# BooleanRegion

BooleanRegion[bfunc,{reg1,reg2,}]

represents the Boolean combination bfunc of regions reg1, reg2, .

# Details and Options

• A point p belongs to BooleanRegion[bfunc,{reg1,reg2,}] if bfunc[preg1,preg2,] is True.
• For BoundaryMeshRegion regi, BooleanRegion represents the smallest BoundaryMeshRegion that contains the Boolean combination of regions regi.
• For MeshRegion regi, BooleanRegion gives the smallest MeshRegion that contains the Boolean combination of regions regi.
• The following functions are equivalent:
•  RegionIntersection[reg1,reg2,…] BooleanRegion[And, {reg1,reg2,…}] RegionUnion[reg1,reg2,…] BooleanRegion[Or, {reg1,reg2,…}] RegionDifference[reg1,reg2] BooleanRegion[And[#1,Not[#2]]&, {reg1,reg2}] RegionSymmetricDifference[reg1,…] BooleanRegion[Xor, {reg1,…}]
• BooleanRegion takes the same options as Region.

# Examples

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## Basic Examples(2)

The Boolean Xor of two disks:

Visualize it:

A Boolean function applied to MeshRegion objects:

## Scope(11)

### Special Regions(5)

A Boolean Or of Line regions:

Visualize it:

A BooleanCountingFunction applied to Polygon regions:

Compute its Area:

A Boolean Xor of two Disk regions:

Visualize it:

A Boolean Or of two Cuboid regions:

Visualize it:

A Boolean And of regions with different RegionDimension:

Visualize it:

### Formula Regions(2)

A Boolean Xor of ImplicitRegion objects is an ImplicitRegion:

2D:

3D:

nD:

A Boolean function applied to ParametricRegion objects:

Visualize it:

### Mesh Regions(2)

A Boolean function of BoundaryMeshRegion objects is a BoundaryMeshRegion:

2D:

3D:

A Boolean function of full dimensional MeshRegion objects is a MeshRegion:

2D:

3D:

### Derived Regions(2)

A Boolean function of BooleanRegion objects:

Visualize it:

A Boolean Or of TransformedRegion objects:

Visualize it:

## Applications(1)

Boolean operations over regions:

## Properties & Relations(2)

RegionUnion is a Boolean combination Or of regions:

The RegionMeasure of a Boolean And obeys a simple formula:

Subtract the measure of the RegionUnion from the sum of the measures:

## Possible Issues(3)

BooleanRegion is defined only for regions with the same RegionEmbeddingDimension:

Components of dimension less than the embedding dimension may be omitted:

Turn on a message:

BooleanRegion may include overlapping lower-dimensional components:

The connected mesh components:

## Neat Examples(1)

The Boolean Xor of two spiral polygons:

Wolfram Research (2014), BooleanRegion, Wolfram Language function, https://reference.wolfram.com/language/ref/BooleanRegion.html (updated 2017).

#### Text

Wolfram Research (2014), BooleanRegion, Wolfram Language function, https://reference.wolfram.com/language/ref/BooleanRegion.html (updated 2017).

#### CMS

Wolfram Language. 2014. "BooleanRegion." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/BooleanRegion.html.

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_booleanregion, author="Wolfram Research", title="{BooleanRegion}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/BooleanRegion.html}", note=[Accessed: 13-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_booleanregion, organization={Wolfram Research}, title={BooleanRegion}, year={2017}, url={https://reference.wolfram.com/language/ref/BooleanRegion.html}, note=[Accessed: 13-August-2024 ]}