Hyperbolic space

Extracted Attributes

• Field

  Mathematics / Differential Geometry

• Manifold Type

  Simply connected, n-dimensional Riemannian manifold

• Alternative Names

  Lobachevsky space, Bolyai–Lobachevsky space, Real hyperbolic space

• Sectional Curvature

  -1 (constant negative curvature)

• Symmetry Properties

  Homogeneous and symmetric space

• Two-Dimensional Name

  Hyperbolic plane (H2)

• Fruit Fly Brain Mapping Dimensions

  64 dimensions

Hyperbolic space

In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant negative sectional curvature, often taken to be −1 for simplicity. It is homogeneous, and satisfies the stronger property of being a symmetric space. There are many ways to construct it as an open subset of R n {\displaystyle \mathbb {R} ^{n}} with an explicitly written Riemannian metric; such constructions are referred to as models. Hyperbolic 2-space, H2, which was the first instance studied, is also called the hyperbolic plane. It is also sometimes referred to as Lobachevsky space or Bolyai–Lobachevsky space after the names of the author who first published on the topic of hyperbolic geometry. Sometimes the qualificative "real" is added to distinguish it from complex hyperbolic spaces. Hyperbolic space serves as the prototype of a Gromov hyperbolic space, which is a far-reaching notion including differential-geometric as well as more combinatorial spaces via a synthetic approach to negative curvature. Another generalisation is the notion of a CAT(−1) space.

Web Search Results

• Hyperbolic space - Wikipedia

  A perspective projection of a dodecahedral tessellation in H3. Four dodecahedra meet at each edge, and eight meet at each vertex, like the cubes of a cubic tessellation in E3 In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant negative sectional curvature, often taken to be −1 for simplicity. It is homogeneous, and satisfies the stronger property of being a symmetric space. There are many ways to construct it as an open subset of {\displaystyle \mathbb {R} ^{n}} with an explicitly written Riemannian metric; such constructions are referred to as models. Hyperbolic 2-space, H2, which was the first instance studied, is also called the hyperbolic plane. [...] ## Geometric properties [edit] ### Parallel lines [edit] Hyperbolic space, developed independently by Nikolai Lobachevsky, János Bolyai and Carl Friedrich Gauss, is a geometric space analogous to Euclidean space, but such that Euclid's parallel postulate is no longer assumed to hold. Instead, the parallel postulate is replaced by the following alternative (in two dimensions): Given any line L and point P not on L, there are at least two distinct lines passing through P that do not intersect L. [...] It is also sometimes referred to as Lobachevsky space or Bolyai–Lobachevsky space after the names of the author who first published on the topic of hyperbolic geometry. Sometimes the qualificative "real" is added to distinguish it from complex hyperbolic spaces. Hyperbolic space serves as the prototype of a Gromov hyperbolic space, which is a far-reaching notion including differential-geometric as well as more combinatorial spaces via a synthetic approach to negative curvature. Another generalisation is the notion of a CAT(−1)_space "CAT(k) space") space. ## Formal definition and models [edit] ### Definition [edit]

• [PDF] Hyperbolic Geometry - UC Davis Mathematics

  If we consider the set of points at constant squared distance from the origin, we obtain in the Euclidean case the spheres of various radii and in Minkowski space hyperboloids of one or two sheets. We may thus define the unit n-dimensional sphere in Euclidean space n+1 by the formula Sn = {x ∈ n+1 : x · x = 1} and n-dimensional hyperbolic space by the formula {x ∈ n+1 : x ∗x = −1}. Thus hyperbolic space is a hyperboloid of two sheets that may be thought of as a “sphere” of squared radius −1 or of radius i = √−1; hence the name hyperbolic geometry. See Figure 1. Usually we deal only with one of the two sheets of the hyperboloid or identify the two sheets projectively. [...] The eigenvalues of I −kO have the form 1 −kλ, where λ is an eigenvalue of O. Since O is orthogonal, its eigenvalues have absolute value 1. Hence if k ￿= 1, then I −kO is invertible, and the equation (I −kO)x = v does indeed have a solution. 13. Curious Facts about Hyperbolic Space Fact 1. In the three conformal models for hyperbolic space, hyperbolic spheres are also Euclidean spheres; however, Euclidean and hyperbolic sphere centers need not coincide. Proof. We work in the hemisphere model J for hyperbolic space and consider the point p = (0, . . . , 0, 1) ∈J. The Riemannian metric ds2 J is clearly rotationally symmetric around p so that a hyperbolic sphere centered at p is a Euclidean sphere. [...] 7. Five Models of Hyperbolic Space We describe here five analytic models of hyperbolic space. The theory of hyperbolic geometry could be built in a unified way within a single model, but with several models it is as if one were able to turn the object that is hyperbolic space about in one’s hands so as to see it first from above, then from the side, and finally from beneath or within; each view supplies its own natural intuitions. Each model has its own metric, geodesics, isometries, and so on. Here are our mnemonic names for the five models: H, the Half-space model. I, the Interior of the disk model. J, the Jemisphere model (pronounce the J as in Spanish). K, the Klein model. L, the ’Loid model (short for hyperboloid).

• 3-Dimensional Space | The hyperbolic space $\mathbb H^3$

  # The hyperbolic space $\mathbb H^3$ Isotropic geometry Hyperbolic surface suspension ## What is $\mathbb H^3 $? The hyperbolic space $\mathbb H^3$ is the three-dimensional analog of the hyperbolic plane. It is an isotropic space (all the directions play the same role). Among the eight geometries, this is probably the one which has the richest class of lattices. Click on the button below to reveal a concrete model of the hyperbolic space.

• Hyperbolic Space

  Although hyperbolic space is an infinite space more voluminous than euclidean space, we can project it into a finite volume of euclidean space. There are two standard projections which map all of hyperbolic space into a ball in euclidean space. The projective model preserves straight lines and distorts angles, while the conformal ball model preserves angles and warps lines. The 2D hyperbolic browser developed at Xerox PARC uses the conformal ball model [LRP95]. We use the projective model in our implementation and in the layout derivation in the Appendix. Transformations in the 3D projective model can be expressed as tex2htm_wrap_inline634 matrices, so we use that model to gain maximum performance. The mapping from projective to conformal coordinates is straightforward, so our layout [...] # Hyperbolic Space Our layout is computed using hyperbolic distances instead of the familiar euclidean distance measure. We use the hyperbolic metric in order to take advantage of the surprising property that hyperbolic space has more room than our familiar euclidean space. Two parallel lines are always the same distance apart in euclidean space. However, in hyperbolic space, parallel lines are not equidistant. We can construct two hyperbolic straight lines which do not intersect yet are separated by increasing distance as we move away from the origin. Figure 5 contains a sketch of both sets of lines. Further explanation of the ramifications of the hyperbolic metric can be found in one of the many mathematical textbooks which cover hyperbolic geometry [Mar75] [Wol45]. external [...] external Figure 5: Left: Parallel lines in euclidean space are always the same distance apart. Middle: In hyperbolic space the distance between two lines that never meet does indeed change. Here we show two geodesics which never meet but are not equidistant: the further they extend away from the origin, the more room there is between them.

• geometry - Relationship Between Hyperbolas and Hyperbolic Spaces - Mathematics Stack Exchange

  ## 2 Answers 2 Reset to default 6 $\begingroup$ Before you read this, if you are confused about the notion of hyperbolic space, consider discarding prior ways you imagined hyperbolic spaces while reading this answer to minimize confusion. The $d$-dimensional hyperbolic space $H^d$ is a simply connected smooth $d$-dimensional Riemannian manifold which has constant negative metric curvature everywhere. Two-dimensional hyperbolic space $H^2$ is called the hyperbolic plane. A sphere is in a way the opposite of hyperbolic space: it has constant positive curvature everywhere. [...] The hyperboloid model of hyperbolic space is a negatively curved analogue of the sphere. It is the surface $$ H = \{ x^2 + y^2 - z^2 = -1 \}. $$ In the same way you define the angle between two unit vectors using the functions $\cos\theta$ and $\sin\theta$, you can define the angle of two vectors from the origin to two points on $H$ using the hyperbolic functions $\cosh \theta$ and $\sinh\theta$ and use this to define the hyperbolic distance between any two points on $H$. The geodesics (analogues of the great circles) are again the intersections of planes through the origin with $H$.