Parametric equation
In mathematics, a parametric equation expresses several quantities, such as the coordinates of a point, as functions of one or several variables called parameters.
[1] In the case of a single parameter, parametric equations are commonly used to express the trajectory of a moving point, in which case, the parameter is often, but not necessarily, time, and the point describes a curve, called a parametric curve.
Parametric representations are generally nonunique (see the "Examples in two dimensions" section below), so the same quantities may be expressed by a number of different parameterizations.
while more complicated cases will give an implicit equation of the form
where p, q, and r are set-wise coprime polynomials, a resultant computation allows one to implicitize.
More precisely, the implicit equation is the resultant with respect to t of xr(t) – p(t) and yr(t) – q(t).
In higher dimensions (either more than two coordinates or more than one parameter), the implicitization of rational parametric equations may by done with Gröbner basis computation; see Gröbner basis § Implicitization in higher dimension.
Consider the unit circle which is described by the ordinary (Cartesian) equation
With the Cartesian equation it is easier to check whether a point lies on the circle or not.
With the parametric version it is easier to obtain points on a plot.
In some contexts, parametric equations involving only rational functions (that is fractions of two polynomials) are preferred, if they exist.
With this pair of parametric equations, the point (−1, 0) is not represented by a real value of t, but by the limit of x and y when t tends to infinity.
An ellipse in canonical position (center at origin, major axis along the x-axis) with semi-axes a and b can be represented parametrically as
Both parameterizations may be made rational by using the tangent half-angle formula and setting
A Lissajous curve is similar to an ellipse, but the x and y sinusoids are not in phase.
where kx and ky are constants describing the number of lobes of the figure.
Note that in the rational forms of these formulae, the points (−a , 0) and (0 , −a), respectively, are not represented by a real value of t, but are the limit of x and y as t tends to infinity.
Some examples: Parametric equations are convenient for describing curves in higher-dimensional spaces.
describes a three-dimensional curve, the helix, with a radius of a and rising by 2πb units per turn.
As u varies from 0 to 2π the point on the surface moves about a short circle passing through the hole in the torus.
As t varies from 0 to 2π the point on the surface moves about a long circle around the hole in the torus.
In kinematics, objects' paths through space are commonly described as parametric curves, with each spatial coordinate depending explicitly on an independent parameter (usually time).
Used in this way, the set of parametric equations for the object's coordinates collectively constitute a vector-valued function for position.
Another important use of parametric equations is in the field of computer-aided design (CAD).
[8] For example, consider the following three representations, all of which are commonly used to describe planar curves.
[9] Numerous problems in integer geometry can be solved using parametric equations.
A classical such solution is Euclid's parametrization of right triangles such that the lengths of their sides a, b and their hypotenuse c are coprime integers.
A system of m linear equations in n unknowns is underdetermined if it has more than one solution.
This occurs when the matrix of the system and its augmented matrix have the same rank r and r < n. In this case, one can select n − r unknowns as parameters and represent all solutions as a parametric equation where all unknowns are expressed as linear combinations of the selected ones.
[10] The standard method for computing a parametric form of the solution is to use Gaussian elimination for computing a reduced row echelon form of the augmented matrix.
