Let $f(t)=(x(t),y(t))'$ for $t\in[0,1]$, represents a parametric function.
Let us consider a parametric equation (straightline) joining two points $a$ and $b$ in 2-dimension: $$f(t)=a(1-t)+bt.$$
Let us consider map $\gamma:t\mapsto t^2$, and consider the re-parametrization $$f(\gamma)=a(1-t^2)+bt^2.$$
My question is: what is the interpretation of this re-parametrization?
Thanks in advance.