RegionIntersection
✖
RegionIntersection
Details and Options

- A point p belongs to RegionIntersection[reg1,reg2,…] if it belongs to all regi.
- RegionIntersection takes the same options as Region.

Examples
open allclose allBasic Examples (2)Summary of the most common use cases

https://wolfram.com/xid/0g7ijzrxagi-bwhjpw

https://wolfram.com/xid/0g7ijzrxagi-kgyvnb

Intersection of two MeshRegion objects:

https://wolfram.com/xid/0g7ijzrxagi-i47t61

Scope (13)Survey of the scope of standard use cases
Special Regions (7)
For some regions, intersection is computed explicitly:

https://wolfram.com/xid/0g7ijzrxagi-d1ggqp

https://wolfram.com/xid/0g7ijzrxagi-1bowl


https://wolfram.com/xid/0g7ijzrxagi-bmt7by

An intersection of an infinite line and a ball:

https://wolfram.com/xid/0g7ijzrxagi-dsd1ix

https://wolfram.com/xid/0g7ijzrxagi-bxmu06


https://wolfram.com/xid/0g7ijzrxagi-bq9s2l

An intersection of Line regions:

https://wolfram.com/xid/0g7ijzrxagi-cl5o55


https://wolfram.com/xid/0g7ijzrxagi-kq6iqq

An intersection of Polygon regions:

https://wolfram.com/xid/0g7ijzrxagi-h9631g

https://wolfram.com/xid/0g7ijzrxagi-i01vvp

https://wolfram.com/xid/0g7ijzrxagi-unxw

An intersection of two Disk regions:

https://wolfram.com/xid/0g7ijzrxagi-hl5gpm

https://wolfram.com/xid/0g7ijzrxagi-eewuc0

An intersection of a cuboid a cone:

https://wolfram.com/xid/0g7ijzrxagi-59fdh

https://wolfram.com/xid/0g7ijzrxagi-v9k9l6

An intersection of regions with different RegionDimension:

https://wolfram.com/xid/0g7ijzrxagi-dq7yx6


https://wolfram.com/xid/0g7ijzrxagi-of250

Formula Regions (2)
An intersection of ImplicitRegion objects is an ImplicitRegion:

https://wolfram.com/xid/0g7ijzrxagi-crtuk1

https://wolfram.com/xid/0g7ijzrxagi-i34nki


https://wolfram.com/xid/0g7ijzrxagi-gc5987

https://wolfram.com/xid/0g7ijzrxagi-bqytv


https://wolfram.com/xid/0g7ijzrxagi-daco4v

https://wolfram.com/xid/0g7ijzrxagi-sa2or


https://wolfram.com/xid/0g7ijzrxagi-bdn04e

https://wolfram.com/xid/0g7ijzrxagi-iw2h19

An intersection of ParametricRegion objects:

https://wolfram.com/xid/0g7ijzrxagi-d62uqc

https://wolfram.com/xid/0g7ijzrxagi-d7q2hj

https://wolfram.com/xid/0g7ijzrxagi-hh12

Mesh Regions (2)
An intersection of BoundaryMeshRegion objects is a BoundaryMeshRegion:

https://wolfram.com/xid/0g7ijzrxagi-k2if8


https://wolfram.com/xid/0g7ijzrxagi-ehdu8m


https://wolfram.com/xid/0g7ijzrxagi-d1hr7s


https://wolfram.com/xid/0g7ijzrxagi-4rlzu


https://wolfram.com/xid/0g7ijzrxagi-e0i5c3


https://wolfram.com/xid/0g7ijzrxagi-co8192

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

https://wolfram.com/xid/0g7ijzrxagi-bnl5n5


https://wolfram.com/xid/0g7ijzrxagi-cadjtl


https://wolfram.com/xid/0g7ijzrxagi-b2vn5v


https://wolfram.com/xid/0g7ijzrxagi-b5oxqo


https://wolfram.com/xid/0g7ijzrxagi-im6jks


https://wolfram.com/xid/0g7ijzrxagi-qu8qa

Derived Regions (2)
An intersection of BooleanRegion objects:

https://wolfram.com/xid/0g7ijzrxagi-fzcaz8

https://wolfram.com/xid/0g7ijzrxagi-i5hp5n

https://wolfram.com/xid/0g7ijzrxagi-b3gvca

An intersection of TransformedRegion objects:

https://wolfram.com/xid/0g7ijzrxagi-fdpyhr

https://wolfram.com/xid/0g7ijzrxagi-dp6ddh

https://wolfram.com/xid/0g7ijzrxagi-c2ee5

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

https://wolfram.com/xid/0g7ijzrxagi-ysokjf

https://wolfram.com/xid/0g7ijzrxagi-7tcgzk

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

https://wolfram.com/xid/0g7ijzrxagi-ndz2ni

https://wolfram.com/xid/0g7ijzrxagi-pqnqkk

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:

https://wolfram.com/xid/0g7ijzrxagi-8t48i
This evaluates an object that is RegionQ and can be used as any other region:

https://wolfram.com/xid/0g7ijzrxagi-o87dci


https://wolfram.com/xid/0g7ijzrxagi-fjl35

Visualize the disk segment together with the disk:

https://wolfram.com/xid/0g7ijzrxagi-52yn5u

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:

https://wolfram.com/xid/0g7ijzrxagi-c2sfwz
Use RegionMember to test membership:

https://wolfram.com/xid/0g7ijzrxagi-i0bat4

https://wolfram.com/xid/0g7ijzrxagi-gq7lk1

RegionIntersection is a Boolean combination And of regions:

https://wolfram.com/xid/0g7ijzrxagi-fgv0t7

https://wolfram.com/xid/0g7ijzrxagi-ehtogm

The RegionMeasure of an intersection obeys a simple formula:

https://wolfram.com/xid/0g7ijzrxagi-hjvs0m
Subtract the measure of the RegionUnion from the sum of the measures:

https://wolfram.com/xid/0g7ijzrxagi-f0wukb

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

https://wolfram.com/xid/0g7ijzrxagi-qehjx

https://wolfram.com/xid/0g7ijzrxagi-ljzl3


https://wolfram.com/xid/0g7ijzrxagi-eww4bw

https://wolfram.com/xid/0g7ijzrxagi-dgrigy

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

https://wolfram.com/xid/0g7ijzrxagi-mgykir

Possible Issues (2)Common pitfalls and unexpected behavior
RegionIntersection is defined only for regions with the same RegionEmbeddingDimension:

https://wolfram.com/xid/0g7ijzrxagi-dqoj2

https://wolfram.com/xid/0g7ijzrxagi-hnwr2q


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

https://wolfram.com/xid/0g7ijzrxagi-p6rhkn

https://wolfram.com/xid/0g7ijzrxagi-s5yjch


https://wolfram.com/xid/0g7ijzrxagi-n45d6y


https://wolfram.com/xid/0g7ijzrxagi-oqb0ib

https://wolfram.com/xid/0g7ijzrxagi-7gcaf2


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
]}
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
]}