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# intersect -- computes the intersection of cones, and polyhedra

- Usage:
    `P = intersection L`
    `C = intersection(C1,C2)`
    `P = intersection(P1,P2)`
- Inputs:
  - L, a list, containing any of the inputs below
  - C1, a convex rational cone
  - C2, a convex rational cone
  - P1, a convex polyhedron
  - P2, a convex polyhedron
- Outputs:
  - P, a convex polyhedron
  - C, a convex rational cone

## Description

If two polyhedra or two cones are inserted, then the intersection of both arguments is computed if both arguments lie in the same ambient space. If both arguments are cones then the output is again a cone. Otherwise intersection returns a polyhedron.

If intersection is called for a list L, then the list may contain a combination of the following in any order.

- Cone
- Polyhedron

    i1 : C = hypercube 2

    o1 = C

    o1 : Polyhedron

    i2 : S = simplex 2

    o2 = S

    o2 : Polyhedron

    i3 : CS = intersection(C,S)

    o3 = CS

    o3 : Polyhedron

## Ways to use intersect:

- `intersect(Cone,Cone)`
- `intersect(Cone,Polyhedron)`
- `intersect(List)` -- see intersect -- compute an intersection
- `intersect(Polyhedron,Cone)`
- `intersect(Polyhedron,Polyhedron)`
- `intersect(Sequence)` -- see intersect -- compute an intersection
- `intersect(Ideal)` -- see intersect(Ideal,Ideal) -- compute an intersection of a sequence of ideals or modules
- intersect(Ideal,Ideal) -- compute an intersection of a sequence of ideals or modules
- `intersect(Module)` -- see intersect(Ideal,Ideal) -- compute an intersection of a sequence of ideals or modules
- `intersect(Module,Module)` -- see intersect(Ideal,Ideal) -- compute an intersection of a sequence of ideals or modules
- intersect(Matrix) -- Deprecated variant of {\tt coneFromHData}
- intersect(Matrix,Matrix) -- Deprecated variants of {\tt polyhedronFromHData} and {\tt coneFromHData}
- `intersect(Matrix,Matrix,Matrix,Matrix)` -- see intersect(Matrix,Matrix) -- Deprecated variants of {\tt polyhedronFromHData} and {\tt coneFromHData}
- intersect(RRi) -- intersection of input intervals
- `intersect(RRi,RRi)` -- see intersect(RRi) -- intersection of input intervals
- intersect(Set,Set) -- set intersection

## For the programmer

The object intersect is an associative binary method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Polyhedra/documentation/old_documentation.m2:558:0.