# The Riemann-Poincaré model

///////

Related KMR-pages:

///////

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.

///////

Other related sources of information:

The Riemann-Poincaré disc model (at Wikipedia)

///////

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

///////

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

///////

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

///////

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

///////

The Riemann-Poincaré model (Rot Z):

///////

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

///////

///////

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):

///////

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

///////

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

///////

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

///////

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

///////

Devil moving rigidly in the Riemann-Poincaré disc:

///////

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

///////

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

///////

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

///////

Playing Sports in Hyperbolic Space