Non-Euclidean geometry is a branch of geometry that explores the properties of curved space. It differs from traditional Euclidean geometry in that it does not follow the parallel postulate, which states that given a line and a point not on the line, there is exactly one line through the point that is parallel to the given line.
The two main types of non-Euclidean geometry are hyperbolic geometry and elliptic geometry. Hyperbolic geometry is characterized by the violation of the parallel postulate, while elliptic geometry is characterized by the violation of the angle sum theorem.
Non-Euclidean geometry has many practical applications, particularly in the fields of physics and engineering. It is used to describe the curved space-time of general relativity, as well as in the design of curved structures such as bridges and domes.
Non-Euclidean geometry differs from Euclidean geometry in several ways. The most significant difference is the violation of the parallel postulate, which leads to the discovery of new properties and theorems. Additionally, non-Euclidean geometry is based on different axioms and has different geometric constructions and proofs.
The development of non-Euclidean geometry was primarily led by mathematicians such as Nikolai Lobachevsky, János Bolyai, and Bernhard Riemann in the 19th century. Later, Albert Einstein's theory of general relativity also heavily relied on the principles of non-Euclidean geometry.