complex plane where the imaginary part (the y coordinate mentioned above) is positive. The Poincaré half-plane model is named after Henri Poincaré, but it...

hyperbolic plane for its world geometry, and also uses the Poincaré disk model. Hyperbolic geometry Beltrami–Klein model Poincaré half-plane model Poincaré metric...

There are four models commonly used for hyperbolic geometry: the Klein model, the Poincaré disk model, the Poincaré half-plane model, and the Lorentz...

hyperbolic geometry, where the Poincaré half-plane model provides a way of examining hyperbolic motions. The Poincaré metric provides a hyperbolic metric...

One is the Poincaré half-plane model, defining a model of hyperbolic space on the upper half-plane. The Poincaré disk model defines a model for hyperbolic...

plane by Möbius transformations. Textbooks on complex functions often mention two common models of hyperbolic geometry: the Poincaré half-plane model...

absolute of the Poincaré disk model and the Klein disk model. The real line forms the Cayley absolute of the Poincaré half-plane model. Pasch's axiom and...

.., an are zero. A half-space is a convex set. Hemisphere (geometry) Line (geometry) Poincaré half-plane model Siegel upper half-space Nef polygon, construction...

the Poincaré disk model or the Poincaré half-plane model. Both the Poincaré disk model and the Klein disk model are models of the hyperbolic plane. An...

Poincaré model can refer to: Poincaré disk model, a model of n-dimensional hyperbolic geometry Poincaré half-plane model, a model of two-dimensional hyperbolic...

{\displaystyle ab} and c d {\displaystyle cd} respectively. Then in the Poincaré half-plane model HP, p {\displaystyle p} and q {\displaystyle q} represent ultraparallel...

is a tiling of the hyperbolic plane, resembling a quadtree over the Poincaré half-plane model of the hyperbolic plane. It was first studied mathematically...

Apollonius' problem where both points are inside the circle. In the Poincaré half-plane model, horocycles are represented by circles tangent to the boundary...

Poincaré half-plane model. The Schwarz–Pick lemma states that every holomorphic function from the unit disk U to itself, or from the upper half-plane...

acting properly discontinuously and freely. In the Poincaré half-plane model for the hyperbolic plane the group of biholomorphic transformations is the...

model, the angles at the vertices of an ideal triangle are not zero, because the Beltrami-Klein model, unlike the Poincaré disk and half-plane models...

half-plane model by the mapping g given above. Both the Poincaré disk and the Poincaré half-plane are conformal models of the hyperbolic plane, which...

manifold Poincaré duality Poincaré disk model Poincaré expansion Poincaré gauge Poincaré group Poincaré half-plane model Poincaré homology sphere Poincaré inequality...

Half-space may refer to: Half-space (geometry), either of the two parts into which a plane divides Euclidean space (Poincaré) Half-space model, a model...

in the upper half-plane. The hyperbolic metric in the quadrant depends on the Poincaré half-plane metric. The motions of the Poincaré model carry over to...

will here designate by Π(p) for AD = p.: 13  In the Poincaré half-plane model of the hyperbolic plane (see Hyperbolic motions), one can establish the relation...

models they have been named after him: the Poincaré disk model and the Poincaré half-plane model. Each model has a group of isometries that is a subgroup...

half-plane model (or rather, the component connected to the identity) is SL ( 2 , R ) {\displaystyle {\text{SL}}(2,\mathbb {R} )} (see Poincaré half-plane...

} By interpreting the upper half of the complex plane as a model of the hyperbolic plane (the Poincaré half-plane model), Ford circles can be interpreted...

the interior of a horocycle. In the Poincaré half-plane model one convenient choice is the portion of the half-plane with y ≥ 1. Then the covering map is...

the better-known models which are described in more detail in their namesake articles: Poincaré half-plane model: this is the upper-half space { ( x 1 ...

upper half-plane is mapped to the unit disk. In terms of the models of hyperbolic geometry, this Cayley transform relates the Poincaré half-plane model to...

Poincaré half-plane model, a model of two-dimensional hyperbolic geometry Poincaré homology sphere Poincaré–Hopf theorem Poincaré inequality Poincaré–Wirtinger...

transformation Modular curve Modular form Kuṭṭaka Poincaré half-plane model Uniform tilings in hyperbolic plane Alperin, Roger C. (April 1993). "PSL2(Z) = Z2...

for this length scale makes formulas simpler. In terms of the Poincaré half-plane model absolute length corresponds to the infinitesimal metric d s =...