Gauss–Bonnet Spherical Excess
Relate spherical angle excess to area and orientation-sensitive parallel transport.
Free theory articleRelate spherical angle excess to area and orientation-sensitive parallel transport.
Concept & interpretation
The Gauss–Bonnet theorem relates the intrinsic geometry of a surface, such as its Gaussian curvature, to its topological properties. For a geodesic triangle on a sphere, this implies that the sum of its interior angles is strictly greater than π. This difference, known as the spherical angle excess, is proportional to the triangle's area and corresponds to the holonomy angle obtained by parallel transporting a vector around the triangle's perimeter.
Experiment with spherical properties
Open Gauss–Bonnet Spherical Excess to visualize how these quantities scale with the sphere's radius and the triangle's longitudinal span.
- Set Radius to 1, Longitude degrees to 90, and Orientation to East. Observe the Angle sum of 4.7124 (3π/2) and the Angle excess of 1.5708 (π/2).
- Decrease the Longitude degrees to 60 (θ = π/3). Notice the Area and Angle excess decrease proportionally to 1.0472 (π/3). The Angle sum decreases to 4.1888.
- Change the Orientation to West. The positive Area remains unchanged, but the Signed area and Holonomy angle become negative, reflecting the reversed path of parallel transport.
- Increase the Radius to 2. The Area increases by a factor of 4 (since it scales with R2), but the Curvature drops to 0.25. As a result, their product remains constant, and the Angle excess and Holonomy angle are unchanged.