PolyhedralSets/PolyhedralCones/SimplicialDecomposition - Maple Help
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PolyhedralSets[PolyhedralCones]

  

SimplicialDecomposition

  

returns a simplicial decomposition of a polyhedral cone

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

SimplicialDecomposition(pc)

SimplicialDecomposition(pc,opts)

Parameters

pc

-

PolyhedralCone

opts

-

(optional) option which can be polyhedralcones or polyhedralsets with the former as default value

Description

• 

The command SimplicialDecomposition(pc) returns a simplicial decomposition, as a list of polyhedral sets, of the polyhedral cone pc.

• 

The command SimplicialDecomposition(pc,polyhedralsets) returns the same as a SimplicialDecomposition(pc).

• 

The command SimplicialDecomposition(pc,polyhedralcones) returns  a simplicial decomposition, as a list of polyhedral cones, of the polyhedral cone pc.

Terminology

• 

A polyhedral cone in dimension d is the solution set of a system of homogeneous linear non-strict inequalities in d variables. Equivalently, this is the conical hull of finitely many vectors with d coordinates. Here, the base field is that of the real numbers.

• 

Suppose that C is  the conical hull of k vectors with d coordinates. Then C is given by the matrix V with k columns  and d columns, whose columns are the  k vectors. The dual cone of C is the polyhedral set in dimension d which is the solution set of the system of homogeneous linear inequalities, whose matrix is the transpose of the matrix V.

• 

The polyhedral cone C in dimension d is called simplicial if it is generated by d linearly independent vectors. A simplicial decomposition of C is a finite set of simplicial cones so that the union of their interiors (in the Euclidean topology) is equal to the interior of C.

• 

Note that a polyhedral cone C, as a polyhedral set, has a single vertex which is the origin. In practice, it is convenient to use the term polyhedral cone  for the translation of a polyhedral cone in the formal sense defined above. With this abuse of terminology, a polyhedral cone is given by a point (its apex, or vertex) and a number of vectors (its generating rays, or simply rays).

Examples

withPolyhedralSets:withPolyhedralCones:

Define a polyhedral set

psPolyhedralSet1x,1y;PolyhedralSets:-Plotps

ps{Coordinates:x,yRelations:y−1,x−1

Define a polyhedral cone from the above polyhedral set

pcPolyhedralConeps

pcpolyhedral cone with vertex 1,1 and rays 0110

(1)

Compute its simplicial decomposition as a list of polyhedral cones

SimplicialDecompositionpc

polyhedral cone with vertex 1,1 and rays 0110

(2)

Compute its simplicial decomposition as a list of polyhedral sets

SimplicialDecompositionpc,polyhedralsets

{Coordinates:x1,x2Relations:x2−1,x1−1

(3)

Define another polyhedral set

psPolyhedralSetx1x2x30,x1+x2+x30,x1x2+x30;PolyhedralSets:-Plotps

ps{Coordinates:x1,x2,x3Relations:x1x2x30,x1+x2+x30,x1x2+x30

Define a polyhedral cone from the above polyhedral set

pcPolyhedralConeps

pcpolyhedral cone with vertex 0,0,0 and rays 11010−101−1

(4)

Compute its simplicial decomposition as a list of polyhedral cones

SimplicialDecompositionpc

polyhedral cone with vertex 0,0,0 and rays 11010−101−1

(5)

Compute its simplicial decomposition as a list of polyhedral sets

SimplicialDecompositionpc,polyhedralsets

{Coordinates:x1,x2,x3Relations:x1x2x30,x1+x2+x30,x1x2+x30

(6)

Define another polyhedral set

psPolyhedralSet0,0,0,1,1,1,1,1,1,1,1,1,1,1,11;PolyhedralSets:-Plotps

ps{Coordinates:x1,x2,x3Relations:x2x30,x1x30,x1+5x26x360,x1+6x25x350

Define a polyhedral cone from the above polyhedral set

pcPolyhedralConeps

pcpolyhedral cone with vertex 0,0,0 and rays 1−11−1−111111111−111

(7)

Compute its simplicial decomposition as a list of polyhedral cones

SimplicialDecompositionpc

polyhedral cone with vertex 0,0,0 and rays −1−111111−111,polyhedral cone with vertex 0,0,0 and rays 1−11−1−111111

(8)

Compute its simplicial decomposition as a list of polyhedral sets

SimplicialDecompositionpc,polyhedralsets

{Coordinates:x1,x2,x3Relations:x2x30,x1x30,x1+x20,{Coordinates:x1,x2,x3Relations:x1x20,x1+5x26x360,x1+6x25x350

(9)

Compatibility

• 

The PolyhedralSets[PolyhedralCones][SimplicialDecomposition] command was introduced in Maple 2025.

• 

For more information on Maple 2025 changes, see Updates in Maple 2025.

See Also

PolyhedralSets[PolyhedralCones][DualCone]

PolyhedralSets[PolyhedralCones][PolyhedralCone]

PolyhedralSets[PolyhedralCones][Rays]

PolyhedralSets[PolyhedralCones][Vertex]

 


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