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)