Hellenica World

# .

In mathematics, parametric equation is a method of defining a relation using parameters. A simple kinematic example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion.

Abstractly, a parametric equation defines a relation as a set of equations. Therefore, it is somewhat more accurately defined as a parametric representation. It is part of regular parametric representation.

2D examples
Parabola

For example, the simplest equation for a parabola,

$$y = x^2\,$$

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

$$x = t\,$$
$$y = t^2.\,$$

Circle

A more sophisticated example might be the following. Consider the unit circle which is described by the ordinary (Cartesian) equation

$$x^2 + y^2 = 1.\,$$

This equation can be parametrized as well, giving

$$(\cos(t), \sin(t))\quad\mathrm{for}\ 0\leq t < 2\pi.\,$$

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.
3D examples
Helix
Parametric helix

Parametric equations are convenient for describing curves in higher-dimensional spaces. For example:

$$x = a \cos(t)\,$$
$$y = a \sin(t)\,$$
$$z = bt\,$$

describes a three-dimensionals curve, the helix, which has a radius of a and rises by 2πb units per turns. Note that the equations are identical in the plane to those for a circle. Such expressions as the one above are commonly written as

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

Parametric surfaces

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

$$x = \cos(t)(R + r \cos(u)),$$
$$y = \sin(t)(R + r \cos(u)),$$
$$z = r \sin(u).$$

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

$$R=2, r=1/2$$

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.
Usefulness

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:

$$v(t) = r'(t) = (x'(t), y'(t), z'(t)) = (-a \sin(t), a \cos(t), b)\,$$

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, see Parametric surface.

Another important use of parametric equations is in the field of computer aided design (CAD).[1] For example, consider the following three representations, all of which are commonly used to describe planar curves.

Type Form Example Description
1. Explicit $$y = f(x) \,\!$$ $$y = mx + b \,\!$$ Line
2. Implicit $$f(x,y) = 0 \,\!$$ $$\left(x - a \right)^2 + \left( y - b \right)^2=r^2$$ Circle
3. Parametric $$x = \frac{x(t)}{w(t)}; y = \frac{y(t)}{w(t)}$$ $$x = a_0 + a_1t; \,\! y = b_0 + b_1t\,\!$$

$$x = a+r\,\cos t; \,\! y = b+r\,\sin t\,\!$$

Line

Circle

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 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.[2]
Conversion from two parametric equations to a single equation

Converting a set of parametric equations to a single equation involves eliminating the variable t from the simultaneous equations x=x(t),\ y=y(t). 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 x(t) and y(t) are rational functions then the techniques of the theory of equations such as resultants can be used to eliminate t. In some cases there is no single equation in closed form that is equivalent to the parametric equations.[3]

To take the example of the circle of radius a above, the parametric equations

$$x = a \cos(t)\,$$
$$y = a \sin(t)\,$$

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

$$x/a = \cos(t)\,$$
$$y/a = \sin(t)\,$$
$$\cos(t)^2 + \sin(t)^2 = 1\,\!$$
$$\therefore (x/a)^2 + (y/a)^2 = 1,$$

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

Parametric Surfaces Gallery

Curve
Parametric estimating
Position vector
Vector-valued function

Notes

^ Stewart, James (2003). Calculus (5th ed.). Belmont, CA: Thomson Learning, Inc.. pp. 687–689. ISBN 0-534-39339-X.
^ 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 0471002143.
^ See "Equation form and Parametric form conversion" for more information on converting from a series of parametric equations to single function.

Graphing Software at the Open Directory Project
Web application to draw parametric curves on the plane