Hyperbolic spaces
The revolution of geometry became an integral part of the mathematical reform which, by developing the group theory and transformation theory, dramatically expanded the world view of the Euclidean legacy. It is instructive how this apparently purely abstract and theoretical reform of thinking influenced the prominent mathematicians of the 19th century who worked and relied on each other. These new and often hard-to-accept paradigm shifts met physical reality and its theoretical and practical facts only in the 20th century.
In mathematical calculations and in geometry, group theory developed the systems of conditions which can be applied to transformations, the handling of calculations related to spaces on the basis of geometrical considerations, which can be applied in our present-day practices. Group theory determined the operations which conserve their properties during transformations, enabling the selection of calculations and spatial transformations which best fit the given tasks.
The impact of the reform is clearly shown by the fact that the relations of angles, and thus the trigonometric rule which was previously considered as a natural law, change during transformations. Transformations also help to form the shapes of practical objects as well.
The possibility to continuously change ordinary forms were illustrated by conic sections, the views of cones of light rays and the resulting images. The modifications of conic sections showed the characteristic features of ellipses, hyperbolas and parabolas. Their calculations made it possible to follow the movement of celestial bodies and—with further transformations—the important steps of understanding and modelling of all kinds of systems in the maze of causes and interactions.