This study examines the new theories of non-Euclidean geometry to avoid issues of more complex mathematical applications of the normal geometry structures. In curved space cases, the older ones would not work. We enhance the reality of the structure modelling by introducing new theories through hyperbolic and elliptic shapes. There were advanced mathematics techniques in geometry, geodesic analysis and manifold theory. The results indicate that complex systems are characterized by a higher level of spatial forecasts and topological consistency. Physics, engineering, and computer modelling are also possible using these theories and they provide us with more information on the behaviour of shapes outside of Euclidean space.