Parametrization along the coordinate axis.

Question

I'm looking for a parametrization along the coordinate axis that goes from $(x,y)=(x_0,0)$ to $(0,0)$ and then from $(0,0)$ to $(0,x_0)$, and I know that my x-component will have to be $x_0-t$ but I somehow can't figure out the y-component.

Answer 1

There are many ways to parametrisize this, but you could use, for example $$ \begin{cases} x = x_0\frac{|t|-t}{2}\\ y = x_0\frac{|t|+t}{2} \end{cases} \qquad \text{with} \qquad -1 \leq t \leq 1 $$