# RegionIntersection

RegionIntersection[reg1,reg2,]

gives the intersection of the regions reg1, reg2, .

# 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

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## Basic Examples(2)

Intersection of two disks:

Visualize it:

Intersection of two MeshRegion objects:

## Scope(13)

### Special Regions(7)

For some regions, intersection is computed explicitly:

Visualize the intersection:

An intersection of an infinite line and a ball:

Visualize the intersection:

An intersection of Line regions:

Visualize it:

An intersection of Polygon regions:

Visualize it:

An intersection of two Disk regions:

Visualize it:

An intersection of a cuboid a cone:

Visualize it:

An intersection of regions with different RegionDimension:

Visualize it:

### Formula Regions(2)

An intersection of ImplicitRegion objects is an ImplicitRegion:

2D:

3D:

nD:

An intersection of ParametricRegion objects:

Visualize it:

### Mesh Regions(2)

An intersection of BoundaryMeshRegion objects is a BoundaryMeshRegion:

2D:

3D:

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

2D:

3D:

### Derived Regions(2)

An intersection of BooleanRegion objects:

Visualize it:

An intersection of TransformedRegion objects:

Visualize it:

## Applications(3)

Intersection of regions:

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

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:

Visualize the disk segment together with the disk:

## Properties & Relations(4)

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

Use RegionMember to test membership:

RegionIntersection is a Boolean combination And of regions:

The RegionMeasure of an intersection obeys a simple formula:

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

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

It can be lower, however:

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

## Possible Issues(2)

RegionIntersection is defined only for regions with the same RegionEmbeddingDimension: Components of dimension less than the embedding dimension may be omitted:

Turn on a message: ## Neat Examples(1)

The intersection of two spiral polygons: