Parametric equation

In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters.^[1] Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, called a parametric curve and parametric surface, respectively. In such cases, the equations are collectively called a parametric representation,^[2] or parametric system,^[3] or parameterization (alternatively spelled as parametrisation) of the object.^[1]^[4]^[5]

For example, the equations ${\begin{aligned}x&=\cos t\\y&=\sin t\end{aligned}}$ form a parametric representation of the unit circle, where $t$ is the parameter: A point $(x, y)$ is on the unit circle if and only if there is a value of $t$ 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.^[1]

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 $t$ ; 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]

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

then its velocity can be found as ${\begin{aligned}\mathbf {v} (t)&=\mathbf {r} '(t)\\&=(x'(t),y'(t),z'(t))\,,\end{aligned}}$

and its acceleration as ${\begin{aligned}\mathbf {a} (t)&=\mathbf {v} '(t)=\mathbf {r} ''(t)\\&=(x''(t),y''(t),z''(t))\,.\end{aligned}}$

Computer-aided design

Another important use of parametric equations is in the field of computer-aided design (CAD).^[7] 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	$f(x,y)=0\,\!$	$\left(x-a\right)^{2}+\left(y-b\right)^{2}=r^{2}$	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
Parametric	$x={\frac {g(t)}{w(t)}};\,\!$ $y={\frac {h(t)}{w(t)}}$	$x=a+r\,\cos t;\,\!$ $y=b+r\,\sin t\,\!$	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.^[8]

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

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

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

By multiplying $a, b$ and $c$ by an arbitrary positive integer, one gets a parametrization of all right triangles whose three sides have integer lengths.

Implicitization

Converting a set of parametric equations to a single implicit equation involves eliminating the variable $t$ from the simultaneous equations $x=f(t),\ y=g(t).$ This process is called implicitization. If one of these equations can be solved for $t$ , the expression obtained can be substituted into the other equation to obtain an equation involving $x$ and $y$ 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 $x={\frac {p(t)}{r(t)}},\qquad y={\frac {q(t)}{r(t)}},$

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.

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

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

${\begin{aligned}{\frac {x}{a}}&=\cos(t)\\{\frac {y}{a}}&=\sin(t)\\\end{aligned}}$ 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.

Examples in two dimensions

Parabola

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

can be (trivially) parameterized by using a free parameter $t$ , and setting $x=t,y=t^{2}\quad \mathrm {for} -\infty <t<\infty .$

Explicit equations

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

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

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: $(x,y)=(\cos(t),\;\sin(t))\quad \mathrm {for} \ 0\leq t<2\pi .$

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 rational parameterization is ${\begin{aligned}x&={\frac {1-t^{2}}{1+t^{2}}}\\y&={\frac {2t}{1+t^{2}}}\,.\end{aligned}}$

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.

Ellipse

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 ${\begin{aligned}x&=a\,\cos t\\y&=b\,\sin t\,.\end{aligned}}$

An ellipse in general position can be expressed as ${\begin{alignedat}{4}x={}&&X_{\mathrm {c} }&+a\,\cos t\,\cos \varphi {}&&-b\,\sin t\,\sin \varphi \\y={}&&Y_{\mathrm {c} }&+a\,\cos t\,\sin \varphi {}&&+b\,\sin t\,\cos \varphi \end{alignedat}}$

as the parameter $t$ varies from $0$ to $2 π$ . Here $(X c, Y c)$ is the center of the ellipse, and $φ$ is the angle between the $x$ -axis and the major axis of the ellipse.

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

Lissajous curve

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

Hyperbola

An east-west opening hyperbola can be represented parametrically by

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

or, rationally

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

A north-south opening hyperbola can be represented parametrically as

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

or, rationally

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

In all these formulae $(h, k)$ are the center coordinates of the hyperbola, $a$ is the length of the semi-major axis, and $b$ 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 of $t$ , but are the limit of $x$ and $y$ as $t$ tends to infinity.

Hypotrochoid

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

A hypotrochoid for which $r = d$
A hypotrochoid for which $R = 5$ , $r = 3$ , $d = 5$

The parametric equations for the hypotrochoids are:

${\begin{aligned}x(\theta )&=(R-r)\cos \theta +d\cos \left({R-r \over r}\theta \right)\\y(\theta )&=(R-r)\sin \theta -d\sin \left({R-r \over r}\theta \right)\,.\end{aligned}}$

Some examples:

$R = 6$ $r = 4$ $d = 1$
$R = 7$ $r = 4$ $d = 1$
$R = 8$ $r = 3$ $d = 2$
$R = 7$ $r = 4$ $d = 2$
$R = 15$ $r = 14$ $d = 1$

Examples in three dimensions

Animated Parametric helix

Helix

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

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

describes a three-dimensional curve, the helix, with a radius of $a$ and rising by $2 π b$ 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

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

where $r$ is a three-dimensional vector.

Parametric surfaces

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

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

where the two parameters $t$ and $u$ both vary between $0$ and $2 π$ .

$R = 2$ , $r = 1/2$

As $u$ varies from $0$ to $2 π$ the point on the surface moves about a short circle passing through the hole in the torus. As $t$ varies from $0$ to $2 π$ the point on the surface moves about a long circle around the hole in the torus.

Example with vectors

The parametric equation of the line through the point $\left(x_{0},y_{0},z_{0}\right)$ and parallel to the vector $a{\hat {\mathbf {i} }}+b{\hat {\mathbf {j} }}+c{\hat {\mathbf {k} }}$ is^[9]

${\begin{aligned}x&=x_{0}+at\\y&=y_{0}+bt\\z&=z_{0}+ct\end{aligned}}$

Underdetermined linear systems

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. That is, if the unknowns are $x_{1},\ldots ,x_{n},$ one can reorder them for expressing the solutions as^[10]

${\begin{aligned}x_{1}&=\beta _{1}+\sum _{j=r+1}^{n}\alpha _{1,j}x_{j}\\\vdots \\x_{r}&=\beta _{r}+\sum _{j=r+1}^{n}\alpha _{r,j}x_{j}\\x_{r+1}&=x_{r+1}\\\vdots \\x_{n}&=x_{n}.\end{aligned}}$

Such a parametric equation is called a parametric form of the solution of the system.^[10]

The standard method for computing a parametric form of the solution it 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.^[10]

Notes

^ ^a ^b ^c Weisstein, Eric W. "Parametric Equations". MathWorld.
^ Kreyszig, Erwin (1972). Advanced Engineering Mathematics (3rd ed.). New York: Wiley. pp. 291, 342. ISBN 0-471-50728-8.
^ Burden, Richard L.; Faires, J. Douglas (1993). Numerical Analysis (5th ed.). Boston: Brookes/Cole. p. 149. ISBN 0-534-93219-3.
^ Thomas, George B.; Finney, Ross L. (1979). Calculus and Analytic Geometry (fifth ed.). Addison-Wesley. p. 91.
^ Nykamp, Duane. "Plane parametrization example". mathinsight.org. Retrieved 2017-04-14.
^ Spitzbart, Abraham (1975). Calculus with Analytic Geometry. Gleview, IL: Scott, Foresman and Company. ISBN 0-673-07907-4. Retrieved August 30, 2015.
^ Stewart, James (2003). Calculus (5th ed.). Belmont, CA: Thomson Learning, Inc. pp. 687–689. ISBN 0-534-39339-X.
^ Shah, Jami J.; Martti Mantyla (1995). Parametric and feature-based CAD/CAM: concepts, techniques, and applications. New York, NY: John Wiley & Sons, Inc. pp. 29–31. ISBN 0-471-00214-3.
^ Calculus: Single and Multivariable. John Wiley. 2012-10-29. p. 919. ISBN 9780470888612. OCLC 828768012.
^ ^a ^b ^c Anton, Howard; Rorres, Chris (2014) [1973]. "1.2 Gaussian Elimination". Elementary Linear Algebra (11th ed.). Wiley. pp. 11–24.

External links

[MathWorld-1] Weisstein, Eric W. "Parametric Equations". MathWorld.

[2] Kreyszig, Erwin (1972). Advanced Engineering Mathematics (3rd ed.). New York: Wiley. pp. 291, 342. ISBN 0-471-50728-8.

[3] Burden, Richard L.; Faires, J. Douglas (1993). Numerical Analysis (5th ed.). Boston: Brookes/Cole. p. 149. ISBN 0-534-93219-3.

[4] Thomas, George B.; Finney, Ross L. (1979). Calculus and Analytic Geometry (fifth ed.). Addison-Wesley. p. 91.

[5] Nykamp, Duane. "Plane parametrization example". mathinsight.org. Retrieved 2017-04-14.

[6] Spitzbart, Abraham (1975). Calculus with Analytic Geometry. Gleview, IL: Scott, Foresman and Company. ISBN 0-673-07907-4. Retrieved August 30, 2015.

[7] Stewart, James (2003). Calculus (5th ed.). Belmont, CA: Thomson Learning, Inc. pp. 687–689. ISBN 0-534-39339-X.

[8] Shah, Jami J.; Martti Mantyla (1995). Parametric and feature-based CAD/CAM: concepts, techniques, and applications. New York, NY: John Wiley & Sons, Inc. pp. 29–31. ISBN 0-471-00214-3.

[9] Calculus: Single and Multivariable. John Wiley. 2012-10-29. p. 919. ISBN 9780470888612. OCLC 828768012.

[anton-10] Anton, Howard; Rorres, Chris (2014) [1973]. "1.2 Gaussian Elimination". Elementary Linear Algebra (11th ed.). Wiley. pp. 11–24.

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