Basic Examples
Basic Examples
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Get the orientation of a simplex:
[Simplex[{a,b}]]
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1
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[Simplex[{b,a}]]
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-1
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Find the permutations of the vertices of a simplex that are equivalent to the original simplex:
s=Simplex[{a,b,c}]
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Simplex[{a,b,c}]
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p=Simplex/@Permutations@@s
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{Simplex[{a,b,c}],Simplex[{a,c,b}],Simplex[{b,a,c}],Simplex[{b,c,a}],Simplex[{c,a,b}],Simplex[{c,b,a}]}
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Selectp,
[#]===
[s]&
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{Simplex[{a,b,c}],Simplex[{b,c,a}],Simplex[{c,a,b}]}
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Get orientations for a list of simplices:
Simplex/@Permutations[{a,b,c}]
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{Simplex[{a,b,c}],Simplex[{a,c,b}],Simplex[{b,a,c}],Simplex[{b,c,a}],Simplex[{c,a,b}],Simplex[{c,b,a}]}
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[%]
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{1,-1,-1,1,1,-1}
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Scope
Scope
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SimplexOrientation
[Point[{1}]]
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1
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SimplexOrientation
[Line[{a,b}]]
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1
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[Line[{b,a}]]
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-1
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SimplexOrientation
[Triangle[{a,b,c}]]
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1
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[Triangle[{a,c,b}]]
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-1
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SimplexOrientation
[Tetrahedron[{a,b,c,d}]]
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1
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[Tetrahedron[{a,c,b,d}]]
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-1
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Visualize orientation using and :
s=Simplex/@Permutations[{{0,0,0},{1,0,0},{0,1,0},{0,0,1}}];
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LabeledGraphics3D[{FaceForm[Red,Blue],#},ImageSizeTiny],
[#]&/@s
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-1 |
1 |
1 |
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-1 |
1 |
1 |
-1 |
-1 |
1 |
1 |
-1 |
-1 |
1 |
1 |
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1 |
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Simplices can have arbitrary expressions as vertices:
Simplex
,
,
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1
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We can see that the cat-dog-bird simplex has the opposite orientation to the cat-bird-dog simplex:
Properties and Relations
Properties and Relations
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