Poincaré Disc Model

Danijela Simić 🌐, Filip Marić 🌐 and Pierre Boutry 📧

December 16, 2019

This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.

Abstract

We describe formalization of the Poincaré disc model of hyperbolic geometry within the Isabelle/HOL proof assistant. The model is defined within the extended complex plane (one dimensional complex projectives space ℂP1), formalized in the AFP entry “Complex Geometry”. Points, lines, congruence of pairs of points, betweenness of triples of points, circles, and isometries are defined within the model. It is shown that the model satisfies all Tarski's axioms except the Euclid's axiom. It is shown that it satisfies its negation and the limiting parallels axiom (which proves it to be a model of hyperbolic geometry).

License

BSD License

Topics

Session Poincare_Disc

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Cite

× 
@article{Poincare_Disc-AFP,
  author  = {Danijela Simić and Filip Marić and Pierre Boutry},
  title   = {Poincaré Disc Model},
  journal = {Archive of Formal Proofs},
  month   = {December},
  year    = {2019},
  note    = {\url{https://isa-afp.org/entries/Poincare_Disc.html},
             Formal proof development},
  ISSN    = {2150-914x},
}
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