On the distinction between one-dimensional Euclidean and hyperbolic spaces

The difference between Euclidean and hyperbolic spaces is clear starting with dimension two. However, the difference between elliptic space and both Euclidean and hyperbolic ones can be described also for dimension one. Does it mean that there is no difference between one-dimensional Euclidean and hyperbolic lines, or it is necessary to better define the difference between them? This paper proposes one possible way to draw clear distinction between one-dimensional Euclidean and hyperbolic lines.