Let $\beta(t)=(x(t),y(t))'$ for $t\in[0,1]$, represents a closed parametric curve.
For example, let $\beta(t)$ is a circle. So one parametrisation may be $\beta(t)=(x(t),y(t))'$ where $x(t)=2\pi \cos t$ and $y(t)=2\pi \sin t$. Again another re-parametrization may be $x(t)=2\pi \sin t$ and $y(t)=2\pi \cos t$.
Similarly I parametrize a triangle with vertices $A(1,2), B(5,2)$ and $C(3,6)$.
In this case, one of continuous parametrization can be defined piecewisely as: $[0,1] \ni t \mapsto \gamma (t) \in \mathbb{R}^2$, $\gamma(t)=3(tB+(\frac{1}{3}-t)A), t\in [0,\frac{1}{3}]$,
$\gamma(t)=3((t-\frac{1}{3})C+(\frac{2}{3}-t)B), t\in [\frac{1}{3},\frac{2}{3}]$, and then
$\gamma(t)=3((t-\frac{2}{3})A+(1-t)C), t\in [\frac{2}{3},1]$.
Could any one please suggest me how to get some other re-parametrizations for the above triangle with vertices $A,B$ and $C$ as in case of circle.
Thanks in advance.