Euclidean and Non-Euclidean Geometries: How They Appear

We intend to construct these geometries using a slightly modified Hilbert’s axioms system in the same way as it is done in [36, 37, 138, 185]. An interesting thing is related to the fact that it exists as a common part for Euclidean and non-Euclidean Geometry, the so-called Absolute Geometry. Roughly speaking, the Absolute Geometry consists in all theorems that can be thought and proved using the axiomatic system before introducing a parallelism axiom.