# Lobachevskian geometry

Hyperbolic triangle

belowground.co.uk/libraries Non-intersecting lines in hyperbolic geometry also have properties that differ from non-intersecting lines in Euclidean geometry :. This implies that there are through P an infinite number of coplanar lines that do not intersect R. Some geometers simply use parallel lines instead of limiting parallel lines, with ultraparallel lines being just non-intersecting. For ultraparallel lines, the ultraparallel theorem states that there is a unique line in the hyperbolic plane that is perpendicular to each pair of ultraparallel lines.

In hyperbolic geometry, there is no line that remains equidistant from another. Instead, the points that all have the same orthogonal distance from a given line lie on a curve called a hypercycle. Another special curve is the horocycle , a curve whose normal radii perpendicular lines are all limiting parallel to each other all converge asymptotically in one direction to the same ideal point , the centre of the horocycle.

Through every pair of points there are two horocycles. The centres of the horocycles are the ideal points of the perpendicular bisector of the line-segment between them. Given any three distinct points, they all lie on either a line, hypercycle , horocycle , or circle. The length of the line-segment is the shortest length between two points. The arc-length of a hypercycle connecting two points is longer than that of the line segment and shorter than that of a horocycle, connecting the same two points.

The arclength of both horocycles connecting two points are equal. The arc-length of a circle between two points is larger the arc-length of a horocycle connecting two points. The difference is referred to as the defect. The area of a hyperbolic triangle is given by its defect in radians multiplied by R 2. As in Euclidean geometry , each hyperbolic triangle has an incircle. In hyperbolic geometry, if all three of its vertices lie on a horocycle or hypercycle , then the triangle has no circumscribed circle.

As in spherical and elliptical geometry , in hyperbolic geometry if two triangles are similar, they must be congruent. A special polygon in hyperbolic geometry is the regular apeirogon , a uniform polygon with an infinite number of sides. In Euclidean geometry , the only way to construct such a polygon is to make the side lengths tend to zero and the apeirogon is indistinguishable from a circle, or make the interior angles tend to degrees and the apeirogon approaches a straight line.

However, in hyperbolic geometry, a regular apeirogon has sides of any length i. The side and angle bisectors will, depending on the side length and the angle between the sides, be limiting or diverging parallel see lines above. If the bisectors are limiting parallel the apeirogon can be inscribed and circumscribed by concentric horocycles. If the bisectors are diverging parallel the apeirogon sometimes called an pseudogon can be inscribed and circumscribed by hypercycles all vertices are the same distance of a line, the axis, Also the midpoint of the side segments are all equidistant to the same axis.

Like the Euclidean plane it is also possible to tessellate the hyperbolic plane with regular polygons as faces. There are also infinitely many uniform tilings that cannot be generated from Schwarz triangles, some for example requiring quadrilaterals as fundamental domains.

In hyperbolic geometry the sum of the angles of a quadrilateral is always less than degrees, and hyperbolic rectangles differ greatly from Euclidean rectangles since there are no equidistant lines, so a proper Euclidean rectangle would need to be enclosed by two lines and two hypercycles. These all complicate coordinate systems. There are however different coordinate systems for hyperbolic plane geometry. All are based around choosing a point the origin on a chosen directed line the x -axis and after that many choices exist.

The Lobachevski coordinates x and y are found by dropping a perpendicular onto the x -axis. Other coordinate systems use the Klein model or the Poincare disk model described below, and take the Euclidean coordinates as hyperbolic. Construct a Cartesian-like coordinate system as follows. For any point in the plane, one can define coordinates x and y by dropping a perpendicular onto the x -axis.

Then the distance between two such points will be [ citation needed ]. This formula can be derived from the formulas about hyperbolic triangles. Since the publication of Euclid's Elements circa BCE, many geometers made attempts to prove the parallel postulate. Some tried to prove it by assuming its negation and trying to derive a contradiction. Their works on hyperbolic geometry had a considerable influence on its development among later European geometers, including Witelo, Gersonides, Alfonso, John Wallis and Saccheri.

In the 18th century, Johann Heinrich Lambert introduced the hyperbolic functions  and computed the area of a hyperbolic triangle. Unlike their predecessors, who just wanted to eliminate the parallel postulate from the axioms of Euclidean geometry, these authors realized they had discovered a new geometry. Gauss called it " non-Euclidean geometry "  causing several modern authors to continue to consider "non-Euclidean geometry" and "hyperbolic geometry" to be synonyms.

Taurinus published results on hyperbolic trigonometry in , argued that hyperbolic geometry is self consistent, but still believed in the special role of Euclidean geometry. In , Eugenio Beltrami provided models see below of hyperbolic geometry, and used this to prove that hyperbolic geometry was consistent if and only if Euclidean geometry was. The term "hyperbolic geometry" was introduced by Felix Klein in The idea used a conic section or quadric to define a region, and used cross ratio to define a metric.

The projective transformations that leave the conic section or quadric stable are the isometries. For more history, see article on non-Euclidean geometry , and the references Coxeter  and Milnor. The discovery of hyperbolic geometry had important philosophical consequences. Before its discovery many philosophers for example Hobbes and Spinoza viewed philosophical rigour in terms of the "geometrical method", referring to the method of reasoning used in Euclid's Elements.

Kant in the Critique of Pure Reason came to the conclusion that space in Euclidean geometry and time are not discovered by humans as objective features of the world, but are part of an unavoidable systematic framework for organizing our experiences. It is said that Gauss did not publish anything about hyperbolic geometry out of fear of the "uproar of the Boeotians ", which would ruin his status as princeps mathematicorum Latin, "the Prince of Mathematicians".

Hyperbolic geometry was finally proved consistent and is therefore another valid geometry. Because Euclidean, hyperbolic and elliptic geometry are all consistent, the question arises: which is the real geometry of space, and if it is hyperbolic or elliptic, what is its curvature? Lobachevsky had already tried to measure the curvature of the universe by measuring the parallax of Sirius and treating Sirius as the ideal point of an angle of parallelism.

The geometrization conjecture gives a complete list of eight possibilities for the fundamental geometry of our space. Special relativity places space and time on equal footing, so that one considers the geometry of a unified spacetime instead of considering space and time separately. In relativity, rather than considering Euclidean, elliptic and hyperbolic geometries, the appropriate geometries to consider are Minkowski space , de Sitter space and anti-de Sitter space ,   corresponding to zero, positive and negative curvature respectively.

Hyperbolic geometry enters special relativity through rapidity , which stands in for velocity , and is expressed by a hyperbolic angle. The study of this velocity geometry has been called kinematic geometry. The space of relativistic velocities has a three-dimensional hyperbolic geometry, where the distance function is determined from the relative velocities of "nearby" points velocities.

The hyperbolic plane is a plane where every point is a saddle point. There exist various pseudospheres in Euclidean space that have a finite area of constant negative Gaussian curvature. By Hilbert's theorem , it is not possible to isometrically immerse a complete hyperbolic plane a complete regular surface of constant negative Gaussian curvature in a three-dimensional Euclidean space.

Other useful models of hyperbolic geometry exist in Euclidean space, in which the metric is not preserved. A particularly well-known paper model based on the pseudosphere is due to William Thurston. In , Keith Henderson demonstrated a quick-to-make paper model dubbed the " hyperbolic soccerball " more precisely, a truncated order-7 triangular tiling. Instructions on how to make a hyperbolic quilt, designed by Helaman Ferguson ,  have been made available by Jeff Weeks. There are different pseudospherical surfaces that have for a large area a constant negative Gaussian curvature, the pseudosphere being the best well known of them.

These models define a hyperbolic plane which satisfies the axioms of a hyperbolic geometry. All these models are extendable to more dimensions. The Beltrami—Klein model , also known as the projective disk model, Klein disk model and Klein model , is named after Eugenio Beltrami and Felix Klein.

For the two dimensions this model uses the interior of the unit circle for the complete hyperbolic plane , and the chords of this circle are the hyperbolic lines. For higher dimensions this model uses the interior of the unit ball , and the chords of this n -ball are the hyperbolic lines. The line B is not included in the model. The hyperboloid model or Lorentz model employs a 2-dimensional hyperboloid of revolution of two sheets, but using one embedded in 3-dimensional Minkowski space.

The hemisphere model is not often used as model by itself, but it functions as a useful tool for visualising transformations between the other models. The hemisphere model is part of a Riemann sphere , and different projections give different models of the hyperbolic plane:.

### Bulletin of the American Mathematical Society

Non-intersecting lines in hyperbolic geometry also have properties that differ from non-intersecting lines in Euclidean geometry :. Kratkii ocherk osnov geometrii Lobachevskogo. We recommend doing some or all of the basic explorations before reading the section. Non-Euclidean Geometry: A critical and historical study of its development. Figure 1 In the darker ages that followed, Euclid's sense of mathematical freedom was lost and philosophers and mathematicians expected geometry to rest on self-evident grounds. Gauss had already worked out the ideas written by Janos Bolyai, but had never published them.

In David Gans proposed a flattened hyperboloid model in the journal American Mathematical Monthly. This model is not as widely used as other models but nevertheless is quite useful in the understanding of hyperbolic geometry. The band model employs a portion of the Euclidean plane between two parallel lines. All models essentially describe the same structure. The difference between them is that they represent different coordinate charts laid down on the same metric space , namely the hyperbolic plane.

The characteristic feature of the hyperbolic plane itself is that it has a constant negative Gaussian curvature , which is indifferent to the coordinate chart used. The geodesics are similarly invariant: that is, geodesics map to geodesics under coordinate transformation. Hyperbolic geometry generally is introduced in terms of the geodesics and their intersections on the hyperbolic plane.

## What makes hyperbolic geometry particularly hard?

Once we choose a coordinate chart one of the "models" , we can always embed it in a Euclidean space of same dimension, but the embedding is clearly not isometric since the curvature of Euclidean space is 0. The hyperbolic space can be represented by infinitely many different charts; but the embeddings in Euclidean space due to these four specific charts show some interesting characteristics. Every isometry transformation or motion of the hyperbolic plane to itself can be realized as the composition of at most three reflections. These are also true for Euclidean and spherical geometries, but the classification below is different.

The white lines in III are not quite geodesics they are hypercycles , but are close to them. It is also possible to see quite plainly the negative curvature of the hyperbolic plane, through its effect on the sum of angles in triangles and squares. Another visible property is exponential growth. In Circle Limit III , for example, one can see that the number of fishes within a distance of n from the center rises exponentially. The fishes have equal hyperbolic area, so the area of a ball of radius n must rise exponentially in n.

HyperRogue is a roguelike game set on various tilings of the hyperbolic plane. Hyperbolic geometry is not limited to 2 dimensions; a hyperbolic geometry exists for every higher number of dimensions. Hyperbolic space of dimension n is a special case of a Riemannian symmetric space of noncompact type, as it is isomorphic to the quotient. The orthogonal group O 1, n acts by norm-preserving transformations on Minkowski space R 1, n , and it acts transitively on the two-sheet hyperboloid of norm 1 vectors.

Timelike lines i.

## Alternative to popular Euclidean Geometry

The stabilizer of any particular line is isomorphic to the product of the orthogonal groups O n and O 1 , where O n acts on the tangent space of a point in the hyperboloid, and O 1 reflects the line through the origin. Many of the elementary concepts in hyperbolic geometry can be described in linear algebraic terms: geodesic paths are described by intersections with planes through the origin, dihedral angles between hyperplanes can be described by inner products of normal vectors, and hyperbolic reflection groups can be given explicit matrix realizations. In small dimensions, there are exceptional isomorphisms of Lie groups that yield additional ways to consider symmetries of hyperbolic spaces.

In both cases, the symmetry groups act by fractional linear transformations, since both groups are the orientation-preserving stabilizers in PGL 2, C of the respective subspaces of the Riemann sphere.

The Cayley transformation not only takes one model of the hyperbolic plane to the other, but realizes the isomorphism of symmetry groups as conjugation in a larger group. This allows one to study isometries of hyperbolic 3-space by considering spectral properties of representative complex matrices. For example, parabolic transformations are conjugate to rigid translations in the upper half-space model, and they are exactly those transformations that can be represented by unipotent upper triangular matrices.

In essence their propositions concerning the properties of quadrangles which they considered assuming that some of the angles of these figures were acute of obtuse, embodied the first few theorems of the hyperbolic and the elliptic geometries. Their other proposals showed that various geometric statements were equivalent to the Euclidean postulate V.

By using our site, you agree to our collection of information through the use of cookies. To learn more, view our Privacy Policy. Log In Sign Up. Georg von Brzeski. The equation relating distance in hyperbolic space, its curvature, and spectral shift is derived by method of horospheres. The active nature of the Lobachevskian vacuum is discussed with applications to physics.

• Petri Nets: Central Models and Their Properties: Advances in Petri Nets 1986, Part I Proceedings of an Advanced Course Bad Honnef, 8.–19. September 1986.
• Lobachevskian Geometry.
• Synchronization.
• Basics.

Our analysis is restricted to purely geometrical properties of hyper- bolic spaces and their relation to an electromagnetic field. It will be shown that the shift in the frequency spectrum of electromagnetic radiation is a consequence of the non-Euclidean geometry of the space under consideration. This is a very subtle point of profound importance and far reaching implications.

Our method extends to any physically realizable representation of hyperbolic space, and its another value is that results are scale independent. We will prove our statements by deriving a general type shift formula, just from the geometry of Lobachevskian space, without any relation to physics. We will then apply our purely geometrical equation to different physical representations. Under the assumption that ambient empty space is a 3 dimensional Lobachevskian space, we will obtain precisely the same formula for the wavelength shift as the one found in a classical analysis of the Doppler effect with a transmitter-receiver pair being in relative motion.

Our exposition will be as much as possible geometrical. We believe that there is an advantage in putting things that way, as we will later show. However, other points of view are equally valid. Since Lobachevskian space is a homogeneous space with a group of motion which is the proper Lorentz group, there is a standard construction where the homogeneous space X with the group of motion G can be described solely in terms of the group G.

In this approach, the homogeneous space is identified with a coset space of group G with respect to a stabilizer Sp of some point p in X. Those facts can be found in [6, 8, 12]. On the historical development of Lobachevskian geometry we refer the reader to [10 ] and on its content to [1, 2, 9] where further references can be found.

A lot of work on the application of Lobachevskian geometry to physics has been done by Soviet physicists [4, 5, 11]. In Euclidean geometry those words are meaningless. Another difference between Lobachevskian and Euclidean geome- tries is that in Lobachevskian space, the volume of a ball grows as an exponential function of the radius, whereas in Euclidean space it grows as a power function of the radius.

There are more distinctive features. For example, there are infinitely many parallel lines through a given point off a given line which are parallel to the line.

Therefore, self-similarity does not hold in a Lobachevskian negatively curved world, and for example, a Sierpinski gasket cannot be constructed there. Finally, two Lobachevskian geometries with distinct curvature constants are not isometric . Regarding physics, the most profound difference between Lobachevskian and Euclidean spaces is the way they relate to the properties of electromagnetic radiation and perhaps other fields propagating at the speed of light , i.

It will be shown that contrary to Euclidean space, a Lobachevskian vacuum is intimately interrelated to electromagnetism and it actively interacts and modifies the parameters of an electromagnetic field. In this respect, Lobachevskian empty space vacuum can be regarded as an active medium, while Euclidean space is inherently passive. In part one of this paper we deal only with the wavelength of electromagnetic radiation in a vacuum.

Both parts are independent and can be read independently of each other. Analysis is carried out from the standpoint of integral geometry, as formulated by I. Gelfand . In integral geometry, several dual spaces can be built in parallel to a given space using as a building blocks geometrical objects of the given space. At the same time a set of functions of interest defined on initial space is transformed into a set of images which are integrals over the objects of the initial space.

For example, in NMR imaging information is sought about the density distribution function over some finite domain in R3. This information is encoded into a Grassman manifold of 2D Euclidean planes in R3. In our case similarly, the information we seek about a given Lobachevskian space is encoded into the objects of a dual space which are horospheres. Therefore, together with Lobachevskian space, we consider a space of geometrical objects on a Lobachevskian space which are horospheres.

We then use sets of parallel horospheres to measure distances in Lobachevskian space and to relate geometry to physics.

Non Euclidean Geometry

In other words, the information we seek about the space is encoded into the objects of dual space- space of horospheres. In the next few sections we will show how information about a hyperbolic space is encoded via an electromagnetic field into geometrical objects of its dual space of horospheres , how it is extracted, and how it is decoded.

A horosphere in Lobachevskian space can be obtained by the following construction. Lobachevskian space with application to kinematics. Horosphere centered at point P. Then we move the center of the sphere to infinity requiring that in the process the sphere passes through the fixed point. As the center of the sphere approaches infinity, the sphere becomes a horosphere.

Alternatively we can define a horosphere as a surface which is orthogonal to the family of parallel lines geodesics converging to one point at infinity and tangent at that point to the boundary at infinity. Two horospheres tangent to the same point at infinity are called parallel. The distance between two parallel horospheres is the distance measured along any ray in a family of parallel rays converging to the tangency point and intersecting the horospheres.

Figure 1 shows some objects in the Lobachevskian space for a Poincare disc model. The internal geometry on horospheres of Lobachevskian space is flat. This means that in the case of a 3D Lobachevskian space, horospheres carry the geometry of a 2D Euclidean plane. We observe that horospheres in Lobachevskian space form a transitive set. This means that any horosphere can be mapped onto any other by some motion in Lobachevskian space.

We will use a family of parallel horospheres which foliate the Lobachevskian space as a distance markers. We will work with homogeneous coordinates in a 3 dimensional projective real space. By taking different normalizations of homoge- neous coordinates we will get different models of Lobachevskian space and we will use them interchangeably without any special notification.

In that realization, Lobachevskian space is an interior of the 3D ball, bounded at infinity by a 2D sphere. This is a geometrical statement independent of coordinate system in which this product is computed. It follows that it will have the same constant value in different coordinate systems frames associated with different points in Lobachevskian space ; however components of those two vectors will undergo change.

At this point, the subscripts T and D have no other meaning than to distinguish two points. This simple looking Equation 2 contains an invariant formulation of spectral shifts that an electromagnetic wave experiences in hyperbolic space, in abstract geometrical form. However it reaches much further and is applicable to all physical phenomena which can be modeled on Lobachevskian geometry, i.

Equation 2 states the almost trivial truth that a real number is an invariant. Below we will illustrate how this equation works in different representations as applicable to physics. In the same way Lobachevskian geometry does not depend on its physical representation, whether it is represented by velocity space, by coordinate configuration space, or by the space of reflection coefficients, which also carries Lobachevskian 2D space geometry Lobachevskian plane.

The mapping of parameter ratio of two parallel horospheres onto Euclidean distance which separates them in the unit ball model is of Lobachevskian space is given by Equation 5 below. The equation we derived relates the parameter ratio tag value ratio for two parallel horospheres in Lobachevskian space to the hyperbolic distance which separates them. No other assumptions have been made beyond those which follow from 3D hyperbolic geometry. In the hyperbolic metric inside the ball, the signed distance between two points has the meaning of uniform relative velocity.

When one point approaches the limiting sphere, the hyperbolic distance between it and a second fixed point goes to infinity. For that reason we say that photon velocities populate the boundary at infinity. To learn more about Lobachevskian geometry of velocity space and its relation to physics we refer reader to paper by Ya. Smorodinsky , N. Chernikov [4, 5], both in Russian and to Klein .

Note one very important fact. Velocity space is not compact. Now we will try to extract some physics by identifying the vectors a and b in Equation 2 with some physical entities. In order to measure distances in Lobachevskian velocity space, we employ an electromagnetic field which labels horospheres as it propagates through free space on its way to us. We could use Equation 5 directly.

Instead, we will do the first example in every detail to give the reader some comfort and to make the bridge between a negatively curved non-compact velocity space and physics more visible. Later we will use Equation 5 directly. First, in a ball of radius c in 3D Euclidean space we introduce homogeneous Weierstrass coordinates in the following way. This means that velocity is a 3D geometrical object. However, due to the invariant and limiting value of c, velocity space is not flat. It is a 3D Lobachevskian hyperbolic non-compact space, and this fact was known as early as to Felix Klein .