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Difference between revisions of "Affine coordinate system"

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A rectilinear coordinate system in an [[Affine space|affine space]]. An affine coordinate system on a plane is defined by an ordered pair of non-collinear vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010970/a0109701.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010970/a0109702.png" /> (an affine basis) and a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010970/a0109703.png" /> (the coordinate origin). The straight lines passing through the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010970/a0109704.png" /> and parallel to the basis vectors are known as the coordinate axes. The vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010970/a0109705.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010970/a0109706.png" /> define the positive direction on the coordinate axes. The axis parallel to the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010970/a0109707.png" /> is called the abscissa (axis), while that parallel to the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010970/a0109708.png" /> is called the ordinate (axis). The affine coordinates of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010970/a0109709.png" /> are given by an ordered pair of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010970/a01097010.png" /> which are the coefficients of the decomposition of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010970/a01097011.png" /> by the basis vectors:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010970/a01097012.png" /></td> </tr></table>
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The first number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010970/a01097013.png" /> is called the abscissa, while the second number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010970/a01097014.png" /> is called the ordinate of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010970/a01097015.png" />.
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A rectilinear coordinate system in an [[Affine space|affine space]]. An affine coordinate system on a plane is defined by an ordered pair of non-collinear vectors  $  \mathbf e _ {1} $
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and  $  \mathbf e _ {2} $(
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an affine basis) and a point  $  O $(
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the coordinate origin). The straight lines passing through the point  $  O $
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and parallel to the basis vectors are known as the coordinate axes. The vectors  $  \mathbf e _ {1} $
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and  $  \mathbf e _ {2} $
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define the positive direction on the coordinate axes. The axis parallel to the vector  $  \mathbf e _ {1} $
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is called the abscissa (axis), while that parallel to the vector  $  \mathbf e _ {2} $
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is called the ordinate (axis). The affine coordinates of a point  $  M $
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are given by an ordered pair of numbers  $  (x, y) $
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which are the coefficients of the decomposition of the vector  $  \overline{OM}\; $
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by the basis vectors:
  
An affine coordinate system in three-dimensional space is defined as an ordered triplet of linearly-independent vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010970/a01097016.png" /> and a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010970/a01097017.png" />. As in the case of the plane, one defines the coordinate axes — abscissa, ordinate and applicate — and the coordinates of a point. Planes passing through pairs of coordinate axes are known as coordinate planes.
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$$
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\overline{OM}\;  =  x \mathbf e _ {1} + y \mathbf e _ {2} .
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$$
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The first number  $  x $
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is called the abscissa, while the second number  $  y $
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is called the ordinate of  $  M $.
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An affine coordinate system in three-dimensional space is defined as an ordered triplet of linearly-independent vectors $  \mathbf e _ {1} , \mathbf e _ {2} , \mathbf e _ {3} $
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and a point $  O $.  
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As in the case of the plane, one defines the coordinate axes — abscissa, ordinate and applicate — and the coordinates of a point. Planes passing through pairs of coordinate axes are known as coordinate planes.

Latest revision as of 16:09, 1 April 2020


A rectilinear coordinate system in an affine space. An affine coordinate system on a plane is defined by an ordered pair of non-collinear vectors $ \mathbf e _ {1} $ and $ \mathbf e _ {2} $( an affine basis) and a point $ O $( the coordinate origin). The straight lines passing through the point $ O $ and parallel to the basis vectors are known as the coordinate axes. The vectors $ \mathbf e _ {1} $ and $ \mathbf e _ {2} $ define the positive direction on the coordinate axes. The axis parallel to the vector $ \mathbf e _ {1} $ is called the abscissa (axis), while that parallel to the vector $ \mathbf e _ {2} $ is called the ordinate (axis). The affine coordinates of a point $ M $ are given by an ordered pair of numbers $ (x, y) $ which are the coefficients of the decomposition of the vector $ \overline{OM}\; $ by the basis vectors:

$$ \overline{OM}\; = x \mathbf e _ {1} + y \mathbf e _ {2} . $$

The first number $ x $ is called the abscissa, while the second number $ y $ is called the ordinate of $ M $.

An affine coordinate system in three-dimensional space is defined as an ordered triplet of linearly-independent vectors $ \mathbf e _ {1} , \mathbf e _ {2} , \mathbf e _ {3} $ and a point $ O $. As in the case of the plane, one defines the coordinate axes — abscissa, ordinate and applicate — and the coordinates of a point. Planes passing through pairs of coordinate axes are known as coordinate planes.

How to Cite This Entry:
Affine coordinate system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Affine_coordinate_system&oldid=45044
This article was adapted from an original article by A.S. Parkhomenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article