Geometry on a Sphere
Explore the mathematics of curved surfaces. From Great Circles to the Cosine Rule, discover how triangles behave when the sum of their angles exceeds 180°.
The Spherical Triangle
A spherical triangle is bounded by three arcs of Great Circles. A Great Circle is the intersection of the sphere with a plane passing through its center.
Key Properties
- ● Sides (a, b, c): Angle magnitudes at the sphere's center.
- ● Angles (A, B, C): Dihedral angles between the bounding planes.
- ● Excess: The total angle sum exceeds 180° by a value proportional to area.
Deriving the Cosine Rule
Vector analysis on a unit sphere allows us to relate angles between surfaces to arcs on the surface.
1. The Unit Vectors
Define u, v, w as vectors from center O to vertices A, B, and C.
u · w = cos(b)
u · v = cos(c)
2. Dihedral Angles
Dihedral angle A is the angle between the plane (OAB) and (OAC).
Napier's Rules
A streamlined system for solving right-angled spherical triangles (C = 90°).
Adjacent Parts
sin(Mid) = tan(Adj₁) tan(Adj₂)
Opposite Parts
sin(Mid) = cos(Opp₁) cos(Opp₂)
Interactive Solver
Calculate unknown sides based on the Law of Cosines.
Result
Degrees
0 Comments