China
Africa
Explore chapters and articles related to this topic
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 KdV Soliton
Published in David S. Ricketts, Donhee Ham, Electrical Solitons, 2011
The derivation of the periodic solution to the KdV equation utilizes elliptic functions, which are unfamiliar to most scientists and engineers. They represent solutions to elliptic integrals which appear in the derivations of many physical systems [43] which have elliptic geometry. Moreover, elliptic integrals provide a means to solve nonlinear, polynomial differential equations. Two prime examples are the large angle solution for the pendulum [44] and also the solution to the KdV equation. Appendix 2.A provides more details and insights into elliptic integrals and functions.
Dynamic hyperbolic geometry: building intuition and understanding mediated by a Euclidean model
Published in International Journal of Mathematical Education in Science and Technology, 2018
Luis Moreno-Armella, Corey Brady, Rubén Elizondo-Ramirez
But there was still a deep question that haunted hyperbolic inquiries: the question of whether or not a physical model existed that realized the new geometry. Euclidean geometry, of course, found its realization in the flat surface of the plane. Similarly, the sphere provided a model for elliptic geometry (a geometry without parallels). People were rather familiar with this geometry through navigational work, in particular, and a theory of spherical trigonometry was already established and in use. The concreteness of the spherical surface as a model that realized elliptic geometry was an incredibly powerful support for that theory.