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Purely numerical realizability with exact collinearity constraints

Develop a purely numerical method over point coordinates that guarantees exact satisfaction of orientation constraints involving collinear triples when solving planar realizability problems for combinatorial orientation assignments.

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Background

The realizability problem asks for coordinates realizing a given orientation assignment; it is ∃ℝ‑complete. The authors introduce a fast local‑search solver (Localizer) for general position, but they note that exact collinearity is brittle and difficult for purely numeric approaches because infinitesimal perturbations break collinearity. For configurations requiring collinearities (as in everywhere‑unbalanced sets), they resort to an ad‑hoc approach, highlighting an unresolved methodological challenge for exact numerical realizability in the presence of collinear constraints.

References

As collinearity is an exact condition where arbitrarily small perturbations will result in non-collinear points, it is not clear how purely numerical methods over the variables ${x_1, y_1, \ldots, x_n, y_n}$ can be used to satisfy the desired orientation constraints exactly.

— Automated Symmetric Constructions in Discrete Geometry (2506.00224 - Subercaseaux et al., 30 May 2025) in Section 5.2 (Realizing Collinear Configurations for Everywhere‑Unbalanced)