What is Poincaré line?
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which the points of the geometry are inside the unit disk, and the straight lines consist of all circular arcs contained within that disk that are orthogonal to the boundary of the disk, plus …
Are the Poincaré disk model and upper half-plane models of hyperbolic geometry isomorphic?
The fact that the disk model and the upper half-plane model of Hyperbolic Geometry are isomorphic, is usually proved through a formula using the Möbius transformation [1, p.
Who created hyperbolic geometry?
The two mathematicians were Euginio Beltrami and Felix Klein and together they developed the first complete model of hyperbolic geometry. This description is now what we know as hyperbolic geometry (Taimina). In Hyperbolic Geometry, the first four postulates are the same as Euclids geometry.
What is elliptic geometry used for?
Applications. One way that elliptic geometry is used is to determine distances between places on the surface of the earth. The earth is roughly spherical, so lines connecting points on the surface of the earth are naturally curved as well.
Is hyperbolic geometry consistent?
Hyperbolic geometry was finally proved consistent and is therefore another valid geometry.
What is a circle with an infinite radius?
A horocycle can also be described as the limit of the circles that share a tangent in a given point, as their radii go towards infinity. In Euclidean geometry, such a “circle of infinite radius” would be a straight line, but in hyperbolic geometry it is a horocycle (a curve).
Do circles exist in hyperbolic geometry?
A circle in the hyperbolic plane is the locus of all points a fixed distance from the center, just as in the Euclidean plane. Therefore, the hyperbolic plane still satisfies Euclid’s third axiom. A hyperbolic circle turns out to be a Euclidean circle after it is flattened out in the Poincare half-plane model.
How do you calculate hyperbolic distance?
One may compute the hyperbolic distance between p and q by first finding the ideal points u and v of the hyperbolic line through p and q and then using the formula dH(p,q)=ln((p,q;u,v)).
What is hyperbolic line?
The hyperbolic lines are half-circles orthogonal to the boundary of the hemisphere. The hemisphere model is part of a Riemann sphere, and different projections give different models of the hyperbolic plane: Stereographic projection from onto the plane projects corresponding points on the Poincaré disk model.
Why is it called the Poincaré model?
It is named after Henri Poincaré, because his rediscovery of this representation fourteen years later became better known than the original work of Beltrami. The Poincaré ball model is the similar model for 3 or n -dimensional hyperbolic geometry in which the points of the geometry are in the n -dimensional unit ball .
What is the Poincaré ball model in geometry?
The Poincaré ball model is the similar model for 3 or n -dimensional hyperbolic geometry in which the points of the geometry are in the n -dimensional unit ball . Hyperbolic straight lines consist of all arcs of Euclidean circles contained within the disk that are orthogonal to the boundary of the disk, plus all diameters of the disk.
What is the Poincaré disk model?
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which the points of the geometry are inside the unit disk, and the straight lines consist of all circular arcs contained within that disk that are orthogonal to the boundary of the disk, plus all diameters of the disk.
How to train a Poincaré embedding?
This module allows training Poincaré Embeddings from a training file containing relations of graph in a csv-like format, or from a Python iterable of relations. Initialize and train a model from a list Initialize and train a model from a file containing one relation per line