Comparing and Contrasting Euclidean, Spherical, and Hyperbolic Geometries

With respect to Euclidean geometry, spherical geometry and hyperbolic geometry, there are many similarities and differences between them. For example, for spherical or hyperbolic geometry, Euclidean geometry may not be correct. For one or two geometries, there are a number of things that apply, not other geometries. However, the properties of all three geometries may be correct. From these points, we were able to achieve the purpose of this article. This article is my opportunity to demonstrate my understanding of Euclidean geometry, spherical geometry, hyperbolic geometry.

In this article, I will learn about two types of non-Euclidean geometry. The first type, called hyperbolic geometry, is the geometry discovered by Bolyai and Lobachevsky. (The great Carl Friedrich Gauss (1777-1855) also discovered this geometry; but he did not announce his work as he was afraid that this would controversial for establishment The fifth hypothesis of Jared is replaced by this. Since the famous Pythagorean theorem depends on a parallel hypothesis, it is Euclidean geometry's theorem. However, we will propose the Pythagorean theorem encountered in non - Euclidean variations of this theorem in Chapter 5 and Chapter 6 and unified in Chapter 7. This is recently 7.4.7. so

The Pythagorean theorem is derived from the axiom of Euclidean geometry and in fact the above Pythagorean theorem does not apply to non-Euclidean geometry. (In fact, the Pythagorean theorem proves to be equivalent to the Euclidean parallel (5) hypothesis.) In other words, in non-Euclidean geometry, the relationship between the sides of the triangle must be adopted . Non Pythagoras style. For example, in a spherical geometry, all three sides (a, b, c, etc.) of a right triangle define the octant of a unit sphere and its length equals π / 2 and all angles are right angles, I will violate it. Since a 2 + b 2 ≠ c 2, the Pythagorean theorem

These differences between Euclid and its fifth axiom created other "non-Euclidean" geometric forms. One of them is "spherical". Spherical geometry was created and inferred by German mathematician Georg Friedrich Bernhard Riemann. Lehman was born on September 7, 1826. As he grew up, he created many lectures and papers. He is interested in the idea that Euclidean geometry is based on basic planes and surfaces and what happens when they occur in balls or spherical galaxies. Although the theory of Riemann was developed in the mid 1800's, it will not affect mathematics world until a couple of years. Spherical geometry is an alternative to the standard plan view where the plane is the surface of a sphere. The line is defined as the "great circle" of a sphere. A large circle is defined by the plane including any two points on the surface of the sphere and the center of the sphere and intersecting the plane and the surface of the sphere.