Parametric equation

{{short description|Representation of a curve by a function of a parameter}} [[File:Butterfly transcendental curve.svg|thumb|right|The butterfly curve can be defined by parametric equations of {{mvar|x}} and {{mvar|y}}.]] In mathematics, a '''parametric equation''' expresses several quantities, such as the coordinates of a point, as functions of one or more variables called parameters.<ref name="MathWorld">{{Cite web |last=Weisstein |first=Eric W. |title=Parametric Equations |website=MathWorld |url=http://mathworld.wolfram.com/ParametricEquations.html}}</ref>

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''',<ref>{{ cite book | last1 = Kreyszig | first1 = Erwin | authorlink = Erwin Kreyszig | title = Advanced Engineering Mathematics | edition = 3rd | location = New York | publisher = Wiley | year = 1972 | isbn = 0-471-50728-8 | pages=291, 342 }}</ref> or '''parametric system''',<ref>{{cite book | first1 = Richard L. | last1 = Burden | first2 = J. Douglas | last2 = Faires | year = 1993 | isbn = 0-534-93219-3 | title = Numerical Analysis | edition = 5th | publisher = Brookes/Cole | location = Boston | page=149 }}</ref> or '''parameterization''' (also spelled '''parametrization''', '''parametrisation''') of the object.<ref name="MathWorld" /><ref>{{Cite book|last=Thomas|first=George B. |last2=Finney |first2=Ross L.|title=Calculus and Analytic Geometry |publisher=Addison-Wesley|edition=fifth |year=1979 |page=91}}</ref><ref>{{Cite web|url=http://mathinsight.org/plane_parametrization_examples|title=Plane parametrization example|last=Nykamp|first=Duane|website=mathinsight.org|access-date=2017-04-14}}</ref>

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 {{mvar|t}} is the parameter: A point {{math|(''x'', ''y'')}} is on the unit circle if and only if there is a value of {{mvar|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:

Parametric representations are generally not unique (see {{slink||Parametric plane curves}}), so the same quantities may be expressed by a number of different parameterizations.<ref name="MathWorld" />

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 {{mvar|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.<ref>{{cite book |last=Spitzbart |first=Abraham |title=Calculus with Analytic Geometry |url=https://archive.org/details/calculuswithanal0000spit |access-date=August 30, 2015 |year=1975 |publisher=Scott, Foresman and Company |location=Gleview, IL |isbn=0-673-07907-4 |url-access=registration }}</ref>

==Implicitization== Converting a set of parametric equations to a single implicit equation involves eliminating the variable {{mvar|t}} from the simultaneous equations <math>x=f(t),\ y=g(t).</math> This process is called '''{{dfn|implicitization}}'''. If one of these equations can be solved for {{mvar|t}}, the expression obtained can be substituted into the other equation to obtain an equation involving {{mvar|x}} and {{mvar|y}} 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 {{mvar|p}}, {{mvar|q}}, and {{mvar|r}} are set-wise coprime polynomials, a resultant computation allows one to implicitize. More precisely, the implicit equation is the resultant with respect to {{mvar|t}} of {{math|''xr''(''t'') – ''p''(''t'')}} and {{math|''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 {{slink|Gröbner basis|Implicitization in higher dimension}}.

To take the example of the circle of radius {{mvar|a}}, 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 {{math|x}} and {{math|y}} 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 <math display="block">\cos(t)^2 + \sin(t)^2 = 1,</math> we get <math display="block">\left(\frac{x}{a}\right)^2 + \left(\frac{y}{a}\right)^2 = 1,</math> and thus <math display="block">x^2+y^2=a^2,</math>

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

==Parametric plane curves== {{further|Plane curve}}

===Parabola===

The simplest equation for a parabola, <math display="block">y = x^2</math>

can be (trivially) parameterized by using a free parameter {{mvar|t}}, 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 <math display="block">y = f(x)</math>

can be (trivially) parameterized by using a free parameter {{mvar|t}}, 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 <math display="block"> x^2 + y^2 = 1.</math>

This equation can be parameterized as follows: <math display="block">(x,y)=(\cos(t),\; \sin(t))\quad\text{ for } 0\leq t < 2\pi.</math>

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 ''{{dfn|rational parameterization}}'' 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 {{math|(−1, 0)}} is not represented by a real value of {{mvar|t}}, but by the limit of {{mvar|x}} and {{mvar|y}} when {{mvar|t}} tends to infinity.

===Ellipse===

An ellipse in canonical position (center at origin, major axis along the {{mvar|x}}-axis) with semi-axes {{mvar|a}} and {{mvar|b}} 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 {{mvar|t}} varies from {{math|0}} to {{math|2''{{pi}}''}}. Here {{math|(''X''{{sub|c}} , ''Y''{{sub|c}})}} is the center of the ellipse, and {{mvar|φ}} is the angle between the {{mvar|x}}-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 {{math|1=''k{{sub|x}}'' = 3}} and {{math|1=''k{{sub|y}}'' = 2}}. A Lissajous curve is similar to an ellipse, but the {{mvar|x}} and {{mvar|y}} 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 {{mvar|k{{sub|x}}}} and {{mvar|k{{sub|y}}}} 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 {{math|(''h'' , ''k'')}} are the center coordinates of the hyperbola, {{mvar|a}} is the length of the semi-major axis, and {{mvar|b}} is the length of the semi-minor axis. Note that in the rational forms of these formulae, the points {{math|(''−a'' , 0)}} and {{math|(0 , ''−a'')}}, respectively, are not represented by a real value of {{mvar|t}}, but are the limit of {{mvar|x}} and {{mvar|y}} as {{mvar|t}} tends to infinity.

===Hypotrochoid=== A hypotrochoid is a curve traced by a point attached to a circle of radius {{mvar|r}} rolling around the inside of a fixed circle of radius {{mvar|R}}, where the point is at a distance {{mvar|d}} from the center of the interior circle. <gallery> Image:2-circles hypotrochoid.gif|<div class="center">A hypotrochoid for which {{math|1= ''r'' = ''d''}}</div> Image:HypotrochoidOutThreeFifths.gif|<div class="center">A hypotrochoid for which {{math|1= ''R'' = 5}}, {{math|1= ''r'' = 3}}, {{math|1= ''d'' = 5}}</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: <gallery> Image: Param1a 6 4 1 a2.svg|<div class="center">{{math|1= ''R'' = 6}} {{math|1= ''r'' = 4}} {{math|1= ''d'' = 1}}</div> Image: Param1a 7 4 1 a4.svg|<div class="center">{{math|1= ''R'' = 7}} {{math|1= ''r'' = 4}} {{math|1= ''d'' = 1}}</div> Image: Param1a 8 3 2 a3.svg|<div class="center">{{math|1= ''R'' = 8}} {{math|1= ''r'' = 3}} {{math|1= ''d'' = 2}}</div> Image: Param1a 7 4 2 a4.svg|<div class="center">{{math|1= ''R'' = 7}} {{math|1= ''r'' = 4}} {{math|1= ''d'' = 2}}</div> Image: Param1a 15 14 1 a14.svg|<div class="center">{{math|1= ''R'' = 15}} {{math|1= ''r'' = 14}} {{math|1= ''d'' = 1}}</div> </gallery>

==Parametric space curves== {{further|Space curve}} 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 {{mvar|a}} and rising by {{math|2''{{pi}}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

<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 {{math|'''r'''}} is a three-dimensional vector.

===Parametric surfaces=== {{main| Parametric surface}} A torus with major radius {{mvar|R}} and minor radius {{mvar|r}} 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 {{mvar|t}} and {{mvar|u}} both vary between {{math|0}} and {{math|2''{{pi}}''}}.

<gallery> File:Torus.svg|{{math|1=''R'' = 2}}, {{math|1=''r'' = 1/2}} </gallery>

As {{mvar|u}} varies from {{math|0}} to {{math|2π}}, the point on the surface moves about a short circle passing through the hole in the torus. As {{mvar|t}} varies from {{math|0}} to {{math|2π}}, the point on the surface moves about a long circle around the hole in the torus.

===Straight line=== {{further|Linear equation}}

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<ref>{{Cite book|title=Calculus: Single and Multivariable.|date=2012-10-29|publisher=John Wiley|isbn=9780470888612|oclc=828768012|pages=919}}</ref>

<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).<ref>{{cite book | last=Stewart | first=James | year=2003 | title=Calculus | edition=5th | publisher=Thomson Learning, Inc. | location=Belmont, CA | isbn=0-534-39339-X | pages=[https://archive.org/details/calculus0000stew/page/687 687–689] | title-link=Calculus }}</ref> 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.<ref>{{cite book | last=Shah | first=Jami J. |author2=Martti Mantyla |author-link1=Jami Shah | year=1995 | title=Parametric and feature-based CAD/CAM: concepts, techniques, and applications | publisher=John Wiley & Sons, Inc. | location=New York, NY | isbn=0-471-00214-3 | pages= 29–31 }}</ref>

===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 {{math|''a'', ''b''}} and their hypotenuse {{math|''c''}} are coprime integers. As {{mvar|a}} and {{mvar|b}} are not both even (otherwise {{math|''a'', ''b''}} and {{math|''c''}} would not be coprime), one may exchange them to have {{mvar|a}} 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 {{mvar|m}} and {{mvar|n}} are positive coprime integers that are not both odd.

By multiplying {{math|''a'', ''b''}} and {{mvar|c}} 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 {{mvar|m}} linear equations in {{mvar|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 {{mvar|r}} and {{math|''r'' < ''n''}}. In this case, one can select {{math|''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 <math>x_1, \ldots, x_n,</math> one can reorder them for expressing the solutions as<ref name=anton> {{cite book |last=Anton |first1=Howard |last2=Rorres |first2=Chris |year=2014 |orig-year=1973 |title=Elementary Linear Algebra |edition=11th |publisher=Wiley |chapter=1.2 Gaussian Elimination |pages=11–24 |chapter-url=https://archive.org/details/LinearAlgebraAntonApplicationsVersion11thC2014Txtbk/page/n24/}}</ref>

<math display="block"> \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 '''{{dfn|parametric form}}''' of the solution of the system.<ref name=anton/>

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.<ref name=anton/>

==Notes== <references/>

==External links== *[http://danher6.100webspace.net/curvas/ Web application to draw parametric curves on the plane]

{{DEFAULTSORT:Parametric Equation}} Category:Multivariable calculus Category:Equations Category:Geometry processing