Parametric Equation - Wikipedia, The Free Encyclopedia

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https://en.wikipedia.org/wiki/Parametric_equation 1/9

The butterfly curve can be defined by parametric

equations of x and y.

Parametric equationFrom Wikipedia,the free encyclopedia

In mathematics, parametric equationsdefine a

group of quantities as functions of one or more

independent variables called parameters.[1]

Parametricequations are commonlyused to expressthe coordinates of the points that make up a

geometric object such as a curveor surface, in which

case the equations are collectively called a

parametric representationor parameterizationof

the object.[2][3]For example, the equations

form a parametric representation of the unit circle,where tis the parameter.

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 oneand oneparameter is used, for

surfaces dimension twoand twoparameters, etc.).

Parametric equations are commonlyused 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 otherphysical quantities (such as geometricvariables) or

can be selected arbitrarily for convenience. Parameterizations are non-unique more thanone set of

parametric equations can specify the same curve.[4]

Contents

1 Applications1.1 Kinematics1.2 Computer-aided design1.3 Integer geometry

2 Implicitization3 Examples in two dimensions

3.1 Parabola3.2 Explicit equations3.3 Circle

3.4 Ellipse3.5 Lissajous Curve3.6 Hyperbola3.7 Hypotrochoid3.8 Some sophisticated functions
https://en.wikipedia.org/wiki/Algebraic_varietyhttps://en.wikipedia.org/wiki/Dimension_of_a_manifoldhttps://en.wikipedia.org/wiki/Algebraic_varietyhttps://en.wikipedia.org/wiki/Butterfly_curve_(transcendental)https://en.wikipedia.org/wiki/Butterfly_curve_(transcendental)https://en.wikipedia.org/wiki/Unit_circlehttps://en.wikipedia.org/wiki/File:Butterfly_trans01.svghttps://en.wikipedia.org/wiki/File:Butterfly_trans01.svghttps://en.wikipedia.org/wiki/File:Butterfly_trans01.svghttps://en.wikipedia.org/wiki/File:Butterfly_trans01.svghttps://en.wikipedia.org/wiki/File:Butterfly_trans01.svghttps://en.wikipedia.org/wiki/Mathematicshttps://en.wikipedia.org/wiki/Trajectoryhttps://en.wikipedia.org/wiki/Kinematicshttps://en.wikipedia.org/wiki/Dimension_of_a_manifoldhttps://en.wikipedia.org/wiki/Algebraic_varietyhttps://en.wikipedia.org/wiki/Manifoldhttps://en.wikipedia.org/wiki/Unit_circlehttps://en.wikipedia.org/wiki/Surface_(mathematics)https://en.wikipedia.org/wiki/Curvehttps://en.wikipedia.org/wiki/Coordinateshttps://en.wikipedia.org/wiki/Parameterhttps://en.wikipedia.org/wiki/Independent_variableshttps://en.wikipedia.org/wiki/Function_(mathematics)https://en.wikipedia.org/wiki/Mathematicshttps://en.wikipedia.org/wiki/Butterfly_curve_(transcendental)https://en.wikipedia.org/wiki/File:Butterfly_trans01.svg


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4 Examples in three dimensions4.1 Helix4.2 Parametric surfaces

5 See also6 Notes7 External links

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

then its velocity can be found as

and its acceleration as

.

Computer-aided design

Another important use of parametric equations is in the field of computer-aided design (CAD).[5]For

example, consider the following three representations, all of which are commonly used to describe

planar curves.

Type Form Example Description

1.Explicit Line

2. Implicit Circle

3. Parametric Line

Circle

The first two types are known as analytical or non-parametric representations of curves, and, in general

tend to be unsuitable for use in CAD applications. For instance, the first one is dependent upon the

choice of a coordinate system and does not lend itself well to geometric transformations, such asrotations, translations, and scaling. In addition, with the implicit representation, it is more difficult of

generating points on a curve. These problems are made easier by rewriting the equations in parametric

form.[6]
https://en.wikipedia.org/wiki/Transformation_(geometry)https://en.wikipedia.org/wiki/Plane_curvehttps://en.wikipedia.org/wiki/Computer-aided_designhttps://en.wikipedia.org/wiki/Accelerationhttps://en.wikipedia.org/wiki/Velocityhttps://en.wikipedia.org/wiki/Derivativehttps://en.wikipedia.org/wiki/Integralhttps://en.wikipedia.org/wiki/Vector-valued_functionhttps://en.wikipedia.org/wiki/Kinematics


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Integer geometry

Numerous problems in integer geometry can be solved using parametric equations. The most widely

known such solution is Euclid's solution in integers for the legs a, band the hypotenuse cof a primitive

right triangle:

which is parametric on the coprime integers mand nof opposite parity.

Implicitization

Converting a set of parametric equations to a single equation involves eliminating the variable from the

simultaneous equations 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 involvingxandyonly.

If the parametrization is given by rational functions

wherep, q, rare set-wise coprime polynomials, a resultant computation allows to implicitize. Moreprecisely, the implicit equation is the resultant with respect to tofxr(t) p(t)andyr(t) q(t)

In higher dimension (either more than two coordinates of more than one parameter), the implicitization

of rational parametric equations may by done with Grbner basis computation see Grbner basis Implicitization in higher dimension.

In some cases there is no single equation in closed form that is equivalent to the parametric equations. [7]

To take the example of the circle of radius aabove, the parametric equations

can be simply expressed in terms ofxandyby way of the Pythagorean trigonometric identity:

which is easily identifiable as a type of conic section (in this case, a circle).

Examples in two dimensions
https://en.wikipedia.org/wiki/Conic_sectionhttps://en.wikipedia.org/wiki/Pythagorean_trigonometric_identityhttps://en.wikipedia.org/wiki/Gr%C3%B6bner_basis#Implicitization_in_higher_dimensionhttps://en.wikipedia.org/wiki/Gr%C3%B6bner_basishttps://en.wikipedia.org/wiki/Resultanthttps://en.wikipedia.org/wiki/Coprimehttps://en.wikipedia.org/wiki/Rational_functionhttps://en.wikipedia.org/wiki/Parity_(mathematics)https://en.wikipedia.org/wiki/Coprimehttps://en.wikipedia.org/wiki/Right_trianglehttps://en.wikipedia.org/wiki/Euclidhttps://en.wikipedia.org/wiki/Integer_triangle


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Parabola

The simplest equation for a parabola,

can be (trivially) parameterized by using a free parameter t, and setting

Explicit equations

More generally, any curve given by an explicit equation

can be (trivially) parameterized by using a free parameter t, and setting

Circle

A more sophisticated example is the following. Consider the unit circle which is described by the

ordinary (Cartesian) equation

This equation can be parameterized as follows:

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 parameterizationis

With this parametric equation, the point (-1, 0)is not represented by a real value of t, but by the limit ofandywhen ttends to infinity.

Ellipse

An ellipse in canonical position (center at origin, major axis along the X-axis) with semi-axes aand bcan be represented parametrically as
https://en.wikipedia.org/wiki/Ellipsehttps://en.wikipedia.org/wiki/Infinityhttps://en.wikipedia.org/wiki/Limit_(mathematics)https://en.wikipedia.org/wiki/Real_numberhttps://en.wikipedia.org/wiki/Polynomialhttps://en.wikipedia.org/wiki/Rational_functionhttps://en.wikipedia.org/wiki/Parabola


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A Lissajous curve where and

.

An ellipse in general position can be expressed as

as the parameter tvaries from 0 to 2. Here is the center of the ellipse, and is the angle

between the -axis and the major axis of the ellipse.

Both parametrizations may be made rational by using tangent half-angle formula and setting

Lissajous Curve

A Lissajous curve is similar to an ellipse, but the x and y

sinusoids are not in phase. In canonical position, a Lissajouscurve is given by

where and are constants describing the number of lobes of

the figure.

Hyperbola

An east-west opening hyperbola can be represented

parametrically by

or, rationally

A north-south opening hyperbola can be represented parametrically as

In all formulae (h,k) are the center coordinates of the hyperbola, ais the length of the semi-major axis,

and bis the length of the semi-minor axis.

Hypotrochoid

A hypotrochoid is a curve traced by a point attached to a circle of radius rrolling around the inside of a

fixed circle of radiusR, where the point is at a distance dfrom the center of the interior circle.
https://en.wikipedia.org/wiki/Hypotrochoidhttps://en.wikipedia.org/wiki/Rational_functionhttps://en.wikipedia.org/wiki/Hyperbolahttps://en.wikipedia.org/wiki/Lissajous_curvehttps://en.wikipedia.org/wiki/Tangent_half-angle_formulahttps://en.wikipedia.org/wiki/Rational_functionhttps://en.wikipedia.org/wiki/File:Lissajous_curve_3by2.svg


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Several graphs by variation ofk

A hypotrochoid forwhich r= d

A hypotrochoid forwhichR= 5, r= 3, d=

5

The parametric equations for the hypotrochoids are:

Some sophisticated functions

Other examples are shown:

j=3 k=3

j=3 k=3

j=3 k=4

j=3 k=4

j=3 k=4
https://en.wikipedia.org/wiki/File:Param34_3.jpghttps://en.wikipedia.org/wiki/File:Param34_2.jpghttps://en.wikipedia.org/wiki/File:Param34_1.jpghttps://en.wikipedia.org/wiki/File:Param33_1.jpghttps://en.wikipedia.org/wiki/File:Param_03.jpghttps://en.wikipedia.org/wiki/File:HypotrochoidOutThreeFifths.gifhttps://en.wikipedia.org/wiki/File:2-circles_hypotrochoid.gifhttps://en.wikipedia.org/wiki/File:Param_02.jpg


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Parametric helix

i=1 j=2

Examples in three dimensions

Helix

Parametric equations are convenient for describing

curves in higher-dimensional spaces. For example:

describes a three-dimensional curve, the helix, with

a radius of aand rising by 2bunits per turn. Notethat the equations are identical in the plane to those

for a circle. Such expressions as the one above are

commonly written as

where ris a three-dimensional vector.

Parametric surfaces

A torus with major radiusRand minor radius rmay be defined parametrically as

where the two parameters t and u both vary between 0 and 2.
https://en.wikipedia.org/wiki/Torushttps://en.wikipedia.org/wiki/Plane_(mathematics)https://en.wikipedia.org/wiki/Helixhttps://en.wikipedia.org/wiki/Curvehttps://en.wikipedia.org/wiki/File:Param_st_01.jpghttps://en.wikipedia.org/wiki/File:Parametric_Helix.png


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R=2, r=1/2

As u varies from 0 to 2 the point on the surface moves about a short circle passing through the hole in

the torus. As t varies from 0 to 2 the point on the surface moves about a long circle around the hole in

the torus.

See also

CurveParametric estimatingPosition vectorVector-valued functionParametrization by arc lengthParametric derivative

Notes

1. Weisstein, Eric W. "Parametric Equations".MathWorld.

2. Thomas, George B. Finney, Ross L. (1979). Calculus and Analytic Geometry(fifth ed.). Addison-Wesley.

p. 91.

3. Weisstein, Eric W. "Parameterization".MathWorld.

4. Spitzbart, Abraham (1975). Calculus with Analytic Geometry. Gleview, IL: Scott, Foresman and Company.

ISBN 0-673-07907-4. Retrieved August 30, 2015.

5. Stewart, James (2003). Calculus(5th ed.). Belmont, CA: Thomson Learning, Inc. pp. 687689. ISBN 0-534-

39339-X.

6. Shah, Jami J. Martti Mantyla (1995).Parametric and feature-based CAD/CAM: concepts, techniques, and

applications. New York, NY: John Wiley & Sons, Inc. pp. 2931. ISBN 0-471-00214-3.

7. See "Equation form and Parametric form conversion" (http://xahlee.org/SpecialPlaneCurves_dir/CoordinateSystem_dir/coordinateSystem.html) for more information on converting from a series of parametric equations to

single function.

External links

Graphing Software (https://www.dmoz.org/Science/Math/Software/Graphing/) at DMOZWeb application to draw parametric curves on the plane (http://danher6.100webspace.net/curvas/)

Retrieved from "https://en.wikipedia.org/w/index.php?title=Parametric_equation&oldid=720101875"

Categories: Multivariable calculus Equations
https://en.wikipedia.org/wiki/Category:Equationshttps://en.wikipedia.org/wiki/Category:Multivariable_calculushttps://en.wikipedia.org/wiki/Help:Categoryhttps://en.wikipedia.org/w/index.php?title=Parametric_equation&oldid=720101875http://danher6.100webspace.net/curvas/https://en.wikipedia.org/wiki/DMOZhttps://www.dmoz.org/Science/Math/Software/Graphing/http://xahlee.org/SpecialPlaneCurves_dir/CoordinateSystem_dir/coordinateSystem.htmlhttps://en.wikipedia.org/wiki/Special:BookSources/0-471-00214-3https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-534-39339-Xhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Calculushttps://en.wikipedia.org/wiki/Special:BookSources/0-673-07907-4https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://www.worldcat.org/oclc/1287519https://en.wikipedia.org/wiki/MathWorldhttp://mathworld.wolfram.com/Parameterization.htmlhttps://en.wikipedia.org/wiki/Addison-Wesleyhttps://en.wikipedia.org/wiki/MathWorldhttp://mathworld.wolfram.com/ParametricEquations.htmlhttps://en.wikipedia.org/wiki/Parametric_derivativehttps://en.wikipedia.org/wiki/Parametrization_by_arc_lengthhttps://en.wikipedia.org/wiki/Vector-valued_functionhttps://en.wikipedia.org/wiki/Position_vectorhttps://en.wikipedia.org/wiki/Parametric_estimatinghttps://en.wikipedia.org/wiki/Curvehttps://en.wikipedia.org/wiki/File:Torus.png


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