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  2. Hyperbolic geometry - Wikipedia

    en.wikipedia.org/wiki/Hyperbolic_geometry

    Compared to Euclidean geometry, hyperbolic geometry presents many difficulties for a coordinate system: the angle sum of a quadrilateral is always less than 360°; there are no equidistant lines, so a proper rectangle would need to be enclosed by two lines and two hypercycles; parallel-transporting a line segment around a quadrilateral causes ...

  3. Portal:Mathematics/Featured picture archive - Wikipedia

    en.wikipedia.org/wiki/Portal:Mathematics/...

    In the 19th century mathematicians began to seriously question the parallel postulate and found that other forms of geometry are possible. For example in elliptical geometry: Given a line and a point not on that line, all lines drawn through that point will intersect the original line. (case 2) And in hyperbolic geometry:

  4. Constructions in hyperbolic geometry - Wikipedia

    en.wikipedia.org/wiki/Constructions_in...

    Hyperbolic geometry is a non-Euclidean geometry where the first four axioms of Euclidean geometry are kept but the fifth axiom, the parallel postulate, is changed.The fifth axiom of hyperbolic geometry says that given a line L and a point P not on that line, there are at least two lines passing through P that are parallel to L. [1]

  5. Poincaré disk model - Wikipedia

    en.wikipedia.org/wiki/Poincaré_disk_model

    Poincaré disk with hyperbolic parallel lines Poincaré disk model of the truncated triheptagonal tiling.. In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or ...

  6. Hyperbolic motion - Wikipedia

    en.wikipedia.org/wiki/Hyperbolic_motion

    Textbooks on complex functions often mention two common models of hyperbolic geometry: the Poincaré half-plane model where the absolute is the real line on the complex plane, and the Poincaré disk model where the absolute is the unit circle in the complex plane. Hyperbolic motions can also be described on the hyperboloid model of hyperbolic ...

  7. Hyperbolic triangle - Wikipedia

    en.wikipedia.org/wiki/Hyperbolic_triangle

    In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called sides or edges and three points called angles or vertices . Just as in the Euclidean case, three points of a hyperbolic space of an arbitrary dimension always lie on the same plane.

  8. Hyperboloid model - Wikipedia

    en.wikipedia.org/wiki/Hyperboloid_model

    In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of n-dimensional hyperbolic geometry in which points are represented by points on the forward sheet S + of a two-sheeted hyperboloid in (n+1)-dimensional Minkowski space or by the displacement vectors from the origin to those points, and m ...

  9. Hyperbolic coordinates - Wikipedia

    en.wikipedia.org/wiki/Hyperbolic_coordinates

    For example, in thermodynamics the isothermal process explicitly follows the hyperbolic path and work can be interpreted as a hyperbolic angle change. Similarly, a given mass M of gas with changing volume will have variable density δ = M / V , and the ideal gas law may be written P = k T δ so that an isobaric process traces a hyperbola in the ...

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