Hyperbolic Geometry

The present text brings one of the important existing definitions in Hyperbolic Geometry: of Ideal Point. This result also can be used as reply of a question proposal for Joo Lucas Marques Barbosa in its book ' ' Hyperbolic geometry Ed. of UFG, 2002; Chapter 6; Section 6.2; p 65. DEFINITION (Of ideal point) In plan (plain Euclidean) with the axioms of Hyperbolic Geometry, either R= r r is straight line in and. It considers in R the relation: r/s pertaining the R, rR*s if, and only if, r = s or r is parallel the s in the same felt. It observes that R* is a equivalence relation, a time that is valid the properties reflexiva, symmetry and transitiva. He is accurately to each pertaining equivalence classroom rd or reverse speed the R/ that we call ideal point (of straight line r).

R/ it is the set of the ideal points or points in the infinite of the hyperbolic plan. Obs: rd is a point that if exactly identifies with a beam of straight lines parallel bars in one felt to some given straight line r. We also denote rd for. Fixed a straight line r we can associate the r colon ideal? + e? -, which can be juxtaposed the r, forming a straight line ' ' longa' ' or a straight line with the points in the infinite ' ' anexados' '. Soon, if r* is one of such straight lines fixed, has: r* = r U rd U reverse speede r* is contained in and U R/. It notices that, R/ is an abstract space and that H = U R/, where H represents the hyperbolic space Perceives finally, that r* is straight line of H if r* = r U rd U reverse speed, where pertaining r the R and rd and reverse speed are the classrooms of straight lines parallel bars to the right and the left respectively.