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 -dimensional vector space has the form

(1)

where and are fixed vectors: is the initial vector and is a directed vector parallel to the line. If a basis in is given and if the coordinates of the vectors and are denoted by and , respectively, then (1) in coordinate form becomes

The parametric representation of an -dimensional affine subspace in has the form

(2)

where is the initial vector corresponding to the value 0 of the parameters and the form a linearly independent system of vectors parallel to the affine subspace in question. In coordinate form (2) becomes

The parametric representation of an -dimensional surface in has the form

(3)

where is, for example, the closure of a certain domain in and is a mapping of a certain class: continuous, differentiable, continuously differentiable, twice differentiable, etc.; accordingly, the -dimensional surface is also called continuous, differentiable, etc. (The rank of the Jacobian matrix is supposed to be .) In the case the set is an interval, , and (3) becomes the parametric representation of a curve: , , in . For example, , , , is a parametric representation in the plane of the circle of radius 1 with centre at the coordinate origin.

For the set on which the parametric representation is given one sometimes takes instead of the closure of an -dimensional domain a subset of of another kind.

Comments

A parametric equation or parametric representation for an -dimensional surface in (or ) need not be of dimension . I.e. any surjective mapping with as image (an open piece of) the surface is a (local) parametric representation of .

A chart is a local parametric representation (equation) for of dimension . Given a chart of a surface in , the curves , fixed, , and , fixed, , are called parametric curves.

References

[a1]	D.J. Struik, "Lectures on classical differential geometry" , Dover, reprint (1988)