How to parametrise a closed curve if we have points:
$A(0,0,0), B(0,1,0), C(1,1,1), D(1,0,0)$
In the orientation $A\rightarrow B\rightarrow C \rightarrow D \rightarrow A$
What I would do (I tried, don't know if it is correct):
$r_{AB} = (0,t,0), t \in [0,1]\\ r_{BC} = (t,1,t), t \in [0,1]\\ r_{CD} = (1, 1-t, 1-t), t\in [0,1]\\ r_{DA} = (1-t,0,0), t\in[0,1]\\$
However I don't know how to put it all together to get a closed curve. Can somebody please explain how to do that.
If I’ve understood correctly, maybe you want something like this:
Your curve can be written like $\gamma \colon [0,1]\to \mathbb{R}^3$, $\gamma(t)=(\gamma_1(t),\gamma_2(t), \gamma_3(t))$.
You are asking to impose some conditions on $\gamma$ and so on the real valued functions $\gamma_i$.
The condition is to go trough these points, at some time $t$. Doesn’t matter at which time $t$ the curve will go trough that point, so our conditions become
$\gamma(0)=A=0$
$\gamma(\frac{1}{4})=B$
$\gamma(\frac{1}{2})=C$
$\gamma(\frac{3}{4})=D$
$\gamma(1)=A=0$
To getting an explicit example you can try using segments between the points, as you tried to do.
Please note that to have a closed curve you have also to impose the continuity of $\gamma$, and so the continuity of the real values functions $\gamma_i$.