Namespaces
Variants
Actions

Difference between revisions of "Parametric equation"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (fix tex)
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
 +
<!--
 +
p0715001.png
 +
$#A+1 = 62 n = 0
 +
$#C+1 = 62 : ~/encyclopedia/old_files/data/P071/P.0701500 Parametric equation
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
''of a set of points in a space''
 
''of a set of points in a space''
  
 
The specification of the points of the set or of their coordinates by the values of functions of certain variables called parameters.
 
The specification of the points of the set or of their coordinates by the values of functions of certain variables called parameters.
  
The parametric representation of a straight line in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p0715001.png" />-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p0715002.png" /> has the form
+
The parametric representation of a straight line in the $  n $-
 
+
dimensional vector space $  \mathbf R  ^ {n} $
<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/p/p071/p071500/p0715003.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
has the form
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p0715004.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p0715005.png" /> are fixed vectors: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p0715006.png" /> is the initial vector and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p0715007.png" /> is a directed vector parallel to the line. If a basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p0715008.png" /> is given and if the coordinates of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p0715009.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150010.png" /> are denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150012.png" />, respectively, then (1) in coordinate form becomes
 
  
<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/p/p071/p071500/p07150013.png" /></td> </tr></table>
+
$$ \tag{1 }
 +
= x  ^ {( 0)} + at ,\ \
 +
x  ^ {( 0)} , a \in \mathbf R  ^ {n} ,\ \
 +
- \infty < t < + \infty ,
 +
$$
  
The parametric representation of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150014.png" />-dimensional affine subspace in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150015.png" /> has the form
+
where  $  x  ^ {( 0)} $
 +
and  $  a $
 +
are fixed vectors: $  x  ^ {( 0)} $
 +
is the initial vector and  $  a \neq 0 $
 +
is a directed vector parallel to the line. If a basis in $  \mathbf R  ^ {n} $
 +
is given and if the coordinates of the vectors  $  x $
 +
and  $  a $
 +
are denoted by  $  x _ {1} \dots x _ {n} $
 +
and  $  a _ {1} \dots a _ {n} $,
 +
respectively, then (1) in coordinate form becomes
  
<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/p/p071/p071500/p07150016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$
 +
x _ {k}  = x _ {k}  ^ {( 0)} + a _ {k} t ,\ \
 +
- \infty < t < + \infty ,\ \
 +
k = 1 \dots n.
 +
$$
  
<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/p/p071/p071500/p07150017.png" /></td> </tr></table>
+
The parametric representation of an  $  m $-
 +
dimensional affine subspace in  $  \mathbf R  ^ {n} $
 +
has the form
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150018.png" /> is the initial vector corresponding to the value 0 of the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150019.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150020.png" /> form a linearly independent system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150021.png" /> vectors parallel to the affine subspace in question. In coordinate form (2) becomes
+
$$ \tag{2 }
 +
= x  ^ {( 0)} + a  ^ {( 1)} t _ {1} + \dots + a ^ {( m)} t _ {m} ,
 +
$$
  
<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/p/p071/p071500/p07150022.png" /></td> </tr></table>
+
$$
 +
x  ^ {( 0)} , a  ^ {( j)}  \in  \mathbf R  ^ {n} ,\ \
 +
- \infty  < t _ {j}  < + \infty ,\  j = 1 \dots m,
 +
$$
  
<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/p/p071/p071500/p07150023.png" /></td> </tr></table>
+
where  $  x  ^ {( 0)} $
 +
is the initial vector corresponding to the value 0 of the parameters  $  t _ {j} $
 +
and the  $  a  ^ {( 1)} \dots a  ^ {( m)} $
 +
form a linearly independent system of  $  m $
 +
vectors parallel to the affine subspace in question. In coordinate form (2) becomes
  
The parametric representation of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150024.png" />-dimensional surface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150025.png" /> has the form
+
$$
 +
x _ {k}  = x _ {k}  ^ {( 0)} +
 +
a _ {k}  ^ {( 1)} t _ {1} + \dots + a _ {k}  ^ {( m)} t _ {m} ,
 +
$$
  
<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/p/p071/p071500/p07150026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$
 +
- \infty  < t _ {j}  < + \infty ,\  j = 1 \dots m; \  k  = 1 \dots n.
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150027.png" /> is, for example, the closure of a certain domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150029.png" /> is a mapping of a certain class: continuous, differentiable, continuously differentiable, twice differentiable, etc.; accordingly, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150030.png" />-dimensional surface is also called continuous, differentiable, etc. (The rank of the Jacobian matrix is supposed to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150031.png" />.) In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150032.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150033.png" /> is an interval, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150034.png" />, and (3) becomes the parametric representation of a curve: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150036.png" />, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150037.png" />. For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150040.png" />, is a parametric representation in the plane of the circle of radius 1 with centre at the coordinate origin.
+
The parametric representation of an  $  m $-
 +
dimensional surface in $  \mathbf R  ^ {n} $
 +
has the form
  
For the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150041.png" /> on which the parametric representation is given one sometimes takes instead of the closure of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150042.png" />-dimensional domain a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150043.png" /> of another kind.
+
$$ \tag{3 }
 +
= x( t)  = x( t _ {1} \dots t _ {m} ),\ \
 +
t = ( t _ {1} \dots t _ {m} ) \in E \subset \mathbf R  ^ {m} ,
 +
$$
  
 +
where  $  E $
 +
is, for example, the closure of a certain domain in  $  \mathbf R  ^ {m} $
 +
and  $  x:  E \rightarrow \mathbf R  ^ {n} $
 +
is a mapping of a certain class: continuous, differentiable, continuously differentiable, twice differentiable, etc.; accordingly, the  $  m $-
 +
dimensional surface is also called continuous, differentiable, etc. (The rank of the Jacobian matrix is supposed to be  $  m $.)
 +
In the case  $  m= 1 $
 +
the set  $  E $
 +
is an interval,  $  E = [ a, b] $,
 +
and (3) becomes the parametric representation of a curve:  $  x = x( t) $,
 +
$  a \leq  t \leq  b $,
 +
in  $  \mathbf R  ^ {n} $.
 +
For example,  $  x _ {1} = \cos  t $,
 +
$  x _ {2} = \sin  t $,
 +
$  0 \leq  t \leq  2 \pi $,
 +
is a parametric representation in the plane of the circle of radius 1 with centre at the coordinate origin.
  
 +
For the set  $  E $
 +
on which the parametric representation is given one sometimes takes instead of the closure of an  $  m $-
 +
dimensional domain a subset of  $  \mathbf R  ^ {m} $
 +
of another kind.
  
 
====Comments====
 
====Comments====
A parametric equation or parametric representation for an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150044.png" />-dimensional surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150045.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150046.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150047.png" />) need not be of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150048.png" />. I.e. any surjective mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150049.png" /> with as image (an open piece of) the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150050.png" /> is a (local) parametric representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150051.png" />.
+
A parametric equation or parametric representation for an $  m $-
 +
dimensional surface $  S $
 +
in $  \mathbf R  ^ {n} $(
 +
or $  \mathbf C  ^ {n} $)  
 +
need not be of dimension $  m $.  
 +
I.e. any surjective mapping $  \mathbf R  ^ {n} \supset E \rightarrow \mathbf R  ^ {n} $
 +
with as image (an open piece of) the surface $  S $
 +
is a (local) parametric representation of $  S $.
  
A [[Chart|chart]] is a local parametric representation (equation) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150052.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150053.png" />. Given a chart <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150054.png" /> of a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150055.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150056.png" />, the curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150058.png" /> fixed, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150059.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150061.png" /> fixed, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071500/p07150062.png" />, are called parametric curves.
+
A [[Chart|chart]] is a local parametric representation (equation) for $  S $
 +
of dimension $  \mathop{\rm dim} ( S) $.  
 +
Given a chart $  r( u, v) $
 +
of a surface $  S $
 +
in $  \mathbf R  ^ {3} $,  
 +
the curves $  r( u _ {0} , v) $,  
 +
$  u _ {0} $
 +
fixed, $  v \in \mathbf R $,  
 +
and $  r( u, v _ {0} ) $,  
 +
$  v _ {0} $
 +
fixed, $  u \in \mathbf R $,  
 +
are called parametric curves.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.J. Struik,  "Lectures on classical differential geometry" , Dover, reprint  (1988)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.J. Struik,  "Lectures on classical differential geometry" , Dover, reprint  (1988)</TD></TR></table>

Latest revision as of 17:54, 11 January 2021


of a set of points in a space

The specification of the points of the set or of their coordinates by the values of functions of certain variables called parameters.

The parametric representation of a straight line in the $ n $- dimensional vector space $ \mathbf R ^ {n} $ has the form

$$ \tag{1 } x = x ^ {( 0)} + at ,\ \ x ^ {( 0)} , a \in \mathbf R ^ {n} ,\ \ - \infty < t < + \infty , $$

where $ x ^ {( 0)} $ and $ a $ are fixed vectors: $ x ^ {( 0)} $ is the initial vector and $ a \neq 0 $ is a directed vector parallel to the line. If a basis in $ \mathbf R ^ {n} $ is given and if the coordinates of the vectors $ x $ and $ a $ are denoted by $ x _ {1} \dots x _ {n} $ and $ a _ {1} \dots a _ {n} $, respectively, then (1) in coordinate form becomes

$$ x _ {k} = x _ {k} ^ {( 0)} + a _ {k} t ,\ \ - \infty < t < + \infty ,\ \ k = 1 \dots n. $$

The parametric representation of an $ m $- dimensional affine subspace in $ \mathbf R ^ {n} $ has the form

$$ \tag{2 } x = x ^ {( 0)} + a ^ {( 1)} t _ {1} + \dots + a ^ {( m)} t _ {m} , $$

$$ x ^ {( 0)} , a ^ {( j)} \in \mathbf R ^ {n} ,\ \ - \infty < t _ {j} < + \infty ,\ j = 1 \dots m, $$

where $ x ^ {( 0)} $ is the initial vector corresponding to the value 0 of the parameters $ t _ {j} $ and the $ a ^ {( 1)} \dots a ^ {( m)} $ form a linearly independent system of $ m $ vectors parallel to the affine subspace in question. In coordinate form (2) becomes

$$ x _ {k} = x _ {k} ^ {( 0)} + a _ {k} ^ {( 1)} t _ {1} + \dots + a _ {k} ^ {( m)} t _ {m} , $$

$$ - \infty < t _ {j} < + \infty ,\ j = 1 \dots m; \ k = 1 \dots n. $$

The parametric representation of an $ m $- dimensional surface in $ \mathbf R ^ {n} $ has the form

$$ \tag{3 } x = x( t) = x( t _ {1} \dots t _ {m} ),\ \ t = ( t _ {1} \dots t _ {m} ) \in E \subset \mathbf R ^ {m} , $$

where $ E $ is, for example, the closure of a certain domain in $ \mathbf R ^ {m} $ and $ x: E \rightarrow \mathbf R ^ {n} $ is a mapping of a certain class: continuous, differentiable, continuously differentiable, twice differentiable, etc.; accordingly, the $ m $- dimensional surface is also called continuous, differentiable, etc. (The rank of the Jacobian matrix is supposed to be $ m $.) In the case $ m= 1 $ the set $ E $ is an interval, $ E = [ a, b] $, and (3) becomes the parametric representation of a curve: $ x = x( t) $, $ a \leq t \leq b $, in $ \mathbf R ^ {n} $. For example, $ x _ {1} = \cos t $, $ x _ {2} = \sin t $, $ 0 \leq t \leq 2 \pi $, is a parametric representation in the plane of the circle of radius 1 with centre at the coordinate origin.

For the set $ E $ on which the parametric representation is given one sometimes takes instead of the closure of an $ m $- dimensional domain a subset of $ \mathbf R ^ {m} $ of another kind.

Comments

A parametric equation or parametric representation for an $ m $- dimensional surface $ S $ in $ \mathbf R ^ {n} $( or $ \mathbf C ^ {n} $) need not be of dimension $ m $. I.e. any surjective mapping $ \mathbf R ^ {n} \supset E \rightarrow \mathbf R ^ {n} $ with as image (an open piece of) the surface $ S $ is a (local) parametric representation of $ S $.

A chart is a local parametric representation (equation) for $ S $ of dimension $ \mathop{\rm dim} ( S) $. Given a chart $ r( u, v) $ of a surface $ S $ in $ \mathbf R ^ {3} $, the curves $ r( u _ {0} , v) $, $ u _ {0} $ fixed, $ v \in \mathbf R $, and $ r( u, v _ {0} ) $, $ v _ {0} $ fixed, $ u \in \mathbf R $, are called parametric curves.

References

[a1] D.J. Struik, "Lectures on classical differential geometry" , Dover, reprint (1988)
How to Cite This Entry:
Parametric equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parametric_equation&oldid=13299
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article