- Parametric equation
In

butterfly curve.mathematics ,**parametric equations**are a method of defining a curve. A simple kinematical example is when one uses a time parameter to determine theposition ,velocity , and other information about a body in motion.Abstractly, a relation is given in the form of an

equation , and it is shown also to be the image of functions from items such as "R"^{"n"}. It is therefore somewhat more accurately defined as a**parametric representation**. It is part ofregular parametric representation .**Examples**For example, the simplest equation for a

parabola ,:$y\; =\; x^2,$

can be parametrized by using a free parameter "t", and setting

:$x\; =\; t,$:$y\; =\; t^2,$

Although the preceding example appears somewhat trivial, consider the following parametrization of a

circle ofradius "a"::$x\; =\; a\; cos(t),$:$y\; =\; a\; sin(t),$

Parametric equations are convenient for describing

curve s in higher-dimensional spaces. For example::$x\; =\; a\; cos(t),$:$y\; =\; a\; sin(t),$:$z\; =\; bt,$

describes a three-dimensional curve, the

helix , which has a radius of "a" and rises by 2π"b" units per turn. (Note that the equations are identical in the plane to those for a circle; in fact, a helix is just "a circle whose ends don't have the same "z"-value".)Such expressions as the one above are commonly written as

:$r(t)\; =\; (x(t),\; y(t),\; z(t))\; =\; (a\; cos(t),\; a\; sin(t),\; b\; t),$

This way of expressing curves is practical as well as efficient; for example, one can integrate and differentiate such curves termwise. Thus, one can describe the

velocity of a particle following such a parametrized path as::$v(t)\; =\; r\text{'}(t)\; =\; (x\text{'}(t),\; y\text{'}(t),\; z\text{'}(t))\; =\; (-a\; sin(t),\; a\; cos(t),\; b),$

and the

acceleration as::$a(t)\; =\; r"(t)\; =\; (x"(t),\; y"(t),\; z"(t))\; =\; (-a\; cos(t),\; -a\; sin(t),\; 0),$

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, seeParametric surface .**Conversion from two parametric equations to a single equation**Converting a set of parametric equations to a single equation involves solving one of the equations (usually the simplest of the two) for the parameter. Then the solution of the parameter is substituted into the remaining equation, and the resulting equation is usually simplified. It should be noted that the parameter is "never" present when the equation is in singular form (i.e., it must "cancel out" during conversion). Or, the process put simply: the

simultaneous equations need to be solved for the parameter, and the result will be one equation. Additional steps need to be performed if there are restrictions on the value of the parameter. [*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 parametrics equations to single function.**ee also***

Curve

*Parametric estimating

*Parametric surface

*Position vector

*Vector-valued function **Notes****External links***dmoz|Science/Math/Software/Graphing/|Graphing Software

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**parametric equation**— parametrinė lygtis statusas T sritis fizika atitikmenys: angl. parametric equation vok. Parametergleichung, f rus. параметрическое уравнение, n pranc. équation paramétrique, f … Fizikos terminų žodynas**parametric equation**— noun Date: 1909 any of a set of equations that express the coordinates of the points of a curve as functions of one parameter or that express the coordinates of the points of a surface as functions of two parameters … New Collegiate Dictionary**parametric equation**— Math. one of two or more equations expressing the location of a point on a curve or surface by determining each coordinate separately. [1905 10] * * * … Universalium**parametric equation**— noun a set of equations that defines the coordinates of the dependent variables (x, y and z) of a curve or surface in terms of one or more independent variables or parameters … Wiktionary**parametric equation**— paramet′ric equa′tion n. math. one of two or more equations expressing the location of a point on a curve or surface by determining each coordinate separately • Etymology: 1905–10 … From formal English to slang**parametric equation**— /pærəˌmɛtrɪk əˈkweɪʒən/ (say paruh.metrik uh kwayzhuhn) noun one of two or more mathematical equations in which the coordinates of points on a curve or surface are given in terms of one or more parameters (def. 3) of that curve or surface … Australian English dictionary**parametric equation**— noun : any of a set of equations that express the coordinates of the points of a curve as functions of one parameter or that express the coordinates of the points of a surface as functions of two parameters * * * Math. one of two or more… … Useful english dictionary**Parametric**— may refer to:*Parametric equation *Parametric statistics *Parametric derivative *Parametric plot *Parametric model *Parametric resonance *Parametric contract *Parametric insurance *Parametric feature based modeler *Spontaneous parametric down… … Wikipedia**Parametric surface**— A parametric surface is a surface in the Euclidean space R3 which is defined by a parametric equation with two parameters. Parametric representation is the most general way to specify a surface. Surfaces that occur in two of the main theorems of… … Wikipedia**Parametric derivative**— In calculus, a parametric derivative is a derivative that is taken when both the x and y variables (traditionally independent and dependent, respectively) depend on an independent third variable t , usually thought of as time .For example,… … Wikipedia