PolyhedralSets/PolyhedralCones/Vertex - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.

Online Help

Home : Support : Online Help : PolyhedralSets/PolyhedralCones/Vertex

PolyhedralSets[PolyhedralCones]

  

Vertex

  

returns the vertex of a polyhedral cone

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

Vertex(pc)

Parameters

pc

-

PolyhedralCone

Description

• 

The command Vertex(pc) returns the apex, as a list of rational numbers, 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 cone from its vertex and rays

pcPolyhedralCone1,1,0,1,1,0

pcpolyhedral cone with vertex 1,1 and rays 0110

(1)

Obtain its vertex

Vertexpc

1,1

(2)

Obtain its rays

Rayspc

0,1,1,0

(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)

Obtain its vertex

Vertexpc

0,0,0

(5)

Obtain its rays

Rayspc

1,1,0,1,0,−1,0,1,−1

(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)

Obtain its vertex

Vertexpc

0,0,0

(8)

Obtain its rays

Rayspc

1,−1,1,−1,−1,1,1,1,11,−1,1,1

(9)

Compatibility

• 

The PolyhedralSets[PolyhedralCones][Vertex] 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][SimplicialDecomposition]

 


Download Help Document