On Rational Trigonometry in Euclidean and Non-Euclidean Geometries
Abstract
Basic concepts and rules of rational trigonometry for Euclidean geometry were first formulated in 2005 by N.J. Wildberger. Later, he expands its concepts for hyperbolic geometry. The essence of the "new" trigonometry is to override the trigonometric ratios without the usage of trigonometric functions by introducing the traditional distances and angles of such concepts as quadrance and spread instead. This approach eliminates the usage of trigonometric tables and, as a result, approximate calculations. This means that it is often more accurate. Despite the fact that the ideas of rational trigonometry caused a mixed impression in the mathematical community, methods of rational trigonometry have been used in solving problems in geometry, combinatorics, and robotics. In this paper, formulas of the inner product and the module of cross product of the vectors of Euclidean space in terms of rational trigonometry are obtained; the basic rules of rational spherical and Lobachevsky’s trigonometry are derived.
DOI 10.14258/izvasu(2015)1.1-17
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