WOLFRAM

Wolfram 语言与系统参考资料中心

BooleanRegion[bfunc,{reg1,reg2,…}]

表示区域 reg1, reg2, … 的布尔组合 bfunc.

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  
衍生区域  
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
按以下格式引用：

BooleanRegion

BooleanRegion[bfunc,{reg1,reg2,…}]

表示区域 reg1, reg2, … 的布尔组合 bfunc.

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)

两个圆盘的布尔 Xor：

Wolfram Language code: BooleanRegion[Xor, {Disk[{-1 / 3, 0}, 1], Disk[{1 / 3, 0}, 1]}];

对其进行可视化：

Wolfram Language code: Region[%]

应用于 MeshRegion 对象的布尔函数：

Wolfram Language code: BooleanRegion[¬#2∧#1&, {[image], [image]}]

范围  (11)

特殊区域  (5)

Line 区域的布尔 Or：

Wolfram Language code: ℛ = BooleanRegion[Or, {Line[{{1}, {2}}], Line[{{3}, {4}}], Line[{{5}, {6}}]}];

对其进行可视化：

Wolfram Language code: Region[ℛ]

应用于 Polygon 区域的 BooleanCountingFunction：

Wolfram Language code: pts = {{-5, 0}, {1, -2}, {-1, 0}, {1, 2}}; {p1, p2, p3, p4} = Polygon /@ {pts, -pts, Reverse /@ pts, -Reverse /@ pts};
Wolfram Language code: ℛ = BooleanRegion[BooleanCountingFunction[{1, 2}, 4], {p1, p2, p3, p4}];

计算它的 Area：

Wolfram Language code: Area[ℛ]

两个 Disk 区域的布尔 Xor：

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

对其进行可视化：

Wolfram Language code: Region[ℛ]

两个 Cuboid 区域的布尔 Or：

Wolfram Language code: ℛ = BooleanRegion[Or, {Cuboid[], Cuboid[{0.5, 0.5, 0.5}]}];

对其进行可视化：

Wolfram Language code: Region[ℛ]

具有不同 RegionDimension 的区域的布尔 And：

Wolfram Language code: ℛ = BooleanRegion[And, {Disk[{0, 0}, 1], Circle[{0, 1}, 1]}];

对其进行可视化：

Wolfram Language code: Region[ℛ]

公式区域  (2)

ImplicitRegion 对象的布尔 Xor 是 ImplicitRegion：

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

二维：

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: BooleanRegion[Xor, {Subscript[ℛ, 1], Subscript[ℛ, 2]}]

三维：

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: BooleanRegion[Xor, {Subscript[ℛ, 1], Subscript[ℛ, 2]}]

n 维：

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: BooleanRegion[Xor, {Subscript[ℛ, 1], Subscript[ℛ, 2]}]

应用于 ParametricRegion 对象的布尔函数：

Wolfram Language code: Subscript[ℛ, 1] = ParametricRegion[{u, v, w}, {{u, 0, 2}, {v, 0, 2}, {w, 0, 2}}]; Subscript[ℛ, 2] = ParametricRegion[{u - 1, v + 1, w - 1}, {{u, 0, 2}, {v, 0, 2}, {w, 0, 2}}];
Wolfram Language code: ℛ = BooleanRegion[¬#1∧#2&, {Subscript[ℛ, 1], Subscript[ℛ, 2]}];

将其可视化：

Wolfram Language code: Region[ℛ]

网格区域  (2)

BoundaryMeshRegion 对象的布尔函数是 BoundaryMeshRegion：

Wolfram Language code: BooleanRegion[¬#2∧#1&, {[image], [image]}]
Wolfram Language code: BoundedRegionQ[%]

二维：

Wolfram Language code: BooleanRegion[¬#2∧#1&, {[image], [image]}]
Wolfram Language code: BoundedRegionQ[%]

三维：

Wolfram Language code: BooleanRegion[¬#2∧#1&, {[image], [image]}]
Wolfram Language code: BoundedRegionQ[%]

全维 MeshRegion 对象的布尔函数是 MeshRegion：

Wolfram Language code: BooleanRegion[¬#2∧#1&, {[image], [image]}]
Wolfram Language code: MeshRegionQ[%]

二维：

Wolfram Language code: BooleanRegion[¬#2∧#1&, {[image], [image]}]
Wolfram Language code: MeshRegionQ[%]

三维：

Wolfram Language code: BooleanRegion[¬#2∧#1&, {[image], [image]}]
Wolfram Language code: MeshRegionQ[%]

衍生区域  (2)

BooleanRegion 对象的布尔函数：

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 = BooleanRegion[¬#2∧#1&, {Subscript[ℛ, 1], Subscript[ℛ, 2]}];

将它可视化：

Wolfram Language code: Region[ℛ]

TransformedRegion 对象的布尔 Or：

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 = BooleanRegion[Or, {Subscript[ℛ, 1], Subscript[ℛ, 2]}];

将它可视化：

Wolfram Language code: Region[ℛ]

应用  (1)

区域上的布尔运算：

Wolfram Language code: {a, b} = {[image], [image]};
Wolfram Language code: Multicolumn[Table[BooleanRegion[op, {a, b}, PlotLabel -> op, BaseStyle -> Opacity[0.7]], {op, {Or, And, ¬#2∧#1&, #1⊻#2&}}], 2]

属性和关系  (2)

RegionUnion 是区域的布尔组合 Or：

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

RegionIntersection：

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

RegionDifference：

Wolfram Language code: RegionDifference[Subscript[ℛ, 1], Subscript[ℛ, 2]] == BooleanRegion[¬#2∧#1&, {Subscript[ℛ, 1], Subscript[ℛ, 2]}]

RegionSymmetricDifference：

Wolfram Language code: RegionSymmetricDifference[Subscript[ℛ, 1], Subscript[ℛ, 2]] == BooleanRegion[Xor, {Subscript[ℛ, 1], Subscript[ℛ, 2]}]

布尔 And 的 RegionMeasure 遵守一个简单的公式：

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

将 RegionUnion 的度量从度量综合中减去：

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

可能存在的问题  (3)

BooleanRegion 仅对具有相同 RegionEmbeddingDimension 的区域定义：

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

维度小于嵌入维度的分量可能会被省略：

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: BooleanRegion[And, {Subscript[ℛ, 1], Subscript[ℛ, 2]}]

打开消息：

Wolfram Language code: On[BooleanRegion::drc]
Wolfram Language code: BooleanRegion[And, {Subscript[ℛ, 1], Subscript[ℛ, 2]}]

BooleanRegion 可能包括重叠的低维分量：

Wolfram Language code: Subscript[ℛ, 1] = Sphere[]; Subscript[ℛ, 2] = Ball[{1 / 2, 0, 0}];
Wolfram Language code: Graphics3D[{Yellow, Subscript[ℛ, 1], Green, Opacity@.5, Subscript[ℛ, 2]}]
Wolfram Language code: Subscript[ℛ, 3] = BooleanRegion[Or, {DiscretizeGraphics[Subscript[ℛ, 1]], BoundaryDiscretizeGraphics[Subscript[ℛ, 2]]}]

连通网格分量：

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

巧妙范例  (1)

两个螺旋多边形的布尔 Xor：

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

文本

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

CMS

Wolfram 语言. 2014. "BooleanRegion." 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/BooleanRegion.html.

APA

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

BibTeX

@misc{reference.wolfram_2026_booleanregion, author="Wolfram Research", title="{BooleanRegion}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/BooleanRegion.html}", note=[Accessed: 09-August-2026]}

BibLaTeX

@online{reference.wolfram_2026_booleanregion, organization={Wolfram Research}, title={BooleanRegion}, year={2017}, url={https://reference.wolfram.com/language/ref/BooleanRegion.html}, note=[Accessed: 09-August-2026]}