Hyperbolic triangle

A three-sided polygon in the hyperbolic plane bounded by geodesics. Its interior angles sum to less than 180 degrees, with the deficit proportional to its area.

Hyperbolic law of cosines: A relation among side lengths and angles of a triangle in hyperbolic geometry. It calculates a missing side or angle while incorporating the plane’s negative curvature.
Hyperbolic geometry: Geometry of spaces with constant negative curvature, where Euclid’s parallel postulate does not hold. It supplies the plane and distance rules in which hyperbolic triangles are defined.
Euclidean triangle: A three-sided polygon in the flat Euclidean plane whose angles sum to 180 degrees. Its fixed angle sum contrasts with the hyperbolic triangle’s curvature-dependent deficit.
Hyperbolic tiling: A repeated arrangement of polygons that covers the hyperbolic plane without gaps or overlaps. Hyperbolic triangles serve as tiles in regular and irregular tessellations.
Gauss–Bonnet theorem: A theorem relating a surface’s curvature and boundary geometry to its topology. For a hyperbolic triangle, it equates angle deficit with curvature-weighted area.
Hyperbolic plane: A simply connected two-dimensional space of constant negative curvature. The triangle’s sides and interior angles are measured within this curved plane.
Spherical triangle: A three-sided figure on a sphere whose sides are arcs of great circles. Positive curvature makes its angle sum exceed 180 degrees, opposite the hyperbolic case.
Hyperbolic triangulation: A decomposition of a hyperbolic surface into triangles with geodesic sides. It builds complex hyperbolic surfaces from the geometry of individual triangles.
Geodesic: A locally distance-minimizing curve that generalizes a straight line to curved spaces. Each side is a geodesic, rather than an arbitrary curve in the hyperbolic plane.
Hyperbolic metric: A distance rule that gives a space its hyperbolic geometry. It determines lengths and angles, including the triangle’s area and angle deficit.

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