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We prove that the following problem has the same computational complexity asthe existential theory of the reals: Given a generic self-intersecting closedcurve $\gamma$ in the plane and an integer $m$, is there a polygon with $m$vertices that is isotopic to $\gamma$? Our reduction implies implies twostronger results, as corollaries of similar results for pseudolinearrangements. First, there are isotopy classes in which every $m$-gon withinteger coordinates requires $2^{\Omega(m)}$ bits of precision. Second, for anysemi-algebraic set $V$, there is an integer $m$ and a closed curve $\gamma$such that the space of all $m$-gons isotopic to $\gamma$ is homotopy equivalentto $V$.

Optimal Curve Straightening is $\exists\mathbb{R}$-Complete