Imagine that you would like to describe the motion of an object whose motion is confined to a plane. If you know how its Cartesian coordinates, \(x\) and \(y\text{,}\) change as time passes, you will be able to draw a curve that represents the trajectory of its motion, as shown in Figure 1.2. Mathematically, we will need two equations to demonstrate how the \(x\)- and \(y\)-coordinates evolve in time \(t\text{,}\) which are known as the parametric equations.
If \(x\) and \(y\) are given as functions of a third variable \(t\text{,}\) called a parameter, by
\begin{equation}
x = f(t), \qquad y = g(t),\tag{1.1}
\end{equation}
then the points \((x,y) = (f(t),\,g(t))\) trace out a curve in the plane, called a parametric curve. The equations \(x=f(t)\) and \(y=g(t)\) are the parametric equations of the curve.
Figure1.2.A point moves in the plane. As time \(t\) passes, its coordinates \(x = f(t)\) and \(y = g(t)\) change, and the point traces out the trajectory of the motion.
Plotting these points and connecting them, we recognize the parabola \(y = x^2\text{,}\) traced from left to right as \(t\) increases from \(-3\) to \(3\text{.}\) See Figure 1.5 and Figure 1.6.
Figure1.5.The parametric curve \(x = t\text{,}\)\(y = t^2\text{:}\) the points from the table appear first, and then the moving point traces out the parabola \(y = x^2\) in the direction of increasing \(t\text{.}\)
Figure1.6.The curve \(x = t\text{,}\)\(y = t^2\) is the parabola \(y = x^2\text{.}\) The marked points correspond to the integer values \(t = -3, \dots, 3\) from the table, and the arrows show the direction of increasing \(t\text{.}\)
Here, we would like to parametrize \(x\) and \(y\) in terms of a single parameter \(t\) in such a way that by varying the value of \(t\text{,}\) we can reconstruct the circle \(x^2+y^2=1\text{.}\) Remember that \(\cos^2(t) + \sin^2(t) = 1\text{,}\) so one way to achieve our goal is to write
\begin{equation}
x = \cos(t), \qquad y = \sin(t),\tag{1.3}
\end{equation}
where \(t \in [0, 2\pi)\text{.}\) The parametrization is illustrated in Figure 1.7 and Figure 1.8.
Figure1.7.Parametrizing the unit circle: as \(t\) increases from \(0\) to \(2\pi\text{,}\) the point \((\cos(t), \sin(t))\) travels counterclockwise around the circle, passing through the marked points at \(t = 0\text{,}\)\(\pi/2\text{,}\)\(\pi\text{,}\) and \(3\pi/2\text{.}\)
Figure1.8.Parametrizing the unit circle \(x^2+y^2=1\) by \(x=\cos(t)\text{,}\)\(y=\sin(t)\text{,}\)\(t \in [0,2\pi)\text{.}\) A generic point of the circle corresponds to the parameter value \(t_0\text{.}\)
