Parametric Form of Conics

Beyond the Cartesian equation $F(x,y)=0$ and the focus–directrix description, a curve can also be traced by expressing both $x$ and $y$ as functions of a third variable $t$: $x=f(t)$, $y=g(t)$. As $t$ varies, the point $(f(t),g(t))$ sweeps out the curve. This third variable is called a parameter, and $x=f(t),y=g(t)$ are the curve's parametric equations. One natural reading of $t$ is time — the equations then give the position of a moving object at time $t$. The parameter need not be called $t$; any convenient letter works, though $t$ (and, for angle-based parametrisations, $\theta$) is the usual choice. A parametric description effectively trades two coordinates $(x,y)$ for one parameter, which is often the …