545: Orientated Affine Simplex

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definition of orientated affine simplex

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Topics

About: vectors space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Note

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Starting Context

• The reader knows a definition of affine simplex.

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Target Context

• The reader will have a definition of orientated affine simplex.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$V$: $\in \{\text{ the real vectors spaces }\}$
$\{p_0,...,p_n\}$: $\subseteq V$, $\in \{\text{ the affine-independent sets of base points on }V\}$
$\ast (p_0,...,p_n)$: $=[p_0,...,p_n]$ with the parity of the order of $(p_0,...,p_n)$
//

Conditions:
//

$-(p_0,...,p_n)$ is $[p_0,...,p_n]$ with the opposite of the parity of the order of $(p_0,...,p_n)$.

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2: Natural Language Description

For any real vectors space, $V$, and any affine-independent set of base points, $\{p_0,...,p_n\}\subseteq V$, the affine simplex, $[p_0,...,p_n]$, with the parity of the order of $(p_0,...,p_n)$

$-(p_0,...,p_n)$ is $[p_0,...,p_n]$ with the opposite of the parity of the order of $(p_0,...,p_n)$.

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3: Note

For example, $(p_0,p_1,p_2)=(p_1,p_2,p_0)=(p_2,p_0,p_1)$ and $(p_0,p_2,p_1)=(p_1,p_0,p_2)=(p_2,p_1,p_0)$, but $(p_0,p_1,p_2)\ne (p_0,p_2,p_1)$, etc. and $(p_0,p_1,p_2)=-(p_0,p_2,p_1)$, etc..

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References

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