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RegionDisjoint
✖
RegionDisjoint

✖
RegionDisjoint[reg1,reg2]

returns True if the regions reg1 and reg2 are disjoint.

✖
RegionDisjoint[reg1,reg2,reg3,…]

returns True if the regions reg1, reg2, reg3, … are pairwise disjoint.

Details and Options

  • The regions reg1 and reg2 are disjoint if there are no points that belong to both reg1 and reg2.
  • If all regi are parameter-free regions, i.e. ConstantRegionQ[regi] is True, the regions are point sets, and typically True or False is returned.
  • If some regi depend on parameters, i.e. ConstantRegionQ[regi] is False, then regi represents a family of regions, and RegionDisjoint will attempt to compute conditions on parameters such that the regions are disjoint.
  • The following options can be given:
  • Assumptions $Assumptionsassumptions to make about parameters
    GenerateConditions Falsewhether to generate conditions on parameters

Examples

open allclose all

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

Test whether two regions are disjoint:

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-bwhjpw
In[3]:=3
✖
https://wolfram.com/xid/0cps1hws2eafue-2rib4
Out[3]=3

Visualize them:

In[4]:=4
✖
https://wolfram.com/xid/0cps1hws2eafue-ch8cm5
Out[4]=4

Generate conditions for which regions are disjoint:

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-bjcd6
In[2]:=2
✖
https://wolfram.com/xid/0cps1hws2eafue-bgxf3h
Out[2]=2

Scope  (17)Survey of the scope of standard use cases

Basic Uses  (5)

Show two regions are disjoint:

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-w6qsve
In[3]:=3
✖
https://wolfram.com/xid/0cps1hws2eafue-02ahd7
Out[3]=3

Visualize them:

In[4]:=4
✖
https://wolfram.com/xid/0cps1hws2eafue-30xgaw
Out[4]=4

Show two regions intersect:

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-rjig01
In[2]:=2
✖
https://wolfram.com/xid/0cps1hws2eafue-h0qhxp
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cps1hws2eafue-q6upsg
Out[3]=3

Find conditions that make regions disjoint:

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-tnzyl9
In[2]:=2
✖
https://wolfram.com/xid/0cps1hws2eafue-w8dsx6
Out[2]=2

Show multiple regions are pairwise disjoint:

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-skw31u
In[2]:=2
✖
https://wolfram.com/xid/0cps1hws2eafue-7m7agj
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cps1hws2eafue-4c84kt
Out[3]=3

Show multiple regions are not pairwise disjoint:

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-4yd22o
In[2]:=2
✖
https://wolfram.com/xid/0cps1hws2eafue-2gdpbp
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cps1hws2eafue-79mtfs
Out[3]=3

Basic Regions  (4)

Regions in including Line and Interval:

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-6v3c9q
Out[1]=1

Point:

In[2]:=2
✖
https://wolfram.com/xid/0cps1hws2eafue-uiw53u
Out[2]=2

Ball:

In[3]:=3
✖
https://wolfram.com/xid/0cps1hws2eafue-0a693w
Out[3]=3

InfiniteLine:

In[4]:=4
✖
https://wolfram.com/xid/0cps1hws2eafue-586b1i
Out[4]=4

Regions in including Point:

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-i2fp1e
In[2]:=2
✖
https://wolfram.com/xid/0cps1hws2eafue-fcyt2v
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cps1hws2eafue-rvnbj1
Out[3]=3

Line:

In[4]:=4
✖
https://wolfram.com/xid/0cps1hws2eafue-gnymc0
In[5]:=5
✖
https://wolfram.com/xid/0cps1hws2eafue-fw05sb
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0cps1hws2eafue-vkk3wa
Out[6]=6

Polygon:

In[7]:=7
✖
https://wolfram.com/xid/0cps1hws2eafue-45f5fx
In[8]:=8
✖
https://wolfram.com/xid/0cps1hws2eafue-if4go9
Out[8]=8
In[9]:=9
✖
https://wolfram.com/xid/0cps1hws2eafue-df44np
Out[9]=9

Disk and Ellipsoid:

In[10]:=10
✖
https://wolfram.com/xid/0cps1hws2eafue-0aeidt
In[11]:=11
✖
https://wolfram.com/xid/0cps1hws2eafue-te9loc
Out[11]=11

Rectangle and RegularPolygon:

In[12]:=12
✖
https://wolfram.com/xid/0cps1hws2eafue-3el84j
Out[12]=12

Regions in including Point:

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-0q3l4s
In[2]:=2
✖
https://wolfram.com/xid/0cps1hws2eafue-bc0xud
Out[2]=2

Line:

In[3]:=3
✖
https://wolfram.com/xid/0cps1hws2eafue-43jbwg
In[4]:=4
✖
https://wolfram.com/xid/0cps1hws2eafue-pqbwax
Out[4]=4

Polygon:

In[5]:=5
✖
https://wolfram.com/xid/0cps1hws2eafue-t559ku
In[6]:=6
✖
https://wolfram.com/xid/0cps1hws2eafue-uaqco4
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0cps1hws2eafue-66pdr2
Out[7]=7

Cuboid and Hexahedron:

In[8]:=8
✖
https://wolfram.com/xid/0cps1hws2eafue-35c0r8
In[9]:=9
✖
https://wolfram.com/xid/0cps1hws2eafue-g2ry70
Out[9]=9

Ball and Ellipsoid:

In[10]:=10
✖
https://wolfram.com/xid/0cps1hws2eafue-g1mnn1
In[11]:=11
✖
https://wolfram.com/xid/0cps1hws2eafue-3j14kp
Out[11]=11

Tetrahedron and Simplex:

In[12]:=12
✖
https://wolfram.com/xid/0cps1hws2eafue-ilyuwb
In[13]:=13
✖
https://wolfram.com/xid/0cps1hws2eafue-o1w6u8
Out[13]=13

Regions in including Cuboid and Parallelepiped in :

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-02h4ee
In[2]:=2
✖
https://wolfram.com/xid/0cps1hws2eafue-29ru1s
Out[2]=2

Ellipsoid and Ball in :

In[3]:=3
✖
https://wolfram.com/xid/0cps1hws2eafue-ufst9o
In[4]:=4
✖
https://wolfram.com/xid/0cps1hws2eafue-8t7at1
Out[4]=4

Formula Regions  (4)

Implicit regions:

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-yhhbit
In[3]:=3
✖
https://wolfram.com/xid/0cps1hws2eafue-uj1qf1
Out[3]=3

Parametric regions:

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-hukmm5
In[2]:=2
✖
https://wolfram.com/xid/0cps1hws2eafue-jgs1z1
Out[2]=2

Compare two formula regions:

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-eie8c4
In[2]:=2
✖
https://wolfram.com/xid/0cps1hws2eafue-w59lx6
Out[2]=2

Nonconstant formula regions:

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-hic6g2
Out[1]=1

Mesh Regions  (3)

Compare MeshRegion in :

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-5yev98
In[3]:=3
✖
https://wolfram.com/xid/0cps1hws2eafue-ciss1i
Out[3]=3

In :

In[4]:=4
✖
https://wolfram.com/xid/0cps1hws2eafue-6p9mf1
In[6]:=6
✖
https://wolfram.com/xid/0cps1hws2eafue-9eeqmu
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0cps1hws2eafue-pmpsgg
Out[7]=7

In :

In[8]:=8
✖
https://wolfram.com/xid/0cps1hws2eafue-ron1r2
In[10]:=10
✖
https://wolfram.com/xid/0cps1hws2eafue-83vd07
Out[10]=10
In[11]:=11
✖
https://wolfram.com/xid/0cps1hws2eafue-lz7nun
Out[11]=11

Compare BoundaryMeshRegion in :

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-ki5zz3
In[2]:=2
✖
https://wolfram.com/xid/0cps1hws2eafue-bt2ml0
Out[2]=2

In :

In[3]:=3
✖
https://wolfram.com/xid/0cps1hws2eafue-5m2w0g
In[4]:=4
✖
https://wolfram.com/xid/0cps1hws2eafue-mmhs4a
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0cps1hws2eafue-baxiqs
Out[5]=5

In :

In[6]:=6
✖
https://wolfram.com/xid/0cps1hws2eafue-hyseqv
In[7]:=7
✖
https://wolfram.com/xid/0cps1hws2eafue-zty52f
Out[7]=7
In[8]:=8
✖
https://wolfram.com/xid/0cps1hws2eafue-ju4qkk
Out[8]=8

Compare MeshRegion with BoundaryMeshRegion in :

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-eq7jc5
In[2]:=2
✖
https://wolfram.com/xid/0cps1hws2eafue-n3t5rq
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cps1hws2eafue-w76nww
Out[3]=3

In :

In[4]:=4
✖
https://wolfram.com/xid/0cps1hws2eafue-t4fi8f
In[5]:=5
✖
https://wolfram.com/xid/0cps1hws2eafue-xmhkjc
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0cps1hws2eafue-vj7fl4
Out[6]=6

Derived Regions  (1)

Compare BooleanRegion:

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-pk317t
In[4]:=4
✖
https://wolfram.com/xid/0cps1hws2eafue-15qu2j
Out[4]=4

Options  (2)Common values & functionality for each option

Assumptions  (1)

Find all radii where a concentric disk and annulus are disjoint:

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-ml68g3
In[3]:=3
✖
https://wolfram.com/xid/0cps1hws2eafue-raue2j
Out[3]=3

Find only the positive radii:

In[4]:=4
✖
https://wolfram.com/xid/0cps1hws2eafue-beh8xv
Out[4]=4

GenerateConditions  (1)

Find when the unit disk is disjoint with an implicitly described annulus:

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-y7rzg1
In[3]:=3
✖
https://wolfram.com/xid/0cps1hws2eafue-2foycc
Out[3]=3

Show the conditions for which the result is valid:

In[4]:=4
✖
https://wolfram.com/xid/0cps1hws2eafue-ysjw49
Out[4]=4

Explicitly allow for degenerate cases:

In[5]:=5
✖
https://wolfram.com/xid/0cps1hws2eafue-cxiwq
Out[5]=5

Applications  (6)Sample problems that can be solved with this function

Estimate by simulating Buffon's needle problem:

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-2il9vg
In[2]:=2
✖
https://wolfram.com/xid/0cps1hws2eafue-wlrn4t

Create randomly orientated line segments of length :

In[3]:=3
✖
https://wolfram.com/xid/0cps1hws2eafue-ue779m

Select line segments that overlap the grid of lines:

In[5]:=5
✖
https://wolfram.com/xid/0cps1hws2eafue-jod63o

Visualize overlapping line segments (red):

In[6]:=6
✖
https://wolfram.com/xid/0cps1hws2eafue-evtvyh
Out[6]=6

Estimation of :

In[7]:=7
✖
https://wolfram.com/xid/0cps1hws2eafue-vmji9a
Out[7]=7

Detect collisions between an object and a collection of walls:

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-wb0wi4
In[2]:=2
✖
https://wolfram.com/xid/0cps1hws2eafue-xqccep

Color walls that do not collide with the cow green, and red otherwise:

In[3]:=3
✖
https://wolfram.com/xid/0cps1hws2eafue-81bjn2
In[4]:=4
✖
https://wolfram.com/xid/0cps1hws2eafue-ftu5x1
Out[4]=4

Find all countries that share a border with France:

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-w1gvne

The polygons of each country:

In[2]:=2
✖
https://wolfram.com/xid/0cps1hws2eafue-4p25ou

Select the countries whose polygons are not disjoint from France's polygon:

In[3]:=3
✖
https://wolfram.com/xid/0cps1hws2eafue-s9dghf
Out[3]=3

Verify the results:

In[4]:=4
✖
https://wolfram.com/xid/0cps1hws2eafue-x5zsoj
Out[4]=4

View these countries on a map:

In[5]:=5
✖
https://wolfram.com/xid/0cps1hws2eafue-tg8ujy
Out[5]=5

Find and visualize all positions where a unit rectangle is disjoint from an annulus:

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-59poq7
In[2]:=2
✖
https://wolfram.com/xid/0cps1hws2eafue-4rztjs
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cps1hws2eafue-xjaxyl
Out[3]=3

Perform a random walk outside of a region:

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-1eu4a3

Define a function to walk a point in a random direction, staying outside of a region:

In[2]:=2
✖
https://wolfram.com/xid/0cps1hws2eafue-2t78a9

Simulate a random walk from an initial point:

In[3]:=3
✖
https://wolfram.com/xid/0cps1hws2eafue-v10re3
In[4]:=4
✖
https://wolfram.com/xid/0cps1hws2eafue-pootx3

Visualize the walk:

In[5]:=5
✖
https://wolfram.com/xid/0cps1hws2eafue-2vflrc
Out[5]=5

Create a network that connects two US states if they share a border:

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-y5c135
In[2]:=2
✖
https://wolfram.com/xid/0cps1hws2eafue-b398l5

Two state's polygons share a border when RegionDisjoint returns False:

In[3]:=3
✖
https://wolfram.com/xid/0cps1hws2eafue-gl3e7l
In[4]:=4
✖
https://wolfram.com/xid/0cps1hws2eafue-wcikml
Out[4]=4

Style this network atop a map of the United States:

In[5]:=5
✖
https://wolfram.com/xid/0cps1hws2eafue-q69f2g
In[6]:=6
✖
https://wolfram.com/xid/0cps1hws2eafue-iinn7p
Out[6]=6

The largest disconnect is between Maine and the westernmost states:

In[7]:=7
✖
https://wolfram.com/xid/0cps1hws2eafue-f2e723
Out[7]=7

Find and highlight a path from Maine to California:

In[8]:=8
✖
https://wolfram.com/xid/0cps1hws2eafue-6ujoa4
In[9]:=9
✖
https://wolfram.com/xid/0cps1hws2eafue-wvubzf
Out[9]=9

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

A region and its complement are always disjoint:

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-qxxixn
In[2]:=2
✖
https://wolfram.com/xid/0cps1hws2eafue-zxr7hs
Out[2]=2

Disjoint regions share no common point:

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-ga16k4
In[2]:=2
✖
https://wolfram.com/xid/0cps1hws2eafue-pm8r3m
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cps1hws2eafue-szw6pm
Out[3]=3

For non‐empty regions, RegionEqual and RegionWithin return False when RegionDisjoint returns True:

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-8azms6
In[2]:=2
✖
https://wolfram.com/xid/0cps1hws2eafue-dp28x5
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cps1hws2eafue-28i8v5
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0cps1hws2eafue-2vh1ks
Out[4]=4

Use FindInstance to find points that lie in the intersection of two regions:

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-ecv99p
In[2]:=2
✖
https://wolfram.com/xid/0cps1hws2eafue-i0zv76
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cps1hws2eafue-vjwpmf
In[4]:=4
✖
https://wolfram.com/xid/0cps1hws2eafue-mccqg1
Out[4]=4

Use RandomPoint to find a uniform sampling of points that lie in the intersection of two regions:

In[5]:=5
✖
https://wolfram.com/xid/0cps1hws2eafue-gpkt0l
In[6]:=6
✖
https://wolfram.com/xid/0cps1hws2eafue-o0kf7b
Out[6]=6

Use Reduce to find where two regions overlap:

In[7]:=7
✖
https://wolfram.com/xid/0cps1hws2eafue-42jril
Out[7]=7
In[8]:=8
✖
https://wolfram.com/xid/0cps1hws2eafue-ly4r2m
Out[8]=8

Neat Examples  (1)Surprising or curious use cases

Create a scene of randomly placed, disjoint balls:

In[1]:=1
✖
https://wolfram.com/xid/0cps1hws2eafue-8pz9b5
In[2]:=2
✖
https://wolfram.com/xid/0cps1hws2eafue-r7mtxb
In[3]:=3
✖
https://wolfram.com/xid/0cps1hws2eafue-ks2qb0

In 2D:

In[4]:=4
✖
https://wolfram.com/xid/0cps1hws2eafue-fqtjp7
In[5]:=5
✖
https://wolfram.com/xid/0cps1hws2eafue-sf4m1g
Out[5]=5

In 3D:

In[6]:=6
✖
https://wolfram.com/xid/0cps1hws2eafue-4892t8
In[7]:=7
✖
https://wolfram.com/xid/0cps1hws2eafue-69b9tg
Out[7]=7
Wolfram Research (2017), RegionDisjoint, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionDisjoint.html.
✖
Wolfram Research (2017), RegionDisjoint, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionDisjoint.html.

Text

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

✖
Wolfram Research (2017), RegionDisjoint, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionDisjoint.html.

CMS

Wolfram Language. 2017. "RegionDisjoint." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RegionDisjoint.html.

✖
Wolfram Language. 2017. "RegionDisjoint." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RegionDisjoint.html.

APA

Wolfram Language. (2017). RegionDisjoint. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RegionDisjoint.html

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Wolfram Language. (2017). RegionDisjoint. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RegionDisjoint.html

BibTeX

@misc{reference.wolfram_2025_regiondisjoint, author="Wolfram Research", title="{RegionDisjoint}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/RegionDisjoint.html}", note=[Accessed: 04-April-2025 ]}

✖
@misc{reference.wolfram_2025_regiondisjoint, author="Wolfram Research", title="{RegionDisjoint}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/RegionDisjoint.html}", note=[Accessed: 04-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_regiondisjoint, organization={Wolfram Research}, title={RegionDisjoint}, year={2017}, url={https://reference.wolfram.com/language/ref/RegionDisjoint.html}, note=[Accessed: 04-April-2025 ]}

✖
@online{reference.wolfram_2025_regiondisjoint, organization={Wolfram Research}, title={RegionDisjoint}, year={2017}, url={https://reference.wolfram.com/language/ref/RegionDisjoint.html}, note=[Accessed: 04-April-2025 ]}