Curves can be written as recipes in a single parameter t
Parametric equations express several quantities—often coordinates—as functions of parameters. One parameter commonly traces a moving point's path as a parametric curve; time is a favorite choice but not required. The pair x=cos t, y=sin t paints the unit circle as t varies.
Two parameters can sweep surfaces; more build higher-dimensional images. Instead of solving y as a function of x, you let both dance with t, which handles loops and vertical tangents gracefully. Engineers plot trajectories; animators move cameras; physicists track orbits—all speaking parametric language.
A point lies on the unit circle exactly when some t reproduces its coordinates via those trig equations.
When graphs refuse to be single-valued functions, parameters quietly restore motion to the page.
Any curve with an explicit equation y = f(x) can be parametrised trivially by setting x = t and y = f(t), so the parabola y = x² becomes x = t, y = t². Parametrisations are not unique. The unit circle, for instance, can be traced by cosine and sine for t from 0 up to 2π, or by the rational pair x = (1 − t²)/(1 + t²), y = 2t/(1 + t²), which reaches the point (−1, 0) only as t heads to infinity. An ellipse with semi-axes a and b becomes x = a cos t, y = b sin t. Each form has its uses: the Cartesian equation makes it easy to test whether a point lies on the circle, while the parametric version makes it easy to generate points for a plot. Going the other way, eliminating t to recover one equation in x and y, is called implicitization; for a circle of radius a it falls out of the identity cos² t + sin² t = 1, while harder rational cases call for resultants or Gröbner bases.
Source: Parametric equation
