Hi I am trying to find a parametrization of the unit square such that the domain in $t$ is in $[0,1]$.
I know that we can have a parametrization of the unit square like
\begin{equation} \begin{cases} (0,t), t\in [0,1] \\ (t,1-t), t\in [0,1] \\ (1-t,1),t \in [0,1] \\ (0,1-t), t\in[0,1] \end{cases} \end{equation}
The curves are then joined to form the unit square. However I am looking for something like:
\begin{equation} \begin{cases} ... t \in [0,0.25] \\ ...t \in [0.25,0.5] \\ ...t \in [0.5,0.75]\\ ...t \in[0.75,1] \end{cases} \end{equation}
Is this possible by rescaling? I cannot find anything so far.
So the parametrization given by works connecting the square with vertices $(0,0),(1,0),(1,1),(0,1)$ in a counter-clockwise fashion for $t \in [0,1]$.
\begin{equation} f(t)= \begin{cases} (4t,0),t \in [0,0.25] \\ (1,4(t-0.25), t \in [0.25,0.5] \\(4(0.75-t),1), t \in [0.5,0.75]\\ (0,4(1-t), t \in[0.75,1] \end{cases} \end{equation}