Universal Hyperbolic Geometry
A recent approach (2009) has been outlined by N. J. Wildberger, termed universal hyperbolic geometry, based on rational trigonometry, his reformulation of the main metrical properties from Euclidean geometry (quadrance and spread). These two measures, which appear quite distinct in their Euclidean setting, become completely dual when transferred to a projective (hyperbolic) one. All 'points' and 'lines' (projectively, lines and planes) become related simply as 'pole and polar', their correspondences being determined by a 'null circle'. The duality in the metrical properties derived ('hyperbolic quadrance' and 'hyperbolic spread') reflect this complete point-line correspondence from the non-metrical point of view. Though naturally synthetic however, the approach is developed purely algebraically and 'without pictures'. As expected, hyperbolic analogues exist for the five main laws of rational trigonometry.
In contrast to the usual models, universal hyperbolic geometry does not require real numbers, calculus, or transcendental functions. In turn an algebraic approach allows any valid result obtained over one field not of characteristic two (and not merely the usual infinite field of the rational numbers) to in principle apply directly over any other field.
(This 'field independence' of theorems is what qualifies use of the term universal)
Read more about this topic: Hyperbolic Geometry
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