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The butterfly curve can be defined by parametric equations of x and y.

In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters.[1] Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, in which case the equations are collectively called a parametric representation or parameterization (alternatively spelled as parametrization) of the object.[1][2][3]

For example, the equations

form a parametric representation of the unit circle, where t is the parameter: A point (x, y) is on the unit circle if and only if there is a value of 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 nonunique (see the "Examples in two dimensions" section below), so the same quantities may be expressed by a number of different parameterizations.[1]

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 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.[4]

Contents

ApplicationsEdit

KinematicsEdit

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 designEdit

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 analytic, or non-parametric, representations of curves; when compared to parametric representations for use in CAD applications, non-parametric representations have shortcomings. In particular, the non-parametric representation depends on the choice of the coordinate system and does not lend itself well to geometric transformations, such as rotations, translations, and scaling; non-parametric representations therefore make it more difficult to generate points on a curve. These problems can be addressed by rewriting the non-parametric equations in parametric form.[6]

Integer geometryEdit

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 a, b and their hypotenuse c are coprime integers. As a and b are not both even (otherwise a, b and c would not be coprime), one may exchange them to have a even, and the parameterization is then

 

where the parameters m and n are positive coprime integers that are not both odd.

By multiplying a, b and c by an arbitrary positive integer, one gets a parametrization of all right triangles whose three sides have integer lengths.

ImplicitizationEdit

Converting a set of parametric equations to a single implicit 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 involving x and y only.

If the parametrization is given by rational functions

 

where p, q, r are set-wise coprime polynomials, a resultant computation allows one to implicitize. More precisely, the implicit equation is the resultant with respect to t of xr(t) – p(t) and yr(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 Gröbner basis computation; see Gröbner 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 a above, the parametric equations

 

can be implicitized in terms of x and y by way of the Pythagorean trigonometric identity:

As

 

and

 

we get

 

and thus

 

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

Examples in two dimensionsEdit

ParabolaEdit

The simplest equation for a parabola,

 

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

 

Explicit equationsEdit

More generally, any curve given by an explicit equation

 

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

 

CircleEdit

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 parameterization is

 

With this pair of parametric equations, the point (-1, 0) is not represented by a real value of t, but by the limit of x and y when t tends to infinity.

EllipseEdit

An ellipse in canonical position (center at origin, major axis along the X-axis) with semi-axes a and b can be represented parametrically as

 

An ellipse in general position can be expressed as

 

as the parameter t varies from 0 to 2π. Here   is the center of the ellipse, and   is the angle between the  -axis and the major axis of the ellipse.

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

Lissajous CurveEdit

 
A Lissajous curve where   and  .

A Lissajous curve is similar to an ellipse, but the x and y sinusoids are not in phase. In canonical position, a Lissajous curve is given by

 

where   and   are constants describing the number of lobes of the figure.

HyperbolaEdit

An east-west opening hyperbola can be represented parametrically by

  or, rationally  

A north-south opening hyperbola can be represented parametrically as

  or, rationally  

In all these formulae (h,k) are the center coordinates of the hyperbola, a is the length of the semi-major axis, and b is the length of the semi-minor axis.

HypotrochoidEdit

A hypotrochoid is a curve traced by a point attached to a circle of radius r rolling around the inside of a fixed circle of radius R, where the point is at a distance d from the center of the interior circle.

The parametric equations for the hypotrochoids are:

 

Some sophisticated functionsEdit

Other examples are shown:

 
 
Several graphs by variation of k
 
 

Examples in three dimensionsEdit

HelixEdit

 
Parametric 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 a and rising by 2πb units per turn. Note that the equations are identical in the plane to those for a circle. Such expressions as the one above are commonly written as

 

where r is a three-dimensional vector.

Parametric surfacesEdit

A torus with major radius R and minor radius r may be defined parametrically as

 

where the two parameters t and u both vary between 0 and 2π.

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

Examples with vectorsEdit

The parametric equation of the line through the point   and parallel to the vector   is[8]

 

See alsoEdit

NotesEdit

  1. ^ a b c 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. ^ Nykamp, Duane. "Plane parametrization example". mathinsight.org. Retrieved 2017-04-14. 
  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. 687–689. 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. 29–31. ISBN 0-471-00214-3. 
  7. ^ See "Equation form and Parametric form conversion" for more information on converting from a series of parametric equations to single function.
  8. ^ Calculus : Single and multivariable. John wiley. 2012-01-01. p. 919. ISBN 9780470888612. OCLC 828768012. 

External linksEdit