The first postulate states that given two points, there is a unique, straight line segment between them. Here, a straight line is defined to be a geodesic, a curve going from one point to the other using the minimum arc length. On a sheet of paper, this statement is quite basic, as it essentially states the existence of one of the most fundamental tools of Euclidean Geometry: the ruler. However, if you consider curved geometries, this statement is not at all obvious. For example, in spherical geometry, the geodesic between two points is an arc of the circle passing through the two points with the same diameter as the sphere. Typically this circle is unique, meaning that there is only one circle satisfying the required properties, but if the two points are antipodal (on opposite sides of the sphere), there are infinitely many such circles.

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While the first postulate doesn't hold true in spherical geometry, it does in hyperbolic geometry. This interactive demonstrates how geodesics work in hyperbolic geometry, and how projection into the Poincare Disk Model warps the notion of distance and arc length: