Parametric equation

thumb|right|The [[butterfly curve (transcendental)|butterfly curve can be defined by parametric equations of and .]]

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

In the case of a single parameter, parametric equations are commonly used to express the trajectory of a moving point. For this 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

<math display="block">\begin{align}

x &= \cos t \\

y &= \sin t

\end{align}</math>

form a parametric representation of the unit circle, where is the parameter: A point is on the unit circle if and only if there is a value of 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:

Parametric representations are generally not unique (see ), so the same quantities may be expressed by a number of different parameterizations.

Implicitization

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

If the parametrization is given by rational functions

<math display="block">x=\frac{p(t)}{r(t)},\qquad y=\frac{q(t)}{r(t)},</math>

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

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 .

To take the example of the circle of radius , the parametric equations

<math display="block">\begin{align}

x &= a \cos(t) \\

y &= a \sin(t)

\end{align}</math>

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

<math display="block">\begin{align}

\frac{x}{a} &= \cos(t) \\

\frac{y}{a} &= \sin(t) \\

\end{align}</math>

and

we get

<math display="block">\left(\frac{x}{a}\right)^2 + \left(\frac{y}{a}\right)^2 = 1,</math>

and thus

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

Parametric plane curves

Parabola

The simplest equation for a parabola,

can be (trivially) parameterized by using a free parameter , and setting

<math display="block">x = t, y = t^2 \quad \mathrm{for} -\infty < t < \infty.</math>

Explicit equations

More generally, any curve given by an explicit equation

can be (trivially) parameterized by using a free parameter , and setting

<math display="block">x = t, y = f(t) \quad \mathrm{for} -\infty < t < \infty.</math>

Circle

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 only rational functions (that is fractions of two polynomials) are preferred, if they exist. In the case of the circle, such a ' is

<math display="block">\begin{align}

x &= \frac{1 - t^2}{1 + t^2} \\[6pt]

y &= \frac{2t}{1 + t^2}\,.

\end{align}</math>

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

Ellipse

An ellipse in canonical position (center at origin, major axis along the -axis) with semi-axes and can be represented parametrically as

<math display="block">\begin{align}

x &= a\,\cos t \\

y &= b\,\sin t\,.

\end{align}</math>

An ellipse in general position can be expressed as

<math display="block">\begin{alignat}{4}

x ={}&& X_\mathrm{c} &+ a\,\cos t\,\cos \varphi {}&&- b\,\sin t\,\sin\varphi \\

y ={}&& Y_\mathrm{c} &+ a\,\cos t\,\sin \varphi {}&&+ b\,\sin t\,\cos\varphi

\end{alignat}</math>

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

Both parameterizations may be made rational by using the tangent half-angle formula and setting <math display="inline">\tan\frac{t}{2} = u\,.</math>

Lissajous curve

thumbnail|A Lissajous curve where and .

A Lissajous curve is similar to an ellipse, but the and sinusoids are not in phase. In canonical position, a Lissajous curve is given by

<math display="block">\begin{align}

x &= a\,\cos(k_xt) \\

y &= b\,\sin(k_yt)

\end{align}</math>

where and are constants describing the number of lobes of the figure.

Hyperbola

An east-west opening hyperbola can be represented parametrically by

<math display="block">\begin{align}

x &= a\sec t + h \\

y &= b\tan t + k\,,

\end{align}</math>

or, rationally

<math display="block">\begin{align}

x &= a\frac{1 + t^2}{1 - t^2} + h \\

y &= b\frac{2t}{1 - t^2} + k\,.

\end{align}</math>

A north-south opening hyperbola can be represented parametrically as

<math display="block">\begin{align}

x &= b\tan t + h \\

y &= a\sec t + k\,,

\end{align}</math>

or, rationally

<math display="block">\begin{align}

x &= b\frac{2t}{1 - t^2} + h \\

y &= a\frac{1 + t^2}{1 - t^2} + k\,.

\end{align}</math>

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

Hypotrochoid

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

Image:2-circles hypotrochoid.gif|<div class="center">A hypotrochoid for which </div>

Image:HypotrochoidOutThreeFifths.gif|<div class="center">A hypotrochoid for which , , </div>

</gallery>

The parametric equations for the hypotrochoids are:

<math display="block">\begin{align}

x (\theta) &= (R - r)\cos\theta + d\cos\left({R - r \over r}\theta\right) \\

y (\theta) &= (R - r)\sin\theta - d\sin\left({R - r \over r}\theta\right)\,.

\end{align}</math>

Some examples:

Image: Param1a 6 4 1 a2.svg|<div class="center"> </div>

Image: Param1a 7 4 1 a4.svg|<div class="center"> </div>

Image: Param1a 8 3 2 a3.svg|<div class="center"> </div>

Image: Param1a 7 4 2 a4.svg|<div class="center"> </div>

Image: Param1a 15 14 1 a14.svg|<div class="center"> </div>

</gallery>

Parametric space curves

thumb|440x440px|Animated Parametric helix

Helix

thumb|300px|right|Parametric helix

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

<math display="block">\begin{align}

x &= a \cos(t) \\

y &= a \sin(t) \\

z &= bt\,

\end{align}</math>

describes a three-dimensional curve, the helix, with a radius of and rising by 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

<math display="block">\begin{align}

\mathbf{r}(t) &= (x(t), y(t), z(t)) \\

&= (a \cos(t), a \sin(t), b t)\,,

\end{align}</math>

where is a three-dimensional vector.

Parametric surfaces

A torus with major radius and minor radius may be defined parametrically as

<math display="block">\begin{align}

x &= \cos(t)\left(R + r \cos(u)\right), \\

y &= \sin(t)\left(R + r \cos(u)\right), \\

z &= r \sin(u)\,.

\end{align}</math>

where the two parameters and both vary between and .

File:Torus.svg|,

</gallery>

As varies from to , the point on the surface moves about a short circle passing through the hole in the torus. As varies from to , 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 <math>\left( x_0, y_0, z_0 \right)</math> and parallel to the vector <math> a \hat\mathbf{i} + b \hat\mathbf{j} + c \hat\mathbf{k}</math> is

<math display="block">\begin{align}

x & = x_0 +a t \\

y & = y_0 +b t \\

z & = z_0 +c t

\end{align}</math>

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

<math display="block">\mathbf{r}(t) = (x(t), y(t), z(t))\,,</math>

then its velocity can be found as

<math display="block">\begin{align}

\mathbf{v}(t) &= \mathbf{r}'(t) \\

&= (x'(t), y'(t), z'(t))\,,

\end{align}</math>

and its acceleration as

<math display="block">\begin{align}

\mathbf{a}(t) &= \mathbf{v}'(t) = \mathbf{r}(t) \\

&= (x(t), y(t), z(t))\,.

\end{align}</math>

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.

{| class="wikitable"

! scope="col" | Type

! scope="col" | Form

! scope="col" | Example

! scope="col" | Description

! scope="row" | Explicit

|<math>y = f(x) \,\!</math>

|<math>y = mx + b \,\!</math>

|Line

! scope="row" | Implicit

|<math>f(x,y) = 0 \,\!</math>

|<math> \left(x - a \right)^2 + \left( y - b \right)^2=r^2</math>

|Circle

! scope="row" rowspan=2 | Parametric

|rowspan=2 |<math>x = \frac{g(t)}{w(t)}; \,\!</math> <math>y = \frac{h(t)}{w(t)}</math>

|<math>x = a_0 + a_1t; \,\!</math> <math>y = b_0 + b_1t\,\!</math>

|Line

|<math>x = a+r\,\cos t; \,\!</math> <math>y = b+r\,\sin t\,\!</math>

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

<math display="block">\begin{align}

a &= 2mn \\

b &= m^2 - n^2 \\

c &= m^2 + n^2\,,

\end{align}</math>

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

By multiplying and 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 in 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 and . In this case, one can select 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 <math>x_1, \ldots, x_n,</math> one can reorder them for expressing the solutions as

\begin{align}

x_1 &= \beta_1+\sum_{j=r+1}^n \alpha_{1,j}x_j\\

\vdots\\

x_r &= \beta_r+\sum_{j=r+1}^n \alpha_{r,j}x_j\\

x_{r+1} &= x_{r+1}\\

\vdots\\

x_n &= x_n.

\end{align}

</math>

Such a parametric equation is called a of the solution of the system.