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Difference between revisions of "Affine coordinate frame"

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A set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010960/a0109601.png" /> linearly-independent vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010960/a0109602.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010960/a0109603.png" />) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010960/a0109604.png" />-dimensional affine space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010960/a0109605.png" />, and a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010960/a0109606.png" />. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010960/a0109607.png" /> is called the initial point, while the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010960/a0109608.png" /> are the scale vectors. Any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010960/a0109609.png" /> is defined with respect to the affine coordinate frame by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010960/a01096010.png" /> numbers — coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010960/a01096011.png" />, occurring in the decomposition of the position vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010960/a01096012.png" /> by the scale vectors: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010960/a01096013.png" /> (summation convention). The specification of two affine coordinate frames defines a unique affine transformation of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010960/a01096014.png" /> which converts the first frame into the second (see also [[Affine coordinate system|Affine coordinate system]]).
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A set of $n$ linearly-independent vectors $\mathbf{e}_i$ ($i=1,\ldots,n$) of $n$-dimensional affine space $A^n$, and a point $O$. The point $O$ is called the ''initial point'', while the vectors $\mathbf{e}_i$ are the scale vectors. Any point $M$ is defined with respect to the affine coordinate frame by $n$ numbers — coordinates $x^i$, occurring in the decomposition of the position vector $OM$ by the scale vectors: $\overline{OM} = x^i\mathbf{e}_i$ ([[summation convention]]). The specification of two affine coordinate frames defines a unique [[affine transformation]] of the space $A^n$ which converts the first frame into the second (see also [[Affine coordinate system]]).
  
  
  
 
====Comments====
 
====Comments====
An equivalent, and more usual, definition is as follows. An affine coordinate frame in affine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010960/a01096015.png" />-space is a set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010960/a01096016.png" /> points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010960/a01096017.png" /> which are linearly independent in the affine sense, i.e. the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010960/a01096018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010960/a01096019.png" />, are linearly independent in the corresponding vector space. Independence of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010960/a01096020.png" /> in the definition should be understood as independence in a corresponding vector space.
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An equivalent, and more usual, definition is as follows. An affine coordinate frame in affine $n$-space is a set of $n+1$ points $p_0,p_1,\ldots,p_n$ which are linearly independent in the affine sense, i.e. the vectors $p_0p_i$, $i=1,\ldots,n$, are linearly independent in the corresponding vector space. Independence of the vectors $\mathbf{e}_i$ in the definition should be understood as independence in a corresponding vector space.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Snapper,  R.J. Troyer,  "Metric affine geometry" , Acad. Press  (1971)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  E. Snapper,  R.J. Troyer,  "Metric affine geometry" , Acad. Press  (1971)</TD></TR>
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</table>
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Latest revision as of 19:23, 21 January 2016

A set of $n$ linearly-independent vectors $\mathbf{e}_i$ ($i=1,\ldots,n$) of $n$-dimensional affine space $A^n$, and a point $O$. The point $O$ is called the initial point, while the vectors $\mathbf{e}_i$ are the scale vectors. Any point $M$ is defined with respect to the affine coordinate frame by $n$ numbers — coordinates $x^i$, occurring in the decomposition of the position vector $OM$ by the scale vectors: $\overline{OM} = x^i\mathbf{e}_i$ (summation convention). The specification of two affine coordinate frames defines a unique affine transformation of the space $A^n$ which converts the first frame into the second (see also Affine coordinate system).


Comments

An equivalent, and more usual, definition is as follows. An affine coordinate frame in affine $n$-space is a set of $n+1$ points $p_0,p_1,\ldots,p_n$ which are linearly independent in the affine sense, i.e. the vectors $p_0p_i$, $i=1,\ldots,n$, are linearly independent in the corresponding vector space. Independence of the vectors $\mathbf{e}_i$ in the definition should be understood as independence in a corresponding vector space.

References

[1] E. Snapper, R.J. Troyer, "Metric affine geometry" , Acad. Press (1971)
How to Cite This Entry:
Affine coordinate frame. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Affine_coordinate_frame&oldid=37609
This article was adapted from an original article by A.P. Shirokov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article