Intersection of two disks:

Visualize it:

Intersection of two MeshRegion objects:

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:

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

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:

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:

The intersection of two spiral polygons: