returns True if the regions reg1 and reg2 are disjoint.


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
    GenerateConditionsFalsewhether to generate conditions on parameters


open allclose all

Basic Examples  (2)

Test whether two regions are disjoint:

Visualize them:

Generate conditions for which regions are disjoint:

Scope  (17)

Basic Uses  (5)

Show two regions are disjoint:

Visualize them:

Show two regions intersect:

Find conditions that make regions disjoint:

Show multiple regions are pairwise disjoint:

Show multiple regions are not pairwise disjoint:

Basic Regions  (4)

Regions in including Line and Interval:




Regions in including Point:



Disk and Ellipsoid:

Rectangle and RegularPolygon:

Regions in including Point:



Cuboid and Hexahedron:

Ball and Ellipsoid:

Tetrahedron and Simplex:

Regions in including Cuboid and Parallelepiped in :

Ellipsoid and Ball in :

Formula Regions  (4)

Implicit regions:

Parametric regions:

Compare two formula regions:

Nonconstant formula regions:

Mesh Regions  (3)

Compare MeshRegion in :

In :

In :

Compare BoundaryMeshRegion in :

In :

In :

Compare MeshRegion with BoundaryMeshRegion in :

In :

Derived Regions  (1)

Compare BooleanRegion:

Options  (2)

Assumptions  (1)

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

Find only the positive radii:

GenerateConditions  (1)

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

Show the conditions for which the result is valid:

Explicitly allow for degenerate cases:

Applications  (6)

Estimate by simulating Buffon's needle problem:

Create randomly orientated line segments of length :

Select line segments that overlap the grid of lines:

Visualize overlapping line segments (red):

Estimation of :

Detect collisions between an object and a collection of walls:

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

Find all countries that share a border with France:

The polygons of each country:

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

Verify the results:

View these countries on a map:

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

Perform a random walk outside of a region:

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

Simulate a random walk from an initial point:

Visualize the walk:

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

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

Style this network atop a map of the United States:

The largest disconnect is between Maine and the westernmost states:

Find and highlight a path from Maine to California:

Properties & Relations  (4)

A region and its complement are always disjoint:

Disjoint regions share no common point:

For nonempty regions, RegionEqual and RegionWithin return False when RegionDisjoint returns True:

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

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

Use Reduce to find where two regions overlap:

Neat Examples  (1)

Create a scene of randomly placed, disjoint balls:

In 2D:

In 3D:

Wolfram Research (2017), RegionDisjoint, Wolfram Language function,


Wolfram Research (2017), RegionDisjoint, Wolfram Language function,


@misc{reference.wolfram_2020_regiondisjoint, author="Wolfram Research", title="{RegionDisjoint}", year="2017", howpublished="\url{}", note=[Accessed: 10-April-2021 ]}


@online{reference.wolfram_2020_regiondisjoint, organization={Wolfram Research}, title={RegionDisjoint}, year={2017}, url={}, note=[Accessed: 10-April-2021 ]}


Wolfram Language. 2017. "RegionDisjoint." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2017). RegionDisjoint. Wolfram Language & System Documentation Center. Retrieved from