In mathematics, a parametric equation expresses several quantities, such as the coordinates of a point, as functions of one or more variables called parameters. In the case of a single parameter, parametric equations are commonly used to express the trajectory of a moving point. For this case, the parameter is often, but not necessarily, time, and the point describes a curve, called a parametric curve. In the case of two parameters, the point describes a surface, called a parametric surface. In all cases, the equations are collectively called a parametric representation, or parametric system, or parameterization (also spelled parametrization, parametrisation) of the object. For example, the equations
x = cos t y = sin t {\displaystyle {\begin{aligned}x&=\cos t\\y&=\sin t\end{aligned}}}
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:
( x , y ) = ( cos t , sin t ) . {\displaystyle (x,y)=(\cos t,\sin t).}
Parametric representations are generally not unique (see § Parametric plane curves), so the same quantities may be expressed by a number of different parameterizations. 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.
Implicitization Converting a set of parametric equations to a single implicit equation involves eliminating the variable t from the simultaneous equations x = f ( t ) , y = g ( t ) . {\displaystyle x=f(t),\ y=g(t).} 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: Solving y = g ( t ) {\displaystyle y=g(t)} to obtain t = g − 1 ( y ) {\displaystyle t=g^{-1}(y)} and using this in x = f ( t ) {\displaystyle x=f(t)} gives the explicit equation x = f ( g − 1 ( y ) ) , {\displaystyle x=f(g^{-1}(y)),} while more complicated cases will give an implicit equation of the form h ( x , y ) = 0. {\displaystyle h(x,y)=0.}
If the parametrization is given by rational functions
x = p ( t ) r ( t ) , y = q ( t ) r ( t ) , {\displaystyle x={\frac {p(t)}{r(t)}},\qquad y={\frac {q(t)}{r(t)}},}
where p, q, and 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 dimensions (either more than two coordinates or more than one parameter), the implicitization of rational parametric equations may by done with Gröbner basis computation; see Gröbner basis § Implicitization in higher dimension. To take the example of the circle of radius a, the parametric equations
x = a cos ( t ) y = a sin ( t ) {\displaystyle {\begin{aligned}x&=a\cos(t)\\y&=a\sin(t)\end{aligned}}}
can be implicitized in terms of x and y by way of the Pythagorean trigonometric identity. With
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