Editing Parametric equation (section)

{{short description|Representation of a curve by a function of a parameter}}
[[File:Butterfly transcendental curve.svg|thumb|right|The [[butterfly curve (transcendental)|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 (mathematics)|point]], as [[Function (mathematics)|functions]] of one or several [[variable (mathematics)|variable]]s called [[parameter]]s.<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, 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 (mathematics)|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 = [[John Wiley & Sons|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 = [[Cengage Learning|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 (mathematics)|scalar]] output variables are combined into a single parametric equation in [[Euclidean vector|vectors]]:

<math display="block">(x, y)=(\cos t, \sin t).</math>

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.<ref name="MathWorld" />

In addition to curves and surfaces, parametric equations can describe [[manifold]]s and [[algebraic variety|algebraic varieties]] of higher [[dimension of a manifold|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>