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Chapter7Geometry on Surfaces

In hyperbolic geometry \((\mathbb{D},{\cal H})\) and elliptic geometry \((\mathbb{P}^2,{\cal S})\text{,}\) the area of a triangle is determined by the sum of its angles. This is a significant difference from Euclidean geometry, in which a triangle with three given angles can be built to have any desired area. Does this mean that if a bug lives in a world adhering to elliptic geometry, it can never stumble upon a triangle with three right angles having area \(3\pi\text{?}\) Yes and no. In the elliptic geometry as defined in Chapter 6, no such triangle exists because a triangle with 3 right angles must have area \((\frac{\pi}{2} + \frac{\pi}{2} + \frac{\pi}{2})-\pi = \frac{\pi}{2}\text{.}\) So the answer appears to be yes. However, the elliptic geometry \((\mathbb{P}^2,{\cal S})\) models the geometry of the unit sphere, and this choice of sphere radius is somewhat arbitrary. What if the radius of the sphere changes? Imagine a triangle with three right angles having one vertex at the north pole and two vertices on the equator. If the sphere uniformly expands, the angles of the triangle will stay the same, but the area of the triangle will increase. So, if a bug is convinced she lives in a world with elliptic geometry, but is also convinced she has found a triangle with three right angles and area \(3\pi\text{,}\) the bug might be drawn to conclude she lives in a world modeled on a larger sphere than the unit 2-sphere.

The key geometrical property of a space dictating the relationship between the angles of a triangle and its area is called curvature. Curvature also dictates the relationship between the circumference of a circle and its radius.