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| [[File:Butterfly trans01.svg|thumb|300px|right|One example of a sketch defined by parametric equations is
| | Marvella is what you can contact her but it's not the most female title out there. One of the issues she enjoys most is to study comics and she'll be starting something else along with it. Hiring is her working day job now but she's usually needed her personal company. Puerto Rico is exactly where he's been living for years and he will never transfer.<br><br>my web blog: [http://Www.Pinaydiaries.com/blog/6662 pinaydiaries.com] |
| the [[butterfly curve (transcendental)|butterfly curve]].]]
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| In [[mathematics]], a '''parametric equation''' of a [[curve]] is a representation of this curve through [[equation]]s expressing the [[coordinates]] of the points of the curve as functions of a [[variable (mathematics)|variable]] called a [[parameter]]. For example,
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| :<math>\begin{align}
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| x&=\cos t\\
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| y&=\sin t,
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| \end{align}</math>
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| is a parametric equation for the [[unit circle]], where ''t'' is the parameter.
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|
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| A common example occurs in [[kinematics]], where the [[trajectory]] of a point is usually represented by a parametric equation with the time as parameter.
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| The notion of ''parametric equation'' has been generalized to [[surface]]s and [[manifold]]s of higher [[dimension of a manifold|dimension]], with a number of parameters, which is equal to the dimension of the manifold (dimension one and one parameter for curves, dimension two and two parameters for surfaces, etc.).
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| ==2D examples==
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| ===Parabola===
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| For example, the simplest equation for a [[parabola]],
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| :<math>y = x^2\,</math>
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| can be parametrized by using a free parameter ''t'', and setting
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| :<math>x = t, y = t^2 \quad \mathrm{for} -\infty < t < \infty.\,</math>
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| ===Circle===
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| A more sophisticated example might be the following. Consider the unit circle which is described by the ordinary (Cartesian) equation
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| :<math> x^2 + y^2 = 1.\,</math>
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| This equation can be parametrized as well, giving
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| :<math>(\cos(t),\; \sin(t))\quad\mathrm{for}\ 0\leq t < 2\pi.\,</math>
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| 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.
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| [[File:Param 02.jpg|thumb|180px|Several graphs by variation of k]]
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| ===Some sophisticated functions===
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| Other examples are shown:
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| <br>
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| :<math>x = (a - b) \cos(t)\ + b \cos(t ((a / b) - 1))</math>
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| :<math>y = (a - b) \sin(t)\ - b \sin(t ((a / b) - 1)), k = a/b </math>
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| <br>
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| :<math>x = \cos(a t) - \cos(b t)^j</math>
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| :<math>y = \sin(c t) - \sin(d t)^k </math>
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| <gallery>
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| Image:Param 03.jpg|<center>'''j=3 k=3'''</center>
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| Image:Param33 1.jpg| <center>'''j=3 k=3'''</center>
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| Image:Param34 1.jpg| <center>'''j=3 k=4'''</center>
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| Image:Param34 2.jpg| <center>'''j=3 k=4'''</center>
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| Image:Param34 3.jpg| <center>'''j=3 k=4'''</center>
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| </gallery>
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| <br>
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| :<math>x = i \cos(a t) - \cos(b t) \sin(c t)</math>
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| :<math>y = j \sin(d t) - \sin(e t)</math>
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| <gallery>
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| Image:Param st 01.jpg|<center>'''i=1 j=2'''</center>
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| </gallery>
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| ==3D examples==
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| ===Helix===
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| [[File:Parametric Helix.png|thumb|300px|right|Parametric helix]]
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| Parametric equations are convenient for describing [[curve]]s in higher-dimensional spaces. For example:
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| :<math>x = a \cos(t)\,</math>
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| :<math>y = a \sin(t)\,</math>
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| :<math>z = bt\,</math>
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| describes a three-dimensional curve, the [[helix]], which has a radius of ''a'' and rises by 2π''b'' units per turns. Note that the equations are identical in the [[plane (mathematics)|plane]] to those for a circle.
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| Such expressions as the one above are commonly written as
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| :<math>r(t) = (x(t), y(t), z(t)) = (a \cos(t), a \sin(t), b t).\,</math>
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| ==Parametric surfaces==
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| A [[torus]] with major radius ''R'' and minor radius ''r'' may be defined parametrically as
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| :<math>x = \cos(t)(R + r \cos(u)),</math>
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| :<math>y = \sin(t)(R + r \cos(u)),</math>
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| :<math>z = r \sin(u).</math>
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| where the two parameters t and u both vary between 0 and 2π.
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| <gallery>
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| File:Torus.png|R=2, r=1/2
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| </gallery>
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| As u varies from 0 to 2π the point on the surface moves about a short circle passing through the hole in the torus.
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| As t varies from 0 to 2π the point on the surface moves about a long circle around the hole in the torus.
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| ==Usefulness==
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| This way of expressing curves is practical as well as efficient; for example, one can [[Integral|integrate]] and [[derivative|differentiate]] such curves termwise. Thus, one can describe the [[velocity]] of a particle following the parametrized path of a helix as:
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| :<math>v(t) = r'(t) = (x'(t), y'(t), z'(t)) = (-a \sin(t), a \cos(t), b)\,</math>
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| and the [[acceleration]] as: | |
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| :<math>a(t) = r''(t) = (x''(t), y''(t), z''(t)) = (-a \cos(t), -a \sin(t), 0)\,</math>
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| In general, a parametric curve is a function of one independent parameter (usually denoted ''t''). For the corresponding concept with two (or more) independent parameters, see [[Parametric surface]].
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| 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=687–689}}</ref> For example, consider the following three representations, all of which are commonly used to describe [[Plane curve|planar curves]].
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| {|"class=wikitable sortable" border="1" cellpadding="8" cellspacing="1"
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| |-
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| ! scope="col" | Type
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| ! scope="col" | Form
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| ! scope="col" | Example
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| ! scope="col" | Description
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| |-
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| | 1. ''Explicit''
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| |<math>y = f(x) \,\!</math>
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| |<math>y = mx + b \,\!</math>
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| |Line
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| |-
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| |style=white-space:nowrap|''2. Implicit''
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| |<math>f(x,y) = 0 \,\!</math>
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| |<math> \left(x - a \right)^2 + \left( y - b \right)^2=r^2</math>
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| |Circle
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| |-
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| |''3. Parametric''
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| |style=white-space:nowrap|<math>x = \frac{x(t)}{w(t)}</math>; <math>y = \frac{y(t)}{w(t)}</math>
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| |<math>x = a_0 + a_1t; \,\!</math> <math> y = b_0 + b_1t\,\!</math>
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| <br /> | |
| <math>x = a+r\,\cos t; \,\!</math> <math> y = b+r\,\sin t\,\!</math>
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| |Line <br /> <br />Circle
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| |-
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| |}
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| The first two types are known as analytical or nonparametric representations of curves, and, in general tend to be unsuitable for use in CAD applications. For instance, both are dependent upon the choice of coordinate system and do not lend themselves well to [[Transformation (geometry)|geometric transformations]], such as rotations, translations, and scaling. In addition, the implicit representation is awkward for generating points on a curve because x values may be chosen which do not actually lie on the curve. These problems are eliminated by rewriting the equations in parametric form.<ref>{{cite book | last=Shah | first=Jami J. | coauthors=Martti Mantyla | 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>
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| ==Conversion from two parametric equations to a single equation==
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| Converting a set of parametric equations to a single equation involves eliminating the variable <math>t</math> from the simultaneous equations <math>x=x(t),\ y=y(t)</math>. 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. If <math>x(t)</math> and <math>y(t)</math> are rational functions then the techniques of the [[theory of equations]] such as [[resultant]]s can be used to eliminate ''t''. In some cases there is no single equation in closed form that is equivalent to the parametric equations.<ref>See [http://xahlee.org/SpecialPlaneCurves_dir/CoordinateSystem_dir/coordinateSystem.html "Equation form and Parametric form conversion"] for more information on converting from a series of parametric equations to single function.</ref>
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| To take the example of the circle of radius ''a'' [[#Examples|above]], the parametric equations
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| :<math>x = a \cos(t)\,</math>
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| :<math>y = a \sin(t)\,</math>
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| can be simply expressed in terms of ''x'' and ''y'' by way of the [[Pythagorean trigonometric identity]]:
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| :<math>x/a = \cos(t)\,</math>
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| :<math>y/a = \sin(t)\,</math>
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| :<math>\cos(t)^2 + \sin(t)^2 = 1\,\!</math>
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| :<math>\therefore (x/a)^2 + (y/a)^2 = 1,</math>
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| which is easily identifiable as a type of [[conic section]] (in this case, a circle).
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| ==See also==
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| *[[Curve]]
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| *[[Parametric estimating]]
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| *[[Position vector]]
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| *[[Vector-valued function]]
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| ==Notes==
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| <references/>
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| ==External links==
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| *{{dmoz|Science/Math/Software/Graphing/|Graphing Software}}
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| *[http://danher6.100webspace.net/curvas/ Web application to draw parametric curves on the plane]
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| {{DEFAULTSORT:Parametric Equation}}
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| [[Category:Multivariable calculus]]
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| [[Category:Equations]]
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