The Beltrami-Klein model

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

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Related KMR-pages:

The Riemann-Poincaré 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:

• …

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Representation: [ \, l_{ine} \, ]_{B_{eltrami}K_{lein}M_{odel}} = \left< \, l_{ine} \, \right>_{B_{eltrami}K_{lein}M_{odel}}

Representation: [ \, a_{bsolute} \, ]_{B_{eltrami}K_{lein}M_{odel}} = \left< \, c_{onic} \, \right>_{B_{eltrami}K_{lein}M_{odel}}

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The Beltrami-Klein model (from hyperboloid gnomonic projection)
(moving point):

The interactive simulation that created this movie.

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Hyperbolic yardstick:

The interactive simulation that created this movie.

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The Beltrami-Klein model (Rot Z):

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The Beltrami-Klein model (varying a):

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Connecting with the Riemann-Poincaré model:

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

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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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One thought on “The Beltrami-Klein model

  1. It’s such as you learn my mind! You seem to grasp so much about this, like you wrote the e-book in it or something. I think that this is great blog. A fantastic read. I will certainly be back.

    Keep up the good work!

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