ, 1], Subscript[ℛ, 2]]

细分区域  (1)

二维空间中 SubdivisionRegion 对象的交集：

Wolfram Language code: Subscript[ℛ, 1] = SubdivisionRegion[Rectangle[]]; Subscript[ℛ, 2] = SubdivisionRegion[Rectangle[{1 / 2, 1 / 2}]];
Wolfram Language code: RegionIntersection[Subscript[ℛ, 1], Subscript[ℛ, 2]]

三维：

Wolfram Language code: Subscript[ℛ, 1] = SubdivisionRegion[Cube[]]; Subscript[ℛ, 2] = SubdivisionRegion[Cube[{1 / 4, 0, 0}]];
Wolfram Language code: RegionIntersection[Subscript[ℛ, 1], Subscript[ℛ, 2]]

应用  (3)

区域相交的部分：

Wolfram Language code: {Subscript[ℛ, 1], Subscript[ℛ, 2]} = {[image], [image]};
Wolfram Language code: Multicolumn[Table[RegionIntersection[Subscript[ℛ, 1], TransformedRegion[Subscript[ℛ, 2], TranslationTransform[{0, 0, t}]]], {t, 0, 1, 0.2}], 3, Appearance -> "Horizontal" ]

通过圆盘和半平面定义圆盘的一部分：

Wolfram Language code: segment = RegionIntersection[Disk[{0, 0}, 1], HalfPlane[{{0.5, 0}, {0.5, 1}}, {1, 0}]];
Wolfram Language code: Show[{Region[Style[Disk[], LightRed]], Region[segment]}]

定义新的基本区域 diskSegment，使用与 Disk 定义圆盘扇面所用的符号相同，使得 diskSegment[{x,y},r,{θ1,θ2}] 表示从 θ1 到 θ2 的部分圆盘. 用 Disk 和 HalfPlane 的 RegionIntersection 表示：

Wolfram Language code: diskSegment[{x0_, y0_}, r_, {θ1_, θ2_}] := Module[{ϕ1, ϕ2, ϕmid}, If[θ1 < θ2, {ϕ1, ϕ2} = {θ1, θ2}, {ϕ1, ϕ2} = {θ1, θ2 + 2π}]; ϕmid = (ϕ1 + ϕ2) / 2; RegionIntersection[Disk[{x0, y0}, r], HalfPlane[{{x0 + r Cos[ϕ1], y0 + r Sin[ϕ1]}, {x0 + r Cos[ϕ2], y0 + r Sin[ϕ2]}}, {Cos[ϕmid], Sin[ϕmid]}]] ];

这里运行 RegionQ 的对象，并且可以用于任何其他区域：

Wolfram Language code: ℛ = diskSegment[{0, 0}, 1, {π / 4, -π / 2}]
Wolfram Language code: RegionMember[ℛ, {x, y}]//FullSimplify

可视化部分圆盘和圆盘：

Wolfram Language code: Show[{Region[Style[Disk[], LightRed]], Region[ℛ]}]

属性和关系  (4)

如果一个点 p 属于所有 regi，那么它属于 RegionIntersection[reg1,reg2,…]：

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

用 RegionMember 来测试其归属：

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

RegionIntersection 是区域的布尔组合 And：

Wolfram Language code: {Subscript[ℛ, 1], Subscript[ℛ, 2]} = {[image], [image]};
Wolfram Language code: RegionIntersection[Subscript[ℛ, 1], Subscript[ℛ, 2]] == BooleanRegion[And, {Subscript[ℛ, 1], Subscript[ℛ, 2]}]

交集的 RegionMeasure 遵循一个简单的公式：

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

从度量的和中减去 RegionUnion 的度量：

Wolfram Language code: RegionMeasure[Subscript[ℛ, 3]] == RegionMeasure[Subscript[ℛ, 1]] + RegionMeasure[Subscript[ℛ, 2]] - RegionMeasure[RegionUnion[Subscript[ℛ, 1], Subscript[ℛ, 2]]]

交集的 RegionDimension 最大为所有输入维数的最小值：

Wolfram Language code: Subscript[ℛ, 1] = Disk[{0, 0}, 1]; Subscript[ℛ, 2] = Circle[{1, 0}, 1];
Wolfram Language code: RegionDimension[RegionIntersection[Subscript[ℛ, 1], Subscript[ℛ, 2]]] == Min[RegionDimension /@ {Subscript[ℛ, 1], Subscript[ℛ, 2]}]

然而，它可以是更小的值：

Wolfram Language code: Subscript[ℛ, 1] = Disk[{0, 0}, 1]; Subscript[ℛ, 2] = Circle[{2, 0}, 1]; Subscript[ℛ, 3] = RegionIntersection[Subscript[ℛ, 1], Subscript[ℛ, 2]];
Wolfram Language code: Show[Region[Subscript[ℛ, 1]], Region[Subscript[ℛ, 2]], Graphics[{Red, Point[{1, 0}]}]]

这些区域只在一个点上重叠，因此交集的维数是0：

Wolfram Language code: RegionDimension[Subscript[ℛ, 3]]

可能存在的问题  (2)

只有当区域有同样的 RegionEmbeddingDimension 时，RegionIntersection 才有定义：

Wolfram Language code: Subscript[ℛ, 1] = Disk[]; Subscript[ℛ, 2] = Ball[{0, 0, 1}, 1];
Wolfram Language code: RegionIntersection[Subscript[ℛ, 1], Subscript[ℛ, 2]]

维度小于嵌入维度的元素将被忽略：

Wolfram Language code: Subscript[ℛ, 1] = MeshRegion[{{0, 0, 0}, {2, 0, 0}, {2, 2, 0}, {1, 1, 2}}, Tetrahedron[{1, 2, 3, 4}]]; Subscript[ℛ, 2] = MeshRegion[{{0, 0, 0}, {2, 2, 0}, {0, 2, 0}, {1, 1, 2}}, Tetrahedron[{1, 2, 3, 4}], PlotTheme -> "Web"];
Wolfram Language code: Show[{Subscript[ℛ, 1], Subscript[ℛ, 2]}]
Wolfram Language code: RegionIntersection[Subscript[ℛ, 1], Subscript[ℛ, 2]]

显示消息：

Wolfram Language code: On[BooleanRegion::drc]
Wolfram Language code: RegionIntersection[Subscript[ℛ, 1], Subscript[ℛ, 2]]

巧妙范例  (1)

两个螺旋多边形的交集：

Wolfram Language code: RegionIntersection[[image], [image]]
Wolfram Research (2014)，RegionIntersection，Wolfram 语言函数，https://reference.wolfram.com/language/ref/RegionIntersection.html （更新于 2017 年）.

文本

Wolfram Research (2014)，RegionIntersection，Wolfram 语言函数，https://reference.wolfram.com/language/ref/RegionIntersection.html （更新于 2017 年）.

CMS

Wolfram 语言. 2014. "RegionIntersection." Wolfram 语言与系统参考资料中心. Wolfram Research. 最新版本 2017. https://reference.wolfram.com/language/ref/RegionIntersection.html.

APA

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

BibTeX

@misc{reference.wolfram_2026_regionintersection, author="Wolfram Research", title="{RegionIntersection}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/RegionIntersection.html}", note=[Accessed: 04-November-2026]}

BibLaTeX

@online{reference.wolfram_2026_regionintersection, organization={Wolfram Research}, title={RegionIntersection}, year={2017}, url={https://reference.wolfram.com/language/ref/RegionIntersection.html}, note=[Accessed: 04-November-2026]}