Parametric equation

The butterfly curve can be defined by parametric equations of Template:Mvar and Template:Mvar.

In mathematics, a parametric equation expresses several quantities, such as the coordinates of a point, as functions of one or several variables called parameters.

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. In the case of two parameters, the point describes a surface, called a parametric surface. In all cases, the equations are collectively called a parametric representation, or parametric system, or parameterization (also spelled parametrization, parametrisation) of the object.

For example, the equations $$\begin{array}{cc}x& =\cos t\\ y& =\sin t\end{array}$$ form a parametric representation of the unit circle, where Template:Mvar is the parameter: A point Template:Math is on the unit circle if and only if there is a value of Template:Mvar such that these two equations generate that point. Sometimes the parametric equations for the individual scalar output variables are combined into a single parametric equation in vectors:

$$(x,y)=(\cos t,\sin t).$$

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.

In addition to curves and surfaces, parametric equations can describe manifolds and algebraic varieties of higher dimension, 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 is one and one parameter is used, for surfaces dimension two and two parameters, etc.).

Parametric equations are commonly used in kinematics, where the trajectory of an object is represented by equations depending on time as the parameter. Because of this application, a single parameter is often labeled Template:Mvar; 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.

Implicitization

Converting a set of parametric equations to a single implicit equation involves eliminating the variable Template:Mvar from the simultaneous equations $x=f(t),y=g(t).$ This process is called Template:Dfn. If one of these equations can be solved for Template:Mvar, the expression obtained can be substituted into the other equation to obtain an equation involving Template:Mvar and Template:Mvar only: Solving $y=g(t)$ to obtain $t=g^{-1}(y)$ and using this in $x=f(t)$ gives the explicit equation $x=f(g^{-1}(y)),$ while more complicated cases will give an implicit equation of the form $h(x,y)=0.$

If the parametrization is given by rational functions $$x=\frac{p(t)}{r(t)},y=\frac{q(t)}{r(t)},$$

where Template:Mvar, Template:Mvar, and Template:Mvar are set-wise coprime polynomials, a resultant computation allows one to implicitize. More precisely, the implicit equation is the resultant with respect to Template:Mvar of Template:Math and Template:Math.

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 Template:Slink.

To take the example of the circle of radius Template:Mvar, the parametric equations $$\begin{array}{cc}x& =a\cos (t)\\ y& =a\sin (t)\end{array}$$

can be implicitized in terms of Template:Math and Template:Math by way of the Pythagorean trigonometric identity. With

$$\begin{array}{cc}\frac{x}{a}& =\cos (t)\\ \frac{y}{a}& =\sin (t)\\ \end{array}$$ and $$\cos (t{)}^2+\sin (t{)}^2=1,$$ we get $${\left(\frac{x}{a}\right)}^2+{\left(\frac{y}{a}\right)}^2=1,$$ and thus $$x^2+y^2=a^2,$$

which is the standard equation of a circle centered at the origin.

Parametric plane curves

Parabola

The simplest equation for a parabola, $$y=x^2$$

can be (trivially) parameterized by using a free parameter Template:Mvar, and setting $$x=t,y=t^2\mathrm{f}\mathrm{o}\mathrm{r}-\mathrm{\infty }<t<\mathrm{\infty }.$$

Explicit equations

More generally, any curve given by an explicit equation $$y=f(x)$$

can be (trivially) parameterized by using a free parameter Template:Mvar, and setting $$x=t,y=f(t)\mathrm{f}\mathrm{o}\mathrm{r}-\mathrm{\infty }<t<\mathrm{\infty }.$$

Circle

A more sophisticated example is the following. Consider the unit circle which is described by the ordinary (Cartesian) equation $$x^2+y^2=1.$$

This equation can be parameterized as follows: $$(x,y)=(\cos (t),\sin (t))\mathrm{f}\mathrm{o}\mathrm{r}0\le t<2\pi .$$

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. In the case of the circle, such a Template:Dfn is $$\begin{array}{cc}x& =\frac{1-t^2}{1+t^2}\\ y& =\frac{2t}{1+t^2}.\end{array}$$

With this pair of parametric equations, the point Template:Math is not represented by a real value of Template:Mvar, but by the limit of Template:Mvar and Template:Mvar when Template:Mvar tends to infinity.

Ellipse

An ellipse in canonical position (center at origin, major axis along the Template:Mvar-axis) with semi-axes Template:Mvar and Template:Mvar can be represented parametrically as $$\begin{array}{cc}x& =a\cos t\\ y& =b\sin t.\end{array}$$

An ellipse in general position can be expressed as $$\begin{array}{cccccc}x=& & X_{\mathrm{c}}& +a\cos t\cos \phi & & -b\sin t\sin \phi \\ y=& & Y_{\mathrm{c}}& +a\cos t\sin \phi & & +b\sin t\cos \phi \end{array}$$

as the parameter Template:Mvar varies from Template:Math to Template:Math. Here Template:Math is the center of the ellipse, and Template:Mvar is the angle between the Template:Mvar-axis and the major axis of the ellipse.

Both parameterizations may be made rational by using the tangent half-angle formula and setting $\tan \frac{t}{2}=u.$

Lissajous curve

A Lissajous curve where Template:Math and Template:Math.

A Lissajous curve is similar to an ellipse, but the Template:Mvar and Template:Mvar sinusoids are not in phase. In canonical position, a Lissajous curve is given by $$\begin{array}{cc}x& =a\cos (k_{x}t)\\ y& =b\sin (k_{y}t)\end{array}$$ where Template:Mvar and Template:Mvar are constants describing the number of lobes of the figure.

Hyperbola

An east-west opening hyperbola can be represented parametrically by

$$\begin{array}{cc}x& =a\sec t+h\\ y& =b\tan t+k,\end{array}$$

or, rationally

$$\begin{array}{cc}x& =a\frac{1+t^2}{1-t^2}+h\\ y& =b\frac{2t}{1-t^2}+k.\end{array}$$

A north-south opening hyperbola can be represented parametrically as

$$\begin{array}{cc}x& =b\tan t+h\\ y& =a\sec t+k,\end{array}$$

or, rationally

$$\begin{array}{cc}x& =b\frac{2t}{1-t^2}+h\\ y& =a\frac{1+t^2}{1-t^2}+k.\end{array}$$

In all these formulae Template:Math are the center coordinates of the hyperbola, Template:Mvar is the length of the semi-major axis, and Template:Mvar is the length of the semi-minor axis. Note that in the rational forms of these formulae, the points Template:Math and Template:Math, respectively, are not represented by a real value of Template:Mvar, but are the limit of Template:Mvar and Template:Mvar as Template:Mvar tends to infinity.

Hypotrochoid

A hypotrochoid is a curve traced by a point attached to a circle of radius Template:Mvar rolling around the inside of a fixed circle of radius Template:Mvar, where the point is at a distance Template:Mvar from the center of the interior circle.

• 
  A hypotrochoid for which Template:Math
• 
  A hypotrochoid for which Template:Math, Template:Math, Template:Math

The parametric equations for the hypotrochoids are:

$$\begin{array}{cc}x(\theta )& =(R-r)\cos \theta +d\cos \left(\frac{R-r}{r}\theta \right)\\ y(\theta )& =(R-r)\sin \theta -d\sin \left(\frac{R-r}{r}\theta \right).\end{array}$$

Some examples:

• 
  Template:Math Template:Math Template:Math
• 
  Template:Math Template:Math Template:Math
• 
  Template:Math Template:Math Template:Math
• 
  Template:Math Template:Math Template:Math
• 
  Template:Math Template:Math Template:Math

Parametric space curves

Template:Further File:Animated Parametric Function.webm

Helix

Parametric helix

Parametric equations are convenient for describing curves in higher-dimensional spaces. For example:

$$\begin{array}{cc}x& =a\cos (t)\\ y& =a\sin (t)\\ z& =bt\end{array}$$

describes a three-dimensional curve, the helix, with a radius of Template:Mvar and rising by Template:Math units per turn. The equations are identical in the plane to those for a circle. Such expressions as the one above are commonly written as

$$\begin{array}{cc}𝐫(t)& =(x(t),y(t),z(t))\\ & =(a\cos (t),a\sin (t),bt),\end{array}$$

where Template:Math is a three-dimensional vector.

Parametric surfaces

Template:Main A torus with major radius Template:Mvar and minor radius Template:Mvar may be defined parametrically as

$$\begin{array}{cc}x& =\cos (t)(R+r\cos (u)),\\ y& =\sin (t)(R+r\cos (u)),\\ z& =r\sin (u).\end{array}$$

where the two parameters Template:Mvar and Template:Mvar both vary between Template:Math and Template:Math.

• 
  Template:Math, Template:Math

As Template:Mvar varies from Template:Math to Template:Math the point on the surface moves about a short circle passing through the hole in the torus. As Template:Mvar varies from Template:Math to Template:Math the point on the surface moves about a long circle around the hole in the torus.

Straight line

The parametric equation of the line through the point $(x_0,y_0,z_0)$ and parallel to the vector $a\widehat{𝐢}+b\widehat{𝐣}+c\widehat{𝐤}$ is

$$\begin{array}{cc}x& =x_0+at\\ y& =y_0+bt\\ z& =z_0+ct\end{array}$$

Applications

Kinematics

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. Such parametric curves can then be integrated and differentiated termwise. Thus, if a particle's position is described parametrically as $$𝐫(t)=(x(t),y(t),z(t)),$$

then its velocity can be found as $$\begin{array}{cc}𝐯(t)& ={𝐫}^{\prime }(t)\\ & =(x^{\prime }(t),y^{\prime }(t),z^{\prime }(t)),\end{array}$$

and its acceleration as $$\begin{array}{cc}𝐚(t)& ={𝐯}^{\prime }(t)={𝐫}^{″}(t)\\ & =(x^{″}(t),y^{″}(t),z^{″}(t)).\end{array}$$

Computer-aided design

Another important use of parametric equations is in the field of computer-aided design (CAD). For example, consider the following three representations, all of which are commonly used to describe planar curves.

Type	Form	Example	Description
Explicit	$y=f(x)$	$y=mx+b$	Line
Implicit	$f(x,y)=0$	${(x-a)}^2+{(y-b)}^2=r^2$	Circle
Parametric	$x=\frac{g(t)}{w(t)};$ $y=\frac{h(t)}{w(t)}$	$x=a_0+a_{1}t;$ $y=b_0+b_{1}t$	Line
		$x=a+r\cos t;$ $y=b+r\sin t$	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 under geometric transformations, and in particular under rotations. 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.

Integer geometry

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 Template:Math and their hypotenuse Template:Math are coprime integers. As Template:Mvar and Template:Mvar are not both even (otherwise Template:Math and Template:Math would not be coprime), one may exchange them to have Template:Mvar even, and the parameterization is then

$$\begin{array}{cc}a& =2mn\\ b& =m^2-n^2\\ c& =m^2+n^2,\end{array}$$

where the parameters Template:Mvar and Template:Mvar are positive coprime integers that are not both odd.

By multiplying Template:Math and Template:Mvar by an arbitrary positive integer, one gets a parametrization of all right triangles whose three sides have integer lengths.

Underdetermined linear systems

A [[system of linear equations|system of Template:Mvar linear equations]] in Template:Mvar 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 Template:Mvar and Template:Math. In this case, one can select Template:Math unknowns as parameters and represent all solutions as a parametric equation where all unknowns are expressed as linear combinations of the selected ones. That is, if the unknowns are $x_1,\dots ,x_n,$ one can reorder them for expressing the solutions as

$$\begin{array}{cc}{x}_1& ={\beta }_1+\sum _{j=r+1}^n{\alpha }_{1,j}{x}_j\\ ⋮\\ x_r& ={\beta }_r+\sum _{j=r+1}^n{\alpha }_{r,j}{x}_j\\ x_{r+1}& =x_{r+1}\\ ⋮\\ x_n& =x_n.\end{array}$$

Such a parametric equation is called a Template:Dfn of the solution of the system.

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. Then the unknowns that can be used as parameters are the ones that correspond to columns not containing any leading entry (that is the left most non zero entry in a row or the matrix), and the parametric form can be straightforwardly deduced.

See also

• Curve
• Parametric estimating
• Position vector
• Vector-valued function
• Parametrization by arc length
• Parametric derivative

External links

• Web application to draw parametric curves on the plane