Circular arc geodesics, angle sum < π, {7,3} and {5,4} tilings
The Poincaré disk model represents hyperbolic plane H² in the open unit disk. Geodesics are circular arcs perpendicular to the boundary circle (or diameters). The metric ds² = 4(dx²+dy²)/(1−r²)² makes the boundary infinitely far away. Any triangle has angle sum α+β+γ < π, with area = π−(α+β+γ) (Gauss-Bonnet). The {7,3} tiling (7 heptagons around each vertex) and {5,4} tiling are realizations of the infinite group generated by hyperbolic reflections.