definition of orientated affine simplex
Topics
About: vectors space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of affine simplex.
Target Context
- The reader will have a definition of orientated affine simplex.
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.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( V\): \(\in \{\text{ the real vectors spaces }\}\)
\( \{p_0, ..., p_n\}\): \(\subseteq V\), \(\in \{\text{ the affine-independent sets of base points on } V\}\)
\(*(p_0, ..., p_n)\): \(= [p_0, ..., p_n]\) with the parity of the order of \((p_0, ..., p_n)\)
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Conditions:
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\(- (p_0, ..., p_n)\) is \([p_0, ..., p_n]\) with the opposite of the parity of the order of \((p_0, ..., p_n)\).
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 \((p_0, ..., p_n)\).
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 \((p_0, p_1, p_2) \neq (p_0, p_2, p_1)\), etc. and \((p_0, p_1, p_2) = - (p_0, p_2, p_1)\), etc..
