Euclidean Geometry was introduced by the mathematician Euclid in 330 B. C. It is the basic form of geometry based on definitions and points, lines and planes. Euclidean geometry includes various but fixed numbers of postulates like the parallel postulate and is studied in a flat space like a piece of paper or simple board. The main postulate of Euclidean theory is based on fact that to a point in space, there is only one parallel line to that point.
The Riemannian geometry negates the Euclidean postulate by saying that through a point in space lines can be drawn that eventually meet and thus, denies any existence of parallel lines while Lobachevskian geometry postulates that although there is existence of parallel lines however, for a single point in space there can be more than two parallel lines. The difference between Euclidean geometry and Riemannian and Lobachevskian geometry is that the latter ones define geometry in different spaces.
Euclidean geometry studies theories in a flat space while Riemannian geometry considers studies in curved spaces. Unlike both the theories, Lobachevskian geometry revolves around saddle-shaped space. The Euclidean geometry has its limitations of being refrained to a much smaller concept of space and lines and thus, can not be applied in detailed scientific studies while Lobachevskian geometry is used by various scientists in studying orbit predictions of objects, astronomy and many more.
Riemannian geometry is used in studies related to local behavior of space, topological type of manifold and behavior of various points located at large distances. The Euclidean geometry served as the beginning point for Riemannian and Lobachevskian geometry. The systematic study of spatial figures by Euclidean geometry came up with five main postulates among with the fifth postulate regarding the parallel lines initiated the rest of non-Euclidean geometry.
Thus, the main difference between the three theories is the rejection of parallel line postulate of Euclid’s theory by Lobachevskian and Riemannian geometries. Later on, the Riemannian and Lobachevskian geometries differed from each other on basis of concept of parallel lines, where in Riemannian concept, all the lines in the space always meet up and thus create a shape and there are no parallel lines, whereas in Lobachevskian geometry there are several parallel lines that are not always straight but form a saddle like shape up to infinSample Essay of EduBirdie.com