Skip to main content
Log in

Hyperbolic plane geometry revisited

  • Published:
Journal of Geometry Aims and scope Submit manuscript

Abstract

Using the method of C. Vörös, we establish results in hyperbolic plane geometry, related to triangles and circles. We present a model independent construction for Malfatti’s problem and several trigonometric formulas for triangles.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bell A.: Hansen’s Right Triangle Theorem, Its Converse and a Generalization. Forum Geom. 6, 335–342 (2006)

    MATH  MathSciNet  Google Scholar 

  2. Cayley A.: Analytical researches connected with Steiner’s extension of Malfatti’s problem. Philos. Trans. R. Soc. Lond. 142, 253–278 (1852)

    Article  Google Scholar 

  3. Casey, J.: A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples 5th. ed. Hodges, Figgis and Co., Dublin (1888)

  4. Casey, J.: A Treatise on Spherical Trigonometry, and its Application to Geodesy and Astronomy, with Numerous Examples Hodges. Figgis and CO., Grafton-ST. Longmans, Green, and CO., London (1889)

  5. Dörrie H.: Triumph der Mathematik. Physica-Verlag, Würzburg (1958)

    Google Scholar 

  6. G.Horváth, Á.: Malfatti’s problem on the hyperbolic plane. Stud. Sci. Math. Hung. 51/2, 201–212 (2014). doi:10.1556/SScMath.2014.1276

  7. G.Horváth, Á.: Formulas on hyperbolic volume. Aequ. Math. 83/1, 97–116 (2012)

  8. G.Horváth, Á.: Addendum to the paper “Hyperbolic plane geometry revisited”. doi:10.13140/2.1.2138.8483 or http://www.math.bme.hu/~ghorvath/hyperbolicproofs

  9. Hart A.S.: Geometric investigations of Steiner’s construction for Malfatti’s problem. Q. J. Pure Appl. Math. 1, 219–221 (1857)

    Google Scholar 

  10. Johnson, R.A.: Advanced Euclidean Geometry, An Elementary Treatise on the Geometry of the Triangle and the Circle. Dover Publications, Inc. New York (The first edition published by Houghton Mifflin Company in 1929) (1960)

  11. Malfatti G.: Memoria sopra un problema sterotomico. Memorie di Matematica e di Fisica della Società à Italiana delle Scienze. 10, 235–244 (1803)

    Google Scholar 

  12. Molnár, E.: Inversion auf der Idealebene der Bachmannschen metrischen Ebene. Acta Math. Acad. Sci. Hung. 37/4, 451–470 (1981)

  13. Steiner’s gesammelte Werke (herausgegeben von K. Weierstrass), Berlin (1881)

  14. Steiner, J.: Einige geometrische Betrachtungen. Journal für die reine und angewandte Mathematik 1/2, 161–184 (1826), 1/3, 252–288 (1826)

  15. Szász, P.: Introduction to Bolyai–Lobacsevski’s geometry. Akadémiai Kiadó, Budapest (1973) (in Hungarian)

  16. Dr. Vörös, C.: Analytic Bolyai Geometry. Budapest (1909) (in Hungarian)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ákos G. Horváth.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Horváth, Á.G. Hyperbolic plane geometry revisited. J. Geom. 106, 341–362 (2015). https://doi.org/10.1007/s00022-014-0252-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00022-014-0252-0

Mathematics Subject Classification

Keywords

Navigation