A Euclidean Construction for the Cross-Ratio of Hyperbolic Ideal Quadrilaterals

This is a preprint of the manuscript: "A Euclidean Construction for the Cross-Ratio of Hyperbolic Ideal Quadrilaterals" Abstract:We provide an elementary Euclidean coordinate for the moduli space of hyperbolic ideal quadrilaterals. Starting from a non-parallelogram quadrilateral with bridge tilt angles α₁ and α₂, we extend two opposite edges without bound. Via the Klein disk model—in which hyperbolic geodesics are Euclidean straight lines and the Klein and Poincaré boundaries coincide—the four asymptotic edge directions directly define four ideal boundary points. Computing their cross-ratio by complete complex algebra yields the closed-form formula cr(α₁, α₂) = 2 cos((α₁ − α₂)/2) / cos((α₁ − α₂)/2) + sin((α₁ + α₂)/2). In the symmetric case α₁ = α₂ = α this simplifies to cr(α) = 2/(1 + sin α), proved here by complete algebra. This formula is independent of bridge length b and establishes a bijection from α ∈ (0, π/2] onto a natural one-parameter slice of the moduli space, cr ∈ [1, 2): the bridge angle α is an elementary Euclidean coordinate on hyperbolic ideal quadrilateral space. All values are verified to machine precision; a self-contained Python script is included in the Computational Appendix. The manuscript has been submitted to a peer-reviewed journal and is currently under review. Keywords: ideal quadrilateral, hyperbolic geometry, cross-ratio, Klein disk model, Poincaré disk, moduli space, bridge angle, conformal invariant, Euclidean construction, elementary coordinate