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PolyhedralSets

  

Faces

  

get the faces of a polyhedral set

  

ID

  

get the identifier of a polyhedral set

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

Faces(polyset)

Faces(polyset, dimension = d)

Faces(polyset, faceid = id)

ID(polyset)

Parameters

polyset

-

polyhedral set

dimension

-

(optional) integer greater than or equal to 1, dimension of faces to be returned, defaults to one less than the dimension of polyset to return its facets

faceid

-

(optional) integer or a set or list of integers indexing faces of polyset

Description

• 

The calling sequences Faces(polyset) and Faces(polyset, dimension = d) return a list of polyhedral sets that are d-faces of polyset.  Faces(polyset) uses a default value of dimension = Dimension(polyset) - 1, returning the facets of polyset.  If there are no faces of dimension d (e.g. asking for vertices of a half-space), an empty list is returned.

• 

The PolyhedralSets[Vertices] (or, PolyhedralSets[Vertexes]) command is shorthand for Faces(polyset, dimension = 0).  Similarly, PolyhedralSets[Edges] is shorthand for Faces(polyset, dimension = 1), and PolyhedralSets[Facets] is shorthand for Faces(polyset).

• 

A particular face can be retrieved via Faces(polyset, faceid = id).  The identification number id corresponds to those displayed on the graph returned by PolyhedralSets[Graph].

• 

The ID number of a given set can alternatively by obtained with the ID command.  This returns an integer that identifies a set relative to its faces.  Two unrelated polyhedral sets can have the same ID number, but the faces of a given polyhedral set will always have unique ID numbers.

Examples

withPolyhedralSets:

Get the facets of a tetrahedron

tExampleSets:-Tetrahedron:t_facesFacest

t_faces{Coordinates:x1,x2,x3Relations:x31,x2+x30,x21,x1+x2x3=1,{Coordinates:x1,x2,x3Relations:x31,x21,x2x30,x1x2+x3=1,{Coordinates:x1,x2,x3Relations:x31,x2x30,x21,x1x2x3=−1,{Coordinates:x1,x2,x3Relations:x31,x21,x2+x30,x1+x2+x3=−1

(1)

Plot the faces individually (which will give them each a different colour).

Plott_faces

The edges of the 5 dimensional simplex are:

s5ExampleSets:-Simplex5

s5{Coordinates:x1,x2,x3,x4,x5Relations:x50,x40,x30,x20,x10,x1+x2+x3+x4+x51

(2)

s5_edgesFacess5,dimension=1

s5_edges{Coordinates:x1,x2,x3,x4,x5Relations:x50,x51,x4+x5=1,x3=0,x2=0,x1=0,{Coordinates:x1,x2,x3,x4,x5Relations:x50,x51,x4=0,x3+x5=1,x2=0,x1=0,{Coordinates:x1,x2,x3,x4,x5Relations:x5=0,x40,x41,x3+x4=1,x2=0,x1=0,{Coordinates:x1,x2,x3,x4,x5Relations:x50,x51,x4=0,x3=0,x2+x5=1,x1=0,{Coordinates:x1,x2,x3,x4,x5Relations:x5=0,x40,x41,x3=0,x2+x4=1,x1=0,{Coordinates:x1,x2,x3,x4,x5Relations:x5=0,x4=0,x30,x31,x2+x3=1,x1=0,{Coordinates:x1,x2,x3,x4,x5Relations:x50,x51,x4=0,x3=0,x2=0,x1+x5=1,{Coordinates:x1,x2,x3,x4,x5Relations:x5=0,x40,x41,x3=0,x2=0,x1+x4=1,{Coordinates:x1,x2,x3,x4,x5Relations:x5=0,x4=0,x30,x31,x2=0,x1+x3=1,{Coordinates:x1,x2,x3,x4,x5Relations:x5=0,x4=0,x3=0,x20,x21,x1+x2=1,{Coordinates:x1,x2,x3,x4,x5Relations:x50,x51,x4=0,x3=0,x2=0,x1=0,{Coordinates:x1,x2,x3,x4,x5Relations:x5=0,x40,x41,x3=0,x2=0,x1=0,{Coordinates:x1,x2,x3,x4,x5Relations:x5=0,x4=0,x30,x31,x2=0,x1=0,{Coordinates:x1,x2,x3,x4,x5Relations:x5=0,x4=0,x3=0,x20,x21,x1=0,{Coordinates:x1,x2,x3,x4,x5Relations:x5=0,x4=0,x3=0,x2=0,x10,x11

(3)

ID numbers are used to identify the faces of a given set, but different unrelated sets may have the same ID number.

p1PolyhedralSet3x,10y+x,x10,x,y,z;p2PolyhedralSety5,3y+x,x7,x,y,z;IDp1;IDp2

p1{Coordinates:x,y,zRelations:yx−10,x−3,x10

p2{Coordinates:x,y,zRelations:y5,yx−3,x7

26

26

(4)

The faces of a set form a universe, within which the ID numbers uniquely identify members of the graph of the set.

mapID,Facesp1

5,11,25

(5)

mapID,Facesp2

17,23,25

(6)

Compatibility

• 

The PolyhedralSets[Faces] and PolyhedralSets[ID] commands were introduced in Maple 2015.

• 

For more information on Maple 2015 changes, see Updates in Maple 2015.

See Also

PolyhedralSets[VerticesAndRays]

PolyhedralSets[Dimension]

PolyhedralSets[Graph]

GraphTheory

GraphTheory[DrawGraph]

PolyhedralSets[PolyhedralSet]

PolyhedralSets