Parametrizing a set of lines from a parametrization of a curve. Given a parametrization $\gamma(t)=(t,t^2,t^3)$ of a curve, can one similarly parametrize the set of lines that go through the origin and a point on the curve? I wonder if this is impossible... If it is possible, I wonder how one can no such a thing "on sight".

$$\frac{x-t}{x-0}=\frac{y-t^2}{y-0}=\frac{z-t^3}{z-0}=u\text{ (say)}$$

So that $x-t=ux,x=\frac t{1-u}$

Similarly, $y=\frac {t^2}{1-u},z=\frac {t^3}{1-u}$

Clearly, $u\
e1$ as $u=1\implies t=0$