Parametric equation

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)