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Geometry
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
The essential difference between Euclidean and non-Euclidean geometry is the nature of parallel lines. Changing the parallel postulate results in other geometries:(5; for hyperbolic geometry) Through a point not on a given straight line, infinitely many lines can be drawn that never meet the given line. For example, the surface of a hyperboloid is an example of hyperbolic geometry.(5; for elliptic geometry) Through a point not on a given straight line, no lines can be drawn that never meet the given line. For example, the surface of a sphere is an example of elliptic geometry.
The surface theory of Gauss
Published in Martin Vermeer, Antti Rasila, Map of the World, 2019
Johann Carl Friedrich Gauss (1777 – 1855), often referred to as Princeps mathematicorum (“the foremost of mathematicians”), was among the first mathematicians to consider non-Euclidean geometry, like his contemporaries Bolyai1 and Lobachevsky2.
Towards the next-generation GIS: a geometric algebra approach
Published in Annals of GIS, 2019
Linwang Yuan, Zhaoyuan Yu, Wen Luo
Non-Euclidean geometry expands the mathematical and physical space from absolute to relative parameters. This lays the mathematical foundation for the change from Newton’s absolute space view (Euclidean space) to Einstein’s relativistic space view (Minkowski space and Riemannian space). GA is as an important breakthrough in the development of non-Euclidean geometry, which is a unified description language used to link geometry and algebra, mathematics and physics, and ultimately achieve the unified expression of Euclidean, Minkowski and Riemann spaces. GA, as the name implies, is characterized by representing, constructing and manipulating geometric objects with an algebraic language. Various geometric systems (such as projective geometry, affine geometry, conformal geometry, differential geometry, etc.) and algebraic systems (such as calculus, tensor algebra, Boolean algebra, space-time algebra, etc.) can be mapped to GA spaces (Figure 1). Therefore, GA can be used to build an unified framework of geography and space-time, for innovatively constructing unified multidimensional representations, and develop analysis and modelling methods in GIS. It is also compatible with the data models, computational and analytical methods of a more general GIS based on mathematical theory.