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14 - The Poincaré Disk

Gerard A. Venema
Affiliation:
Calvin College
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Summary

In this final chapter we study the Poincaré disk model for hyperbolic geometry. Even though hyperbolic geometry is a non-Euclidean geometry, the topic is nonetheless appropriate for inclusion in a treatment of Euclidean geometry because the Poincaré disk is built within Euclidean geometry. The main tool used in the construction of the Poincaré disk model is inversion in Euclidean circles, so the tools you made in Chapter 13 will be used in this chapter to perform hyperbolic constructions. Many of the constructions in this chapter were inspired by those in the beautiful paper [7] by Chaim Goodman-Strauss.

It was Eugenio Beltrami (1835–1900) who originated the idea of representing hyperbolic geometry within Euclidean geometry. There are many ways in which to construct models of hyperbolic geometry, but the Poincaré disk model is probably the best known. One reason for its popularity is the great beauty of the diagrams that can be constructed in it. The model is named for the French mathematician Henri Poincaré (1854–1912) who first introduced it.

The Poincaré disk model for hyperbolic geometry

A model for a geometry is an interpretation of the technical terms of the geometry (such as point, line, distance, angle measure, etc.) that is consistent with the axioms of the geometry. The usual model for Euclidean geometry is ℝ2, the Cartesian plane, which consists of all ordered pairs of real numbers.

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Publisher: Mathematical Association of America
Print publication year: 2013

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  • The Poincaré Disk
  • Gerard A. Venema, Calvin College
  • Book: Exploring Advanced Euclidean Geometry with GeoGebra
  • Online publication: 05 August 2013
  • Chapter DOI: https://doi.org/10.5948/9781614441113.016
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  • The Poincaré Disk
  • Gerard A. Venema, Calvin College
  • Book: Exploring Advanced Euclidean Geometry with GeoGebra
  • Online publication: 05 August 2013
  • Chapter DOI: https://doi.org/10.5948/9781614441113.016
Available formats
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Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • The Poincaré Disk
  • Gerard A. Venema, Calvin College
  • Book: Exploring Advanced Euclidean Geometry with GeoGebra
  • Online publication: 05 August 2013
  • Chapter DOI: https://doi.org/10.5948/9781614441113.016
Available formats
×