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Understanding Spherical Triangles

Basic Principles

cos(c) = cos(a)cos(b) + sin(a)sin(b)cos(C)

sin(A)/sin(a) = sin(B)/sin(b) = sin(C)/sin(c)

E = A + B + C - 180°

Fundamental Properties

  • Great circle arcs
  • Spherical excess
  • Polar triangles
  • Angle sum > 180°
  • Area formula

Key Relationships

  • Spherical cosine law
  • Spherical sine law
  • Napier's rules
  • Gauss equations
  • Delambre formulas

Special Cases

  • Right spherical triangles
  • Isosceles spherical triangles
  • Equilateral spherical triangles
  • Birectangular triangles
  • Trirectangular triangles

Advanced Concepts

Applications

  • Navigation
  • Astronomy
  • Geodesy
  • Cartography
  • Global positioning

Computational Methods

  • Numerical stability
  • Half-angle formulas
  • Lhuilier's formula
  • L'Huilier's theorem
  • Vincenty's formulae

Theoretical Extensions

  • Non-Euclidean geometry
  • Riemannian geometry
  • Differential geometry
  • Ellipsoidal triangles
  • Hyperbolic triangles