Where is elliptical geometry used

He redefined the notion of a point as a set of antipodal points. With this definition, any two points determine a unique line so that the traditional form of Euclid's first postulate is restored.

Thus modified, spherical geometry became what Klein called elliptical geometry. Spherical trigonometry This is the branch of spherical geometry dealing with the ratios between the sides and angles of spherical triangles.

Spherical astronomy , of importance in positional astronomy and space exploration, is the application of spherical trigonometry to determinations of stellar positions on the celestial sphere. Generalization to elliptical geometry It was Felix Klein who first saw clearly how to rid spherical geometry of its one blemish: the fact that two lines have not one but two common points. My maths channel on Youtube. A great circle is a circle whose center lies at the center of the sphere, as shown in Figure No matter how they are drawn, each pair of great circles will always intersect.

As a result, parallel lines do not exist. The mathematician Bernhard Riemann ? Ironically enough, he was born about the same time that hyperbolic geometry was developed by Bolyai and Lobachevsky, and he was instrumental in convincing the mathematical world of the merits of non-Euclidean geometry.

Although hyperbolic geometry needed only to contradict the Parallel Postulate, spherical geometry doesn't get off so easy. There are three changes in Euclid's axiomatic system necessary to successfully create spherical geometry. Remember that lines in Euclidean geometry are the great circles in spherical geometry. Spherical geometry is important in navigation, because the shortest distance between two points on a sphere is the path along a great circle.

All lines have the same finite length. All great circles have a finite length: the circumference of the circles they describe. It can be visualized as the surface of a sphere on which "lines" are taken as great circles. In elliptic geometry, the sum of angles of a triangle is. Weisstein, Eric W. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more.

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