Parametric equation

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Graph of a butterfly curve (transcendental), a parametric equation discovered by Temple H. Fay

In mathematics, parametric equations bear slight similarity to functions: they allow one to use arbitrary values, called parameters, in place of independent variables in equations, which in turn provide values for dependent variables. 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.

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 of radius a:

[x = a \cos(t)\,]

[y = a \sin(t)\,]

Finally, there are certain geometric forms that are nearly impossible to describe as a single equation but have very elegant expressions in parametric form:

[x = a \cos(t)\,]

[y = a \sin(t)\,]

[z = bt\,]

which describe 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 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'(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). Parametrized surfaces, of great use in such vector calculus applications as Stokes' theorem, are functions of two parameters, most commonly (s, t) or (u,v).

An example of a parametrized surface is the (capless) cylinder given by

[r(u, v) = (x(u, v), y(u, v), z(u, v)) = (a \cos(u), a \sin(u), v)\,]

Considering the equation as representing a circle in the plane, it is evident that this represents a cylinder. It is then allowed to take on values of z.

Conversion from two parametric equations to a singular 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.

Parametric equation

Conversion from two parametric equations to a singular equation

See also