KONSISTENSI AKSIOMA-AKSIOMA TERHADAP ISTILAH-ISTILAH TAKTERDEFINISI GEOMETRI HIPERBOLIK PADA MODEL PIRINGAN POINCARE

Febriyana Putra Pratama, Julan Hernadi
DOI: 10.24269/ed.v2i2.148

Abstract


This research aims to know the interpretation the undefined terms on Hyperbolic geometry and it’s consistence with respect to own axioms of Poincare disk model. This research is a literature study that discusses about Hyperbolic geometry. This study refers to books of Foundation of Geometry second edition by Gerard A. Venema (2012), Euclidean and Non Euclidean Geometry (Development and History)  by Greenberg (1994), Geometry : Euclid and Beyond by Hartshorne (2000) and Euclidean Geometry: A First Course by M. Solomonovich (2010). The steps taken in the study are: (1) reviewing the various references on the topic of Hyperbolic geometry. (2) representing the definitions and theorems on which the Hyperbolic geometry is based. (3) prepare all materials that have been collected in coherence to facilitate the reader in understanding it. This research succeeded in interpret the undefined terms of Hyperbolic geometry on Poincare disk model. The point is coincide point in the Euclid on circle . Then the point onl γ is not an Euclid point. That point interprets the point on infinity. Lines are categoried in two types. The first type is any open diameters of   . The second type is any open arcs of circle. Half-plane in Poincare disk model is formed by Poincare line which divides Poincare field into two parts. The angle in this model is interpreted the same as the angle in Euclid geometry. The distance is interpreted in Poincare disk model defined by the cross-ratio as follows. The definition of distance from  to  is , where  is cross-ratio defined by  . Finally the study also is able to show that axioms of Hyperbolic geometry on the Poincare disk model consistent with respect to associated undefined terms.


Keywords


Geometry, Hyperbolic Geometry, Poincare Disk Model

References


Greenberg, M. J. 1994. Euclidean and Non-Euclidean Geometries (Development and History) (3rd ed.). New York: W. H. Freeman and Company.

Hartshorne, Robin. 2000. Geometry: Euclid and Beyond. New York: Springer-Verlag.

Solomonovuch. M. 2010. Euclidean Geometry : A First Course. New York: iUnivers, Inc.

Venema, G. A. 2012. Foundations of Geometry (2nd ed.).Boston: Pearson.


Full Text: PDF

Refbacks

  • There are currently no refbacks.