In the euclidean plane I want a smooth curve $\gamma (t)$ which satisfy:$$\gamma (0)=(0,0)\quad\quad \gamma '(0)=(1,0)\quad\quad n\in \mathbb{N}\land n\neq 1\Rightarrow \gamma ^{(n)}(0)=(0,0)\\\gamma (s)=(1,1)\quad\quad \gamma '(s)=(0,1)\quad\quad n\in \mathbb{N}\land n\neq 1\Rightarrow \gamma ^{(n)}(s)=(0,0)$$
Where $s>0$ is the arclength of $\gamma$. There are many such, but I think uniqueness is obtained by contraining to those that minimize $\int \kappa (s)^2ds$, and satisfy:$$0\leq t\leq s\Rightarrow \|\gamma '(t)\|=1$$
How can this curve be found/approximated? Does $\gamma$ solve some well-known differential equation that I can look up/solve?