Optimal curve straightening to a target vertex count is ∃R-complete, and isotopy realization spaces of curves are universal up to homotopy equivalence.
Adiprasito, Arnau Padrol, and Louis Theran
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Optimal Curve Straightening is $\exists\mathbb{R}$-Complete
Optimal curve straightening to a target vertex count is ∃R-complete, and isotopy realization spaces of curves are universal up to homotopy equivalence.