What do you mean 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 are examples of hyperbolic geometry?

The best-known example of a hyperbolic space are spheres in Lorentzian four-space. The Poincaré hyperbolic disk is a hyperbolic two-space. Hyperbolic geometry is well understood in two dimensions, but not in three dimensions. Hilbert extended the definition to general bounded sets in a Euclidean space.

Do parallelograms exist in hyperbolic geometry?

A parallelogram is defined to be a quadrilateral in which the lines containing opposite sides are non-intersecting. Show with a generic example that in hyperbolic geometry, the opposite sides of a parallelogram need not be congruent.

Who developed hyperbolic geometry?

Finally, in the 19th century, two mathematicians published two separate descriptions of a geometry that satisfied all but the fifth of Euclids postulates. The two mathematicians were Euginio Beltrami and Felix Klein and together they developed the first complete model of hyperbolic geometry.

What does hyperbolic space look like?

at all points, i.e. a sphere has constant positive Gaussian curvature. Hyperbolic Spaces locally look like a saddle point. . Since each point of hyperbolic space locally looks like an identical saddle, we see that hyperbolic space has constant negative curvature.

Who is the father of hyperbolic geometry?

Carl F. Gauss
Over 2,000 years after Euclid, three mathematicians finally answered the question of the parallel postulate. Carl F. Gauss, Janos Bolyai, and N.I. Lobachevsky are considered the fathers of hyperbolic geometry.

What is the hyperbolic axiom?

Axiom 2.1 (The hyperbolic axiom). Given a line and a point not on the line, there are infinitely many lines through the point that are parallel to the given line. A consistent model of this axiomatic system implies that the parallel pos- tulate is logically independent of the first four postulates.

Do parallel lines exist in hyperbolic geometry?

In Hyperbolic geometry there are infinitely many parallels to a line through a point not on the line. However, there are two parallel lines that contains the limiting parallel rays which are defined as lines criti- cally parallel to a line l through a point P /∈ l.

Are there similar triangles in hyperbolic geometry?

While the sides of hyperbolic triangles can get as large as you want, the area of any triangle is less than pi. There is no concept of similar triangles — if two triangles have the same angles then they are congruent.

What is a good way to visualize hyperbolic geometry?

One way to visualize hyperbolic space is as a collection of points each of which locally looks like a saddle point. If we take a plane containing a normal vector to a saddle point, and intersect it with the saddle, we see that the curve on the saddle radically changes as we rotate the plane.

What are the three basic types of geometry?

There are three basic types of geometry: Euclidean, hyperbolic and elliptical. Although there are additional varieties of geometry, they are all based on combinations of these three basic types.

Who invented hyperbolic geometry?

The term “hyperbolic geometry” was introduced by Felix Klein in 1871. Klein followed an initiative of Arthur Cayley to use the transformations of projective geometry to produce isometries. The idea used a conic section or quadric to define a region, and used cross ratio to define a metric .

What is hyperbolic trigonometry?

hyperbolic function. Any of a set of six functions related, for a real or complex variable x, to the hyperbola in a manner analogous to the relationship of the trigonometric functions to a circle, including: The hyperbolic sine, defined by the equation sinh x = 12(ex – e-x).