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Parametric equation

 
Sci-Tech Dictionary: parametric equation
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(¦par·ə¦me·trik i′kwā·zhən)

(mathematics) An equation where coordinates of points appear dependent on parameters such as the parametric equation of a curve or a surface.

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Sci-Tech Encyclopedia: Parametric equation
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A type of mathematical equation used, typically, to represent curves in a plane or in space of three dimensions, In principle, however, there is no limitation to any particular number of dimensions. A parameter is actually an independent variable. In elementary analytic geometry a curve in the xy plane is often studied, in the first instance, as the locus of an equation y = F(x) or G(x,y) = 0. The form y = F(x) is not adequate for the complete representation of certain curves, whereas the form G(x,y) = 0 may be adequate. The circle x2 + y2 − 16 = 0 affords an example. But the form G(x,y) = 0 is not always convenient. The parametric form x = f(t), y = g(t) is often the most convenient; moreover, it is often the naturally occurring form of representation of the curve. For the circle x2 + y2 − 16 = 0, one possible parametric representation is x = 4 cos t and y = 4 sin t.

A pair of equations x = f(t), y = g(t), where f and g are continuous functions defined for some interval of values of t, for example, a ≤ t ≤ b, is said to define a parametric curve. If one thinks of t as time, the equations define the motion of the point (x, y) as t increases from a to b. Clearly the path can cross itself, double back on itself, or the point may even remain motionless.

A parametric surface in space of three dimensions is defined by x = f(u,v), y = g(u,v), z = h (u,v), where f, g, h are continuous functions of the two parameters, u,v. See also Analytic geometry; Calculus; Partial differentiation.

Wikipedia: Parametric equation
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One example of a curve defined by parametric equations is the butterfly curve.

In mathematics, parametric equations are a method of defining a function using parameters. A simple kinematical example is when one uses a time parameter to determine the position, 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 Rn. It is therefore somewhat more accurately defined as a parametric representation. It is part of regular parametric representation.

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Examples

Parametric helix

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 is a somewhat trivial case, consider the following parametrization of a circle of radius a:

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

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-dimensional curve, the helix, which has a radius of a and rises by 2πb units per turn. (Note that the equations are identical in the plane to those for a circle; in fact, a helix is sometimes humorously described as just "a circle whose ends don't have the same z-value". This is not exactly true, as a circle is by definition a two dimensional curve and a helix is by definition a three dimensional curve. Also there are smooth curves other than the helix that can be described as "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'(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.

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.[1]

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).

See also

Notes

  1. ^ See "Equation form and Parametric form conversion" for more information on converting from a series of parametric equations to single function.

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This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

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