Types of Non-Euclidean Geometry

There are two types of figures classified based on Euclid’s parallel postulate. Figures that deviate from satisfying the parallel postulate are categorized as non-Euclidean. The main types of non-Euclidean figures are the hyperbola and ellipse. Non-Euclidean geometry is further divided based on the shapes of these figures into two branches:

• Hyperbolic Geometry
• Elliptical Geometry

Hyperbolic Geometry

Hyperbolic Geometry, a departure from Euclidean principles, was first conceptualized within Euclid’s postulates. It was established that hyperbolic geometries differ only in scale.

• In the mid-19th century, it was demonstrated that hyperbolic surfaces must possess constant negative curvature.
• Eugenio Beltrami’s pseudosphere, described in 1868, showed constant negative curvature but wasn’t a complete model for hyperbolic geometry.
• David Hilbert, in 1901, showed the impossibility of defining a complete hyperbolic surface using real analytic functions.
• However, Nicolaas Kuiper later proved the existence of such a surface in 1955, and William Thurston provided a construction in the 1970s.
• Three models were Klein-Beltrami, Poincaré disk, and Poincaré upper half-plane, these models helped in visualizing hyperbolic geometry despite some distortion.

Elliptic Geometry

Elliptic Geometry, another departure from Euclidean geometry, was first explored within the framework of Euclid’s postulates. Unlike hyperbolic geometry, elliptic geometries differ only in scale.

• In the mid-19th century, it was established that elliptic surfaces must have constant positive curvature.
• Eugenio Beltrami, in 1868, described a surface called the pseudosphere, displaying constant positive curvature.
• However, this pseudosphere isn’t a complete model for elliptic geometry, as straight lines on it may intersect themselves and can’t be extended beyond a certain point.
• Unlike hyperbolic and Euclidean geometry, elliptic geometry doesn’t have a flat model that can be represented on a plane without distortion.

Primarily only Hyperbolic and Elliptical Geometry are types of Non-Euclidean Geometry but Spherical Geometry also forms a part of Non-Euclidean Geometry. Hence, we will have a look on Spherical Geometry.

Spherical Geometry

Spherical Geometry is a non-Euclidean geometry that focuses on the surface of a sphere. In this geometry, space is represented by the curved surface of a sphere, which exhibits constant positive curvature.

• Unlike Euclidean geometry, spherical geometry lacks parallel lines, and any two great circles on a sphere intersect at two points.
• Spherical triangles, formed by great circle arcs, replace the traditional triangles of Euclidean geometry, with their angles summing to more than 180 degrees.
• Distances on a sphere are measured along great circle arcs, making the shortest distance between two points along a segment of a great circle.
• Spherical geometry finds practical applications in navigation, astronomy, and geography, particularly when considering the Earth’s surface as a sphere for calculations.

Non-Euclidean Geometry refers to the branch of mathematics that deals with the study of geometry on Curved Surfaces. It is a different way of studying shapes compared to what Euclid, an ancient mathematician, taught. There are two main types: hyperbolic and elliptic geometries. In these, we change the working of lines which gives us different shapes than usual. Hyperbolic shapes have a saddle-like curve, and elliptic shapes have a round curve.

In this article, we will understand the various concepts related to non-euclidean geometry like definition, the historical background of non-euclidean geometry, its principles, its application, and the types of noon-euclidean geometry.