# 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 \$Assumptions assumptions to make about parameters GenerateConditions False whether to generate conditions on parameters

# Examples

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:

Ball:

Regions in including Point:

Line:

Disk and Ellipsoid:

Regions in including Point:

Line:

Cuboid and Hexahedron:

Ball and Ellipsoid:

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:

### 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, https://reference.wolfram.com/language/ref/RegionDisjoint.html.

#### Text

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.

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_regiondisjoint, author="Wolfram Research", title="{RegionDisjoint}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/RegionDisjoint.html}", note=[Accessed: 15-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_regiondisjoint, organization={Wolfram Research}, title={RegionDisjoint}, year={2017}, url={https://reference.wolfram.com/language/ref/RegionDisjoint.html}, note=[Accessed: 15-August-2024 ]}