Simplex
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I don't like the ambient RN requirement, look to drop it. Also get more references
Contents
[hide]Definition
Let {a0,…,an}⊆RN be a geometrically independent set in RN[Note 1]. We define the "n-simplex", σ, spanned by {a0,…,an} to be the following set[1]:
- σ:={ x∈RN | x=n∑i=0tiai∧(ti)ni=0⊂R∧∀i∈{0,…,n}⊂N[ti≥0]∧n∑i=0ti=1}
The numbers, (ti)ni=0⊂R are uniquely determined by x and are called the "barycentric coordinates" of x with respect to {a0,…,an}
Elementary properties
- σ is convex
- In fact σ is the convex hull of {a0,…,an}
Terminology
- Dimension: Dim(σ):=|{a0,…,an}|−1[1]
- Vertices: the vertices of σ are the points a0,…,an[1]
- Face: any simplex spanned by A∈(P({a0,…,an})−{∅}) is called a face[1] of σ.
- Face opposite ai∈{a0,…,an}: is the face spanned by {a0,…,an}−{ai} which is sometimes denoted {a0,…,^ai,…,an}[Note 3]
- Boundary: the union of all proper faces is the boundary of σ[1] denoted Bd(σ)[1] or ∂σ
- i.e. ∂σ:=⋃τ∈Kτ where K:=(P({a0,…,an})−{∅,{a0,…,an}})
- Interior: Int(σ):=σ−∂σ[1] - the interior is sometimes called an open simplex
Proof of claims
Grade: A
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The claim that the coordinates are unique and stuff is totally missing
Notes
- Jump up ↑ This means that N>n - certainly. We may be able to go lower (to N≥n) but I don't want to at this time.
- Jump up ↑ σ is a face of itself. Much like {a,b}⊆{a,b} is a subset, but {a}⊂{a,b} is a proper subset of {a,b}
- Jump up ↑ It is quite common to denote "deletion from the array" by a ˆa for a∈ the array
