10 Parametric Representation
Until now, we have only produced curves described by functions in the form \(y=f(x)\). These functions are said to be in the Cartesian form.
One other form of mathematically representing a curve, called implicit, is made by equations \(f(x,y)=0\). An example of a curve described in the implicit form is the equation \(x ^ 2 + y ^ 2-r ^ 2 = 0\), which describes a circumference of radius \(r\) centered at \((0,0)\). The implicit form is more general than the Cartesian one which, in fact, is obtained by solving the implicit form in terms of the ordinate \(y\). Unfortunately, the Cartesian form is not as useful as the implicit one, since it is not always trivial (or possible) to solve the equation in terms of the ordinate. Lines are an exception regarding the difficulty of finding the Cartesian form. In fact, its implicit equation \(ax + by + c = 0, b \neq 0\), can be easily solved in terms of the ordinate: \(y = -\frac{ax+c}{b}\) For example, in the case of the circumference, the best we can do is to produce two functions:
\[y(x)=\pm\sqrt{r^2-x^2}\]
There is, however, a third form for representing curves that becomes particularly useful: the parametric form. The parametric form is based on the idea that a curve can be drawn by a point whose position evolves over time. Time is, in this case, merely a parameter that determines the position of the point on the curve. In the example of the circumference centered at \((0,0)\), its parametric description will be
\[x(t)= r\cos t\]
\[y(t)= r\sin t\]
Obviously, for the point \((x, y)\) to draw the entire circumference, the parameter \(t\) needs to vary in the range \([0,2\pi[\).
Equations like the previous ones are called parametric equations of the curve. If, in a parametric representation, we eliminate the parameter \(t\), we find the curve’s implicit equation. For example, in the case of the circumference, if we add the squares of the parametric equations, we obtain
\[x^2+y^2=r^2(\cos^2 t + \sin^2 t)=r^2\]
The parametric form allows using any coordinate system. For example, the parametric equations of the circumference can become even simpler if we use the polar coordinate system:
\[\left\{ \begin{aligned} \rho(t)&= r\\ \phi(t)&= t \end{aligned}\right.\]