quality:correction, information:classification, information:filtering
Circle

A more sophisticated example is the following. Consider the unit circle which is described by the ordinary (Cartesian) equation x 2 + y 2 = 1. {\displaystyle x^{2}+y^{2}=1.}

This equation can be parameterized as follows: (x, y) = (cos (t), sin (t)) f o r 0 ≤ t < 2 π. {\displaystyle (x,y)=(\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.

In some contexts, parametric equations involving only rational functions (that is fractions of two polynomials) are preferred, if they exist. In the case of the circle, such a rational parameterization is x = 1 − t 2 1 + t 2 y = 2 t 1 + t 2. {\displaystyle {\begin{aligned}x&={\frac {1-t^{2}}{1+t^{2}}}\\y&={\frac {2t}{1+t^{2}}}\,.\end{aligned}}}

With this pair of parametric equations, the point (−1, 0) is not represented by a real value of t, but by the limit of x and y when t tends to infinity.