What is meant by hyperbolic geometry?
What is meant by hyperbolic geometry?
Hyperbolic geometry, also called Lobachevskian Geometry, a non-Euclidean geometry that rejects the validity of Euclid’s fifth, the “parallel,” postulate. In hyperbolic geometry, two parallel lines are taken to converge in one direction and diverge in the other.
What is Euclidean geometry?
Euclidean geometry, the study of plane and solid figures on the basis of axioms and theorems employed by the Greek mathematician Euclid (c. 300 bce). In its rough outline, Euclidean geometry is the plane and solid geometry commonly taught in secondary schools.
What is Playfair’s axiom Class 9?
Playfair axiom is an axiom which can be used in place of the fifth postulate of Euclid. In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.
What is the difference between axiom and postulate?
What is the difference between Axioms and Postulates? An axiom generally is true for any field in science, while a postulate can be specific on a particular field. It is impossible to prove from other axioms, while postulates are provable to axioms.
What is called Play Fair axiom?
The axiom which says that, given a point not on a given line, there is precisely one line through the point parallel to the line. It is equivalent to the axiom of Euclid’s known as the parallel postulate. The name arises from its occurrence in a book by John Playfair (1748–1819), a British geologist and mathematician.
Which is a limiting case of Lobachevskii geometry?
Euclidean geometry can be obtained as a limiting case of Lobachevskii geometry when the two parallels passing through P merge into one, that is, when the set of all lines passing through P and not intersecting the given line A ′ A reduce to a unique line. Then the angle α = π / 2 for any a .
How did Nikolai Lobachevsky contribute to differential geometry?
He developed the angle of parallelism which depends on the distance the point is off the given line. In hyperbolic geometry the sum of angles in a hyperbolic triangle must be less than 180 degrees. Non-Euclidean geometry stimulated the development of differential geometry which has many applications.
Why was Lobachevsky called the Copernicus of geometry?
William Kingdon Clifford called Lobachevsky the ” Copernicus of Geometry” due to the revolutionary character of his work.
When did Poincare create model of Lobachevskii geometry?
Poincaré, in 1887, proposed a model of Lobachevskii geometry as the geometry of plane diametral sections of one of the sheets of a two-sheet hyperboloid, which can also be treated as the geometry of a sphere of purely imaginary radius in a pseudo-Euclidean space. These models can be generalized to the case of an n – dimensional space.