On Rational Trigonometry in Euclidean and Non-Euclidean Geometries
DOI:
https://doi.org/10.14258/izvasu(2015)1.1-17Keywords:
rational trigonometry, spread, quadrance, spherical trigonometry, Lobachevsky’s trigonometryAbstract
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
Downloads
References
Wildberger N.J. Divine Proportions: Rational Trigonometry to Universal Geometry. - Sydney, 2005.
2
Wildberger N.J. Universal Hyperbolic Geometry I: Trigonometry // Geom. Dedicata. - 2013. - V. 163.
3
Коксетер Г.С.М., Грейтцер С.П. Новые встречи с геометрией. - М., 1978. (Серия «Библиотека математического кружка»).
4
Ефимов Н.В. Высшая геометрия : учеб. для вузов. - 5-е изд. - М., 1971.
Downloads
Issue
Section
License
Izvestiya of Altai State University is a golden publisher, as we allow self-archiving, but most importantly we are fully transparent about your rights.
Authors may present and discuss their findings ahead of publication: at biological or scientific conferences, on preprint servers, in public databases, and in blogs, wikis, tweets, and other informal communication channels.
Izvestiya of Altai State University allows authors to deposit manuscripts (currently under review or those for intended submission to Izvestiya of Altai State University) in non-commercial, pre-print servers such as ArXiv.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License (CC BY 4.0) that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
