ALGORITHM 2.

Resolve_extremal_curve.

Input: Initial point $q_0\in {\mathbb{R}}^d$ and velocity $v_0\in {\mathbb{R}}^d$.

Output: Final point $q$ and velocity $v$ of the extremal curve $\gamma$.

Construct numerical continuation curve $\gamma$ for $$\{\begin{array}{l}\mathrm{Hess}(Z)\mathrm{grad}(Z)=\lambda \mathrm{grad}(Z)\\ Z(\gamma (t))=L\end{array}$$ such that $\gamma (0)=q_0$ and $\dot{\gamma }(0)=v_0$. $L$ is the value of the level set at the current time step.

Prolong $\gamma$ until the first instant $t>0$ such that $\gamma (t)\in V$ but $\gamma (t+s)\notin V$ for any $s>0$.

return $\gamma (t)$, $\dot{\gamma }(t)$