This distortion greatly affects the navigation of ships and our observation in physics. In a Riemannian geometry two points are connected with the line of a Great Circle. The Great Circle has the same diameter as the sphere itself. Thus, one and only one line can connect two points. Due to this feature and the nature of the spherical plane, no parallel lines exist in a Riemannian geometry.
Every infinite line intersects another infinite line.
A Poincare geometry contains hyperbolic lines on a Cartesian plane. Since the lines are hyperbolic, no point can exist below or on the y-axis. The Poincare geometry has uses for modeling trajectories of thrown objects and electro-magnetic forces. But, these parallel lines occur within a specific boundary defined by two unique parallel lines. Rather than changing the definition of parallel lines, it changed the definition of distance.