Angular Distance: Spherical and Hyperbolic Geometry

The geometry of the sphere is familiar to us, from everyday life as well as from geography. It is a part of the metric geometry of space, yet it represents something of its own in it. The distance of two points on the unit sphere is its angle, measured from the center; thus the angle takes on a whole new meaning: spherical distance. There is a second geometry which is similarly defined, but has exactly opposite properties in many respects: Here the surrounding Euclidean space is replaced by ℝ n + 1 $$\mathbb {R}^{n+1}$$ with the Lorentzian scalar product, the spacetime of Special Relativity. Relativity The “unit sphere” in this space is a model of the non-Euclidean geometry of Lobachevski and Bolyai, which had caused a great surprise in the early nineteenth century because it contradicted the common belief that Euclidean geometry Geometry was the only conceivable geometry.