WOLFRAM

RegionIntersection[reg1,reg2,]

gives the intersection of the regions reg1, reg2, .

Details and Options

Examples

open allclose all

Basic Examples  (2)Summary of the most common use cases

Intersection of two disks:

Visualize it:

Out[2]=2

Intersection of two MeshRegion objects:

Out[1]=1

Scope  (13)Survey of the scope of standard use cases

Special Regions  (7)

For some regions, intersection is computed explicitly:

Out[3]=3

Visualize the intersection:

Out[4]=4

An intersection of an infinite line and a ball:

Out[2]=2

Visualize the intersection:

Out[3]=3

An intersection of Line regions:

Out[1]=1

Visualize it:

Out[2]=2

An intersection of Polygon regions:

Visualize it:

Out[3]=3

An intersection of two Disk regions:

Visualize it:

Out[2]=2

An intersection of a cuboid a cone:

Visualize it:

Out[2]=2

An intersection of regions with different RegionDimension:

Out[1]=1

Visualize it:

Out[2]=2

Formula Regions  (2)

An intersection of ImplicitRegion objects is an ImplicitRegion:

Out[3]=3

2D:

Out[6]=6

3D:

Out[9]=9

nD:

Out[12]=12

An intersection of ParametricRegion objects:

Visualize it:

Out[3]=3

Mesh Regions  (2)

An intersection of BoundaryMeshRegion objects is a BoundaryMeshRegion:

Out[2]=2
Out[3]=3

2D:

Out[6]=6
Out[8]=8

3D:

Out[9]=9
Out[12]=12

An intersection of full-dimensional MeshRegion objects is a MeshRegion:

Out[1]=1
Out[2]=2

2D:

Out[3]=3
Out[4]=4

3D:

Out[5]=5
Out[6]=6

Derived Regions  (2)

An intersection of BooleanRegion objects:

Visualize it:

Out[4]=4

An intersection of TransformedRegion objects:

Visualize it:

Out[3]=3

Applications  (3)Sample problems that can be solved with this function

Intersection of regions:

Out[6]=6

Define a disk segment as an intersection of a disk and a half-plane:

Out[2]=2

Define a new basic region diskSegment that uses the same notation as Disk does for disk sectors, so that diskSegment[{x,y},r,{θ1,θ2}] represents the disk segment from θ1 to θ2. Do it by writing it as a RegionIntersection of a Disk and a HalfPlane:

This evaluates an object that is RegionQ and can be used as any other region:

Out[2]=2
Out[3]=3

Visualize the disk segment together with the disk:

Out[4]=4

Properties & Relations  (4)Properties of the function, and connections to other functions

A point p belongs to RegionIntersection[reg1,reg2,] if it belongs to all regi:

Use RegionMember to test membership:

Out[12]=12

RegionIntersection is a Boolean combination And of regions:

Out[2]=2

The RegionMeasure of an intersection obeys a simple formula:

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

Out[2]=2

The RegionDimension of an intersection is at most the minimum of all input dimensions:

Out[2]=2

It can be lower, however:

Out[4]=4

These regions overlap only at a point, so the dimension of the intersection is 0:

Out[5]=5

Possible Issues  (2)Common pitfalls and unexpected behavior

RegionIntersection is defined only for regions with the same RegionEmbeddingDimension:

Out[3]=3

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

Out[2]=2
Out[3]=3

Turn on a message:

Out[5]=5

Neat Examples  (1)Surprising or curious use cases

The intersection of two spiral polygons:

Out[1]=1
Wolfram Research (2014), RegionIntersection, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionIntersection.html (updated 2017).
Wolfram Research (2014), RegionIntersection, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionIntersection.html (updated 2017).

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_regionintersection, author="Wolfram Research", title="{RegionIntersection}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/RegionIntersection.html}", note=[Accessed: 19-June-2025 ]}

@misc{reference.wolfram_2025_regionintersection, author="Wolfram Research", title="{RegionIntersection}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/RegionIntersection.html}", note=[Accessed: 19-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_regionintersection, organization={Wolfram Research}, title={RegionIntersection}, year={2017}, url={https://reference.wolfram.com/language/ref/RegionIntersection.html}, note=[Accessed: 19-June-2025 ]}

@online{reference.wolfram_2025_regionintersection, organization={Wolfram Research}, title={RegionIntersection}, year={2017}, url={https://reference.wolfram.com/language/ref/RegionIntersection.html}, note=[Accessed: 19-June-2025 ]}