Sensos-e Vol: II Num: 1  ISSN 2183-1432
URL: http://sensos-e.ese.ipp.pt/?p=7997

The Riemann Sphere in Geogebra

Autor: Jose Manuel Dos Santos Dos Santos Afiliação: Instituto GeoGebra Portugal
Autor: Ana Breda Afiliação: Universidade de Aveiro

Abstract: The stereographic projection is a bijective smooth map which allows us to think the sphere as the extended complex plane. Among its properties it should be emphasized the remarkable property of being angle conformal that is, it is an angle measure preserving map. Unfortunately, this projection map does not preserve areas. Besides being conformal it has also the property of projecting spherical circles in either circles or straight lines in the plane
This type of projection maps seems to have been known since ancient times by Hipparchus (150 BC), being Ptolemy (AD 140) who, in his work entitled "The Planisphaerium", provided a detailed description of such a map. Nonetheless, it is worthwhile to mention that the property of the invariance of angle measure has only been established much later, in the seventeenth century, by Thomas Harriot. In fact, it was exactly in that century that the Jesuit François d’Aguilon introduced the terminology "stereographic projection" for this type of maps, which remained up to our days.
Here, we shall show how we create in GeoGebra, the PRiemannz tool and its potential concerning the visualization and analysis of the properties of the stereographic projection, in addition to the viewing of the amazing relations between Möbius Transformations and stereographic projections.
Keywords: Mathematics, GeoGebra, Riemann Sphere, Möbius Transformations, Stereographic Projections

The Riemann Sphere in Geogebra

Autor: Jose Manuel Dos Santos Dos Santos Afiliação: Instituto GeoGebra Portugal
Autor: Ana Breda Afiliação: Universidade de Aveiro

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Referências


D.N. Arnold, and J. Rogness, Möbius Transformations Revealed. Notices of the AMS 55 (2008), 1226–1231 [ https://www.ima.umn.edu/~arnold/papers/moebius.pdf ].
A. Breda, A. Trocado, J. Santos, O GeoGebra para além da segunda dimensão. In Indagatio Didactica, v. 5, n. 1 (2013) [ http://revistas.ua.pt/index.php/ID/article/view/2421].

 

 

 

This work was supported in part by Portuguese funds through the CIDMA – Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT-Fundacão para a Ciência e a Tecnologia”), within project PEst-OE/MAT/UI4106/2014.