What is difference b/w Euclidean and non Euclidean Geometry

Euclidean geometry:

Euclidean geometry is the study of geometry based on the assumptions made by the ancient Greek mathematician Euclid, where a straight line is the shortest distance between two points, and parallel lines never meet. This geometry is the study of the properties of figures that can be drawn on a plane or in space.

Non-Euclidean geometry:

 Non-Euclidean geometry, on the other hand, is a general term for geometries that are not based on the assumptions made by Euclid. These geometries do not necessarily assume that a straight line is the shortest distance between two points or that parallel lines never meet. Non-Euclidean geometries can be used to study curved surfaces or spaces, and they are often used in modern physics to describe the curvature of space-time. One of the most well-known non-Euclidean geometries is hyperbolic geometry, which describes the properties of figures drawn on a surface with constant negative curvature. Another type of non-Euclidean geometry is elliptic geometry, which describes the properties of figures drawn on a surface with constant positive curvature.

Summary:

 In summary, the main difference between Euclidean and non-Euclidean geometries is that Euclidean geometry is based on the assumptions made by Euclid, while non-Euclidean geometries do not necessarily follow those assumptions and may describe the properties of figures on curved surfaces or spaces.

Why do we need non-Euclidean Geometry?

Non-Euclidean geometry is an essential branch of geometry that has applications in various fields, including physics, astronomy, and computer graphics. It provides a framework for studying geometrical objects and spaces that do not conform to Euclid's axioms of geometry.

One of the main reasons we need non-Euclidean geometry is that it provides a more accurate description of the geometry of the universe. Euclidean geometry assumes that space is flat and that the sum of the angles in a triangle is always 180 degrees. However, in reality, space is curved, and the sum of the angles in a triangle can be greater or less than 180 degrees, depending on the curvature of the space.

Non-Euclidean geometry also has applications in the theory of relativity. In Einstein's theory of general relativity, the curvature of space-time is described by non-Euclidean geometry. Without the framework provided by non-Euclidean geometry, it would be impossible to accurately describe the behavior of gravity and the motion of objects in the universe.

Additionally, non-Euclidean geometry has practical applications in computer graphics and game design fields. It provides a way to represent and manipulate objects in three-dimensional space more accurately and efficiently than traditional Euclidean geometry methods.

In summary, non-Euclidean geometry is essential because it provides a more accurate description of the geometry of the universe, has applications in the theory of relativity, and has practical applications in fields such as computer graphics and game design.