Hyperboloid Model - Term Paper Example

The paper’s aim is to comprehensively discuss the concept of the ‘Hyperboloid Model’, in relation to application in the field of Geometry and Mathematics in general. While focus will be placed more on the geometrical application, influences and effects of the model, the paper will also delve into other applications. Conclusively, it will portray the functional application of the model, essential in gaining required accuracy. Towards better understanding the vital importance of the hyperboloid model, there is need of a historical analysis of the concept, in terms of geometrical application.

To be noted, as Alekseevskij, Vinberg and Solodovnikov (1993) portray, is that the study of prevailing relations amongst hyperbolic, spherical and Euclidean geometries historically dates back to the early 19th century. This was in an attempt at proving Euclid’s fifth postulate. Accordingly, it is towards ascertaining this that C. F. Gauss was able to subsequently discover, in the 1820s, the concept of hyperbolic geometry. Influential is that only a few years were to pass, before this form of geometry was to be independently re-discovered by both J.

Bolyai (1832) and N. Lobacheviski (1829). Notable is that the concept’s founders were in agreement, in terms of providing its strongest evidence for its consistency. This was based upon the duality present, between spherical and hyperbolic trigonometries (Alekseevskij, Vinberg & Solodovnikov, 1993). Initially demonstrated by Lambert – in his [L1770] 1770 memoir – the duality aspect present between the two forms of trigonometries is vivid in a variety of theorems. Inclusive is the ‘law of sines’, which can be affirmed in a form that is applicable in hyperbolic, Euclidean and spherical geometries.

Accordingly, it is towards proving the prevailing consistency of hyperbolic geometry that necessitated the building of diverse analytical models upon the Euclidean plane. This is perhaps the reason why Beltrami E.