Inmathematics, aparametric equation expresses several quantities, such as thecoordinates of apoint, asfunctions of one or severalvariables calledparameters.[1]
In the case of a single parameter, parametric equations are commonly used to express thetrajectory of a moving point, in which case, the parameter is often, but not necessarily, time, and the point describes acurve, called aparametric curve. In the case of two parameters, the point describes asurface, called aparametric surface. In all cases, the equations are collectively called aparametric representation,[2] orparametric system,[3] orparameterization (also spelledparametrization,parametrisation) of the object.[1][4][5]
For example, the equationsform a parametric representation of theunit circle, wheret is the parameter: A point(x,y) is on the unit circleif and only if there is a value oft such that these two equations generate that point. Sometimes the parametric equations for the individualscalar output variables are combined into a single parametric equation invectors:
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.[1]
In addition to curves and surfaces, parametric equations can describemanifolds andalgebraic varieties of higherdimension, with the number of parameters being equal to the dimension of the manifold or variety, and the number of equations being equal to the dimension of the space in which the manifold or variety is considered (for curves the dimension isone andone parameter is used, for surfaces dimensiontwo andtwo parameters, etc.).
Parametric equations are commonly used inkinematics, where thetrajectory of an object is represented by equations depending on time as the parameter. Because of this application, a single parameter is often labeledt; however, parameters can represent other physical quantities (such as geometric variables) or can be selected arbitrarily for convenience. Parameterizations are non-unique; more than one set of parametric equations can specify the same curve.[6]
Converting a set of parametric equations to a singleimplicit equation involves eliminating the variablet from the simultaneous equations This process is calledimplicitization. If one of these equations can be solved fort, the expression obtained can be substituted into the other equation to obtain an equation involvingx andy only: Solving to obtain and using this in gives the explicit equation while more complicated cases will give an implicit equation of the form
If the parametrization is given byrational functions
wherep,q, andr are set-wisecoprime polynomials, aresultant computation allows one to implicitize. More precisely, the implicit equation is theresultant with respect tot ofxr(t) –p(t) andyr(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 withGröbner basis computation; seeGröbner basis § Implicitization in higher dimension.
To take the example of the circle of radiusa, the parametric equations
can be implicitized in terms ofx andy by way of thePythagorean trigonometric identity. With
andwe getand thus
which is the standard equation of a circle centered at the origin.
The simplest equation for aparabola,
can be (trivially) parameterized by using a free parametert, and setting
More generally, any curve given by an explicit equation
can be (trivially) parameterized by using a free parametert, and setting
A more sophisticated example is the following. Consider the unit circle which is described by the ordinary (Cartesian) equation
This equation can be parameterized as follows:
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 onlyrational functions (that is fractions of twopolynomials) are preferred, if they exist. In the case of the circle, such arational parameterization is
With this pair of parametric equations, the point(−1, 0) is not represented by areal value oft, but by thelimit ofx andy whent tends toinfinity.
Anellipse in canonical position (center at origin, major axis along thex-axis) with semi-axesa andb can be represented parametrically as
An ellipse in general position can be expressed as
as the parametert varies from0 to2π. Here(Xc ,Yc) is the center of the ellipse, andφ is the angle between thex-axis and the major axis of the ellipse.
Both parameterizations may be maderational by using thetangent half-angle formula and setting
ALissajous curve is similar to an ellipse, but thex andysinusoids are not in phase. In canonical position, a Lissajous curve is given bywherekx andky are constants describing the number of lobes of the figure.
An east-west openinghyperbola can be represented parametrically by
or,rationally
A north-south opening hyperbola can be represented parametrically as
or, rationally
In all these formulae(h ,k) are the center coordinates of the hyperbola,a is the length of the semi-major axis, andb is the length of the semi-minor axis. Note that in the rational forms of these formulae, the points(−a , 0) and(0 ,−a), respectively, are not represented by a real value oft, but are the limit ofx andy ast tends to infinity.
Ahypotrochoid is a curve traced by a point attached to a circle of radiusr rolling around the inside of a fixed circle of radiusR, where the point is at a distanced from the center of the interior circle.
The parametric equations for the hypotrochoids are:
Some examples:
Parametric equations are convenient for describingcurves in higher-dimensional spaces. For example:
describes a three-dimensional curve, thehelix, with a radius ofa and rising by2πb units per turn. The equations are identical in theplane to those for a circle.Such expressions as the one above are commonly written as
wherer is a three-dimensional vector.
Atorus with major radiusR and minor radiusr may be defined parametrically as
where the two parameterst andu both vary between0 and2π.
Asu varies from0 to2π the point on the surface moves about a short circle passing through the hole in the torus. Ast varies from0 to2π the point on the surface moves about a long circle around the hole in the torus.
The parametric equation of the line through the point and parallel to the vector is[7]
Inkinematics, 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 avector-valued function for position. Such parametric curves can then beintegrated anddifferentiated termwise. Thus, if a particle's position is described parametrically as
then itsvelocity can be found as
and itsacceleration as
Another important use of parametric equations is in the field ofcomputer-aided design (CAD).[8] For example, consider the following three representations, all of which are commonly used to describeplanar curves.
| Type | Form | Example | Description |
|---|---|---|---|
| Explicit | Line | ||
| Implicit | Circle | ||
| Parametric | Line | ||
| Circle |
Each representation has advantages and drawbacks for CAD applications.
The explicit representation may be very complicated, or even may not exist. Moreover, it does not behave well undergeometric transformations, and in particular underrotations. On the other hand, as a parametric equation and an implicit equation may easily be deduced from an explicit representation, when a simple explicit representation exists, it has the advantages of both other representations.
Implicit representations may make it difficult to generate points on the curve, and even to decide whether there are real points. On the other hand, they are well suited for deciding whether a given point is on a curve, or whether it is inside or outside of a closed curve.
Such decisions may be difficult with a parametric representation, but parametric representations are best suited for generating points on a curve, and for plotting it.[9]
Numerous problems ininteger geometry can be solved using parametric equations. A classical such solution isEuclid's parametrization ofright triangles such that the lengths of their sidesa,b and their hypotenusec arecoprime integers. Asa andb are not both even (otherwisea,b andc would not be coprime), one may exchange them to havea even, and the parameterization is then
where the parametersm andn are positive coprime integers that are not both odd.
By multiplyinga,b andc by an arbitrary positive integer, one gets a parametrization of all right triangles whose three sides have integer lengths.
Asystem ofm linear equations inn unknowns isunderdetermined if it has more than one solution. This occurs when thematrix of the system and itsaugmented matrix have the samerankr andr <n. In this case, one can selectn −r unknowns as parameters and represent all solutions as a parametric equation where all unknowns are expressed aslinear combinations of the selected ones. That is, if the unknowns are one can reorder them for expressing the solutions as[10]
Such a parametric equation is called aparametric form of the solution of the system.[10]
The standard method for computing a parametric form of the solution is to useGaussian elimination for computing areduced row echelon form of the augmented matrix. Then the unknowns that can be used as parameters are the ones that correspond to columns not containing anyleading entry (that is the left most non zero entry in a row or the matrix), and the parametric form can be straightforwardly deduced.[10]