The Riemann-Poincaré model

This page is a sub-page of our page on Hyperbolic Geometry.

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Related KMR-pages:

The Beltrami-Klein model
Elliptic Geometry
Euclidean Geometry
Non-Euclidean Geometry
Projective Geometry
Projective Metrics
The Euclidean Degeneration

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Books:

• John Stillwell (2016), Elements of Mathematics – From Euclid to Gödel,
Princeton University Press, ISBN 978-0-691-17854-7.
• John Stillwell (1999, (1989)), Mathematics and Its History, Springer, ISBN 0-387-96981-0.
• Jeremy Gray (2007), Worlds Out of Nothing – A Course in the History of Geometry in the 19th Century, Springer, ISBN 1-84628-632-8.

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Other related sources of information:

The Riemann-Poincaré disc model (at Wikipedia)

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Representation: [ \, l_{ine} \, ]_{P_{oincaré}M_{odel}} = \left< \, c_{ircleSegment} \, \perp \, a_{bsolute} \, \right>_{P_{oincaré}M_{odel}}

Representation: [ \, a_{bsolute} \, ]_{P_{oincaré}M_{odel}} = \left< \, c_{ircle} \, \right>_{P_{oincaré}M_{odel}}

Representation: [ \, a_{ngle} \, ]_{P_{oincaré}M_{odel}} = [ \, a_{ngle} \, ]_{E_{uclidean}G_{eometry}}

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Hyperbolic sterographic projection
(at Wikipedia/Poincaré disk model):

Hyperbolic stereographic projection(Wikipedia)

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The Riemann-Poincaré model from hyperbolic stereographic projection
(moving point):

The interactive simulation that created this movie.

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The Riemann-Poincaré model from hyperbolic stereographic projection (moving point) (rotating around the z-axis):

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The Riemann-Poincaré model (Rot Z):

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The Riemann-Poincaré model (varying the parameter a):

The interactive simulation that created this movie.

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The Beltrami-Klein model

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Relating the Riemann-Poincaré model to the Beltrami-Klein model

Projecting the Beltrami-Klein model into the Riemann-Poincaré model
(mediated by the lower vertical semi-circle):

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Projecting the Beltrami-Klein model into the inside and outside of the Riemann-Poincaré model (mediated by the full vertical circle):

The interactive simulation that created this movie.

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Projecting the Beltrami-Klein model into the Riemann-Poincaré model
(mediated by translating the vertical plane with the lower semi-circle):

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Mapping the Beltrami-Klein model to the Riemann-Poincaré model
(pencil on a boundary point):

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Mapping the Beltrami-Klein model to the Riemann-Poincaré model
(concentric hypercycles):

The interactive simulation that created this movie.

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Devil moving rigidly in the Riemann-Poincaré disc:

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Angels and Devils moving rigidly in the Riemann-Poincaré disc:

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H.S.M. Coxeter discusses the math behind M.C. Escher’s Circle Limit IV:

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Illuminating hyperbolic geometry
(Henry Segerman and Saul Schleimer on YouTube):

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Playing Sports in Hyperbolic Space
(Numberphile on Youtube):

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