Editing Parametric equation (section)

==Implicitization==
Converting a set of parametric equations to a single [[implicit equation]] involves eliminating the variable {{mvar|t}} from the simultaneous equations <math>x=f(t),\ y=g(t).</math> This process is called '''{{dfn|implicitization}}'''. If one of these equations can be solved for {{mvar|t}}, the expression obtained can be substituted into the other equation to obtain an equation involving {{mvar|x}} and {{mvar|y}} only: Solving <math>y=g(t)</math> to obtain <math>t=g^{-1}(y)</math> and using this in <math>x=f(t)</math> gives the explicit equation <math> x=f(g^{-1}(y)),</math> while more complicated cases will give an implicit equation of the form <math>h(x,y)=0.</math>

If the parametrization is given by [[rational function]]s
<math display="block">x=\frac{p(t)}{r(t)},\qquad y=\frac{q(t)}{r(t)},</math>

where {{mvar|p}}, {{mvar|q}}, and {{mvar|r}} are set-wise [[coprime]] polynomials, a [[resultant]] computation allows one to implicitize. More precisely, the implicit equation is the [[resultant]] with respect to {{mvar|t}} of {{math|''xr''(''t'') – ''p''(''t'')}} and {{math|''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 {{slink|Gröbner basis|Implicitization in higher dimension}}.

To take the example of the circle of radius {{mvar|a}}, the parametric equations
<math display="block">\begin{align}
  x &= a \cos(t) \\
  y &= a \sin(t)
\end{align}</math>

can be implicitized in terms of {{math|x}} and {{math|y}}  by way of the [[Pythagorean trigonometric identity]]. With

<math display="block">\begin{align}
\frac{x}{a} &= \cos(t) \\
\frac{y}{a} &= \sin(t) \\
\end{align}</math>
and
<math display="block">\cos(t)^2 + \sin(t)^2 = 1,</math>
we get 
<math display="block">\left(\frac{x}{a}\right)^2 + \left(\frac{y}{a}\right)^2 = 1,</math>
and thus
<math display="block">x^2+y^2=a^2,</math>

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