Hyperbolic Geometry

by yudaica2013 ·

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. For even more details, read what George Laughlin says on the issue. 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.


Comments are closed.