Namespaces
Variants
Actions

Difference between revisions of "Affine parameter"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
''affine arc length''
+
<!--
 +
a0110701.png
 +
$#A+1 = 14 n = 0
 +
$#C+1 = 14 : ~/encyclopedia/old_files/data/A011/A.0101070 Affine parameter,
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
A parameter on a curve which is preserved under transformations of the [[Affine group|affine group]], for the determination of which the derivatives of the position vector of the curve of the lowest order must be known. The best known is the parameter which is invariant with respect to the equi-affine transformations, i.e. with respect to the affine unimodular group. For a plane curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011070/a0110701.png" /> this affine parameter is computed by the formula:
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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/a011/a011070/a0110702.png" /></td> </tr></table>
+
''affine arc length''
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011070/a0110703.png" /> is the skew product of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011070/a0110704.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011070/a0110705.png" />. In particular, for the affine length of an arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011070/a0110706.png" /> of a parabola, the affine parameter is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011070/a0110707.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011070/a0110708.png" /> is the area of the triangle formed by the chord <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011070/a0110709.png" /> and the tangents to the parabola at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011070/a01107010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011070/a01107011.png" />. It is possible to introduce in a similar manner the affine parameter of a space curve in the geometry of the general affine group or any one of its subgroups.
+
A parameter on a curve which is preserved under transformations of the [[Affine group|affine group]], for the determination of which the derivatives of the position vector of the curve of the lowest order must be known. The best known is the parameter which is invariant with respect to the equi-affine transformations, i.e. with respect to the affine unimodular group. For a plane curve $  \mathbf r = \mathbf r (t) $
 +
this affine parameter is computed by the formula:
  
 +
$$
 +
s  =  \int\limits _ {t _ {0} } ^ { t }
 +
| ( \dot{\mathbf r} , \dot{\mathbf r} dot  ) |  ^ {1/3}  dt ,
 +
$$
  
 +
where  $  ( \dot{\mathbf r} , \dot{\mathbf r} dot  ) $
 +
is the skew product of the vectors  $  \dot{\mathbf r} $
 +
and  $  \dot{\mathbf r} dot $.
 +
In particular, for the affine length of an arc  $  M _ {0} M _ {1} $
 +
of a parabola, the affine parameter is  $  s = {2f }  ^ {1/3} $,
 +
where  $  f $
 +
is the area of the triangle formed by the chord  $  M _ {0} M _ {1} $
 +
and the tangents to the parabola at the points  $  M _ {0} $
 +
and  $  M _ {1} $.
 +
It is possible to introduce in a similar manner the affine parameter of a space curve in the geometry of the general affine group or any one of its subgroups.
  
 
====Comments====
 
====Comments====
 
The arc length given by the formula above is sometimes referred to as the special affine arc length.
 
The arc length given by the formula above is sometimes referred to as the special affine arc length.
  
The notion of an  "affine parameter"  is also used in the theory of geodesics. An affine parameter of a geodesic (an autoparallel curve) of an affine connection is a parametrization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011070/a01107012.png" /> of the geodesic such that for the corresponding covariant derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011070/a01107013.png" /> the equation
+
The notion of an  "affine parameter"  is also used in the theory of geodesics. An affine parameter of a geodesic (an autoparallel curve) of an affine connection is a parametrization $  x (t) $
 +
of the geodesic such that for the corresponding covariant derivative $  \nabla $
 +
the equation
  
<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/a011/a011070/a01107014.png" /></td> </tr></table>
+
$$
 +
\nabla _ {\dot{x} (t) }
 +
\dot{x} (t)  = 0
 +
$$
  
 
is valid (cf. [[#References|[a3]]]).
 
is valid (cf. [[#References|[a3]]]).

Latest revision as of 16:09, 1 April 2020


affine arc length

A parameter on a curve which is preserved under transformations of the affine group, for the determination of which the derivatives of the position vector of the curve of the lowest order must be known. The best known is the parameter which is invariant with respect to the equi-affine transformations, i.e. with respect to the affine unimodular group. For a plane curve $ \mathbf r = \mathbf r (t) $ this affine parameter is computed by the formula:

$$ s = \int\limits _ {t _ {0} } ^ { t } | ( \dot{\mathbf r} , \dot{\mathbf r} dot ) | ^ {1/3} dt , $$

where $ ( \dot{\mathbf r} , \dot{\mathbf r} dot ) $ is the skew product of the vectors $ \dot{\mathbf r} $ and $ \dot{\mathbf r} dot $. In particular, for the affine length of an arc $ M _ {0} M _ {1} $ of a parabola, the affine parameter is $ s = {2f } ^ {1/3} $, where $ f $ is the area of the triangle formed by the chord $ M _ {0} M _ {1} $ and the tangents to the parabola at the points $ M _ {0} $ and $ M _ {1} $. It is possible to introduce in a similar manner the affine parameter of a space curve in the geometry of the general affine group or any one of its subgroups.

Comments

The arc length given by the formula above is sometimes referred to as the special affine arc length.

The notion of an "affine parameter" is also used in the theory of geodesics. An affine parameter of a geodesic (an autoparallel curve) of an affine connection is a parametrization $ x (t) $ of the geodesic such that for the corresponding covariant derivative $ \nabla $ the equation

$$ \nabla _ {\dot{x} (t) } \dot{x} (t) = 0 $$

is valid (cf. [a3]).

References

[a1] M. Spivak, "A comprehensive introduction to differential geometry" , 2 , Publish or Perish (1970) pp. 1–5
[a2] S. Buchin, "Affine differential geometry" , Sci. Press and Gordon & Breach (1983)
[a3] B. O'Neill, "Semi-Riemannian geometry (with applications to relativity)" , Acad. Press (1983)
How to Cite This Entry:
Affine parameter. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Affine_parameter&oldid=45047
This article was adapted from an original article by A.P. Shirokov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article