How to graph a curve finding the Cartesian equation

Question

I have to graph the curve $$\gamma(t)=\left(\frac{t^3}3-t,(\ln t)^2\right)$$ with $t>0$. How can I express the curve as a function graph? Maybe I have to put $\frac{t^3}3=x$? I don't think this is the best way, but I only know this method. Can you help me?

If you want to know, the curve is simple and it is regular $\forall t\ne1$.

Answer 1

It's not possible to describe the curve by some expression $y=f(x)$, as we will see, but that's not really necessary. In gnuplot, you'd do something like

gnuplot> set parametric

dummy variable is t for curves, u/v for surfaces
gnuplot> plot [t=0.1:2] t**3/3-t, (log(t))**2

and get a decent graph: As you can see, the point corresponding to $t=1$, i.e. $(x,y)=(-2/3, 0)$, is not a regular one.