Beltrami–Klein model

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Lines in the projective model of the hyperbolic plane.
A hyperbolic triheptagonal tiling in a Beltrami–Klein model projection
The regular hyperbolic dodecahedral honeycomb, {5,3,4}

In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of hyperbolic geometry in which points are represented by the points in the interior of the unit disk (or n-dimensional unit ball) and lines are represented by the chords, straight line segments with ideal endpoints on the boundary sphere.

The Beltrami–Klein model is named after the Italian geometer Eugenio Beltrami and the German Felix Klein while "Cayley" in Cayley–Klein model refers to the English geometer Arthur Cayley.

The Beltrami–Klein model is strongly analogous to the gnomonic projection of spherical geometry, which maps great circles to straight lines; the formulae relating these two to the hyperboloid model and the sphere, respectively, are very similar.

This model is not conformal, meaning that angles and circles are distorted, whereas the Poincaré disk model preserves these.

In this model, lines and segments are straight Euclidean segmments, whereas in the Poincaré disk model, lines are arcs that meet the boundary orthogonally.

Given two distinct points p and q in the open unit ball, the unique straight line connecting them intersects the boundary at two ideal points, a and b, labeled so that the points are, in order, a, p, q, b. so that |aq| > |ap| and |pb| > |qb|. The hyperbolic distance between p and q is then

d(p,q)=\frac{1}{2} \log \frac{|aq||pb|}{|ap||qb|} ,

where the vertical bars indicate the Euclidean distances in the model.

History[edit]

Formally speaking, this model was first constructed in 1859 by the English mathematician Arthur Cayley. Yet he considered it only as a construction in projective geometry and apparently did not notice the connection with non-Euclidean or hyperbolic geometry.


This model made its first appearance for hyperbolic geometry in two memoirs of Eugenio Beltrami published in 1868, first for dimension n = 2 and then for general n, devoted to showing equiconsistency of hyperbolic geometry with ordinary Euclidean geometry.[1][2][3]


In 1869, the young (twenty-year-old) Felix Klein became acquainted with Cayley's work. He recalled that in 1870, and gave a talk on the work of Cayley at the seminar of the mathematician Karl Weierstrass, and, as he writes:

"I finished with a question whether there might exist a connection between the ideas of Cayley and Lobachevsky. I was given the answer that these two systems were conceptually widely separated."

As Klein puts it, "I allowed myself to be convinced by these objections and put aside this already mature idea." However, in 1871, he returned to this idea, formulated it mathematically, and published it.[4]

The distance is given by the Cayley–Klein metric and was first written down by Arthur Cayley in the context of projective and spherical geometry. Felix Klein recognized its importance for non-Euclidean geometry and popularized the subject.

Distance formula[edit]

Arthur Cayley applied the cross-ratio from projective geometry to measurement of distances and angles in spherical geometry.[5] Later, Felix Klein realized that Cayley's ideas give rise to a projective model of the non-Euclidean plane.[6]

Given two distinct points p and q in the open unit ball, the unique straight line connecting them intersects the unit sphere in two ideal points, a and b, labeled so that the points are, in order, a, p, q, b so that |aq| > |ap| and |pb| > |qb|. Then the hyperbolic distance between p and q is expressed as

d(p,q) = \frac{1}{2} \log \frac{ \left| qa \right| \left| bp \right| }{ \left| pa \right| \left| bq \right| } ,

where the vertical bars indicate Euclidean distances. The factor of one half is needed to make the curvature −1.

The associated metric tensor is given by

g (x, dx) = \frac{4 (x \cdot dx)^2}{(1 - \left\Vert x \right\Vert^2)^2} + \frac{4 \left\Vert dx \right\Vert^2}{(1 - \left\Vert x \right\Vert^2)}.

or


ds^2 \;=\; \frac{\|\mathbf{dx}\|^2}{1-\|\mathbf{x}\|^2} + \frac{(\textbf{x}\cdot\textbf{dx})^2}{\bigl(1-\|\mathbf{x}\|^2\bigr)^2}[7][8]

The Klein disk model[edit]

In two dimensions the Beltrami–Klein model is called the Klein disk model. It is a disk and the inside of the disk is a model of the entire hyperbolic plane. Lines in this model are represented by chords of the boundary circle (also called the absolute). The points on the boundary circle are called ideal points; although well defined, they do not belong to the hyperbolic plane. Neither do points outside the disk, which are sometimes called ultra ideal points.

The model is not conformal, meaning that angles are distorted, and circles on the hyperbolic plane are in general not circular in the model. Only circles that have their centre at the centre of the boundary circle are not distorted. All other circles are distorted , as are horocycles and hypercycles.

Properties[edit]

Chords that meet on the boundary circle are limiting parallel lines.

Two chords are perpendicular if, when extended beyond the model, each goes through the pole of the other. (The pole of a chord is an ultra ideal point: a point outside the disk.) Chords that go through the centre of the disk have their pole at infinity, orthogonal to the direction of the chord (meaning that right angles on diameters are not distorted).

Compass and straightedge constructions[edit]

Here is how one can use compass and straightedge constructions in the model to achieve the effect of the basic constructions in the hyperbolic plane.

  • The pole of a line or of a segment. While the pole is not a point in the hyperbolic plane (it is an ultra ideal point) most constructions will use the pole of a line or of a segment in one or more ways.
For a line: construct the tangents to the boundary circle through the ideal (end) points of the line. the point where these tangents intersect is the pole.
For a segment: extend the segment to a line and see above.
For diameters of the disk: the pole is at infinity perpendicular to the diameter.
When the line is a diameter of the disk then the perpendicular is the chord that is (Euclidean) perpendicular to that diameter and going trough the given point.
  • To find the midpoint of given segment  AB: Draw the lines through A and B that are perpendicular to  AB. (see above) Draw the lines connecting the ideal points of these lines, two of these lines will intersect the segment  AB and will do this at the same point. This point is the (hyperbolic) midpoint of AB.[9]
  • To bisect a given angle  \angle BAC: Draw the rays AB and AC. Draw tangents to the circle where the rays intersect the boundary circle. Draw a line from A to the point where the tangents intersect. The part of this line between A and the boundary circle is the bisector.[10]
When one of the chords is a diameter of the boundary circle then the common perpendicular is the chord that is perpendicular to the diameter and that when lengthened goes through the pole of the other chord.
  • To reflect a point P in a line l : From a point R on the line l draw the ray through P. Let X be the idealpoint where the ray intersects the absolute. Draw the ray from the pole of line l through X , let Y be the other intersection point with the absolute. Draw the segment RY. The reflection of point P is the point where the ray from the pole of line l through P intersects RY.[11]

Relation to the Poincaré disk model[edit]

the Beltrami–Klein model (line K), the Poincaré disk model (line P), and their relations with the other models

Both the Poincaré disk model and the Klein disk model are models of the hyperbolic plane. An advantage of the Poincaré disk model is that it is conformal (circles and angles are not distorted); a disadvantage is that lines of the geometry are circular arcs orthogonal to the boundary circle of the disk.

The two models are related through a projection on or from the hemisphere model. The Klein model is an orthographic projection to the hemisphere model while the Poincaré disk model is an stereographic projection.

When projecting the same lines in both models on one disk both lines go through the same two ideal points.(the ideal points remain on the same spot) also the pole of the chord is the centre of the circle that contains the arc.

If u is a vector of norm less than one representing a point of the Poincaré disk model, then the corresponding point of the Beltrami–Klein model is given by:

s = \frac{2u}{1+u \cdot u}.

Conversely, from a vector s of norm less than one representing a point of the Beltrami–Klein model, the corresponding point of the Poincaré disk model is given by:

u = \frac{s}{1+\sqrt{1-s \cdot s}} = \frac{\left(1-\sqrt{1-s \cdot s}\right)s}{s \cdot s}.

Relation to the hyperboloid model[edit]

The hyperboloid model is a model of hyperbolic geometry within (n + 1)-dimensional Minkowski space. The Minkowski inner product is given by

\mathbf{x} \cdot \mathbf{y} = x_0 y_0 - x_1 y_1 - \cdots - x_n y_n \,

and the norm by  \left\| \mathbf{x} \right\| = \sqrt{ \mathbf{x} \cdot \mathbf{x} }. The hyperbolic plane is embedded in this space as the vectors x with ||x|| = 1 and x0 (the "timelike component") positive. The intrinsic distance (in the embedding) between points u and v is then given by

 d ( \mathbf{u} , \mathbf{v} ) = \cosh^{-1} ( \mathbf{u} \cdot \mathbf{v}) .

This may also be written in the homogeneous form

 d ( \mathbf{u} , \mathbf{v} ) = \cosh^{-1} \left( \frac{ \mathbf{u} }{ \left\| \mathbf{u} \right\| } \cdot \frac{ \mathbf{v} }{ \left\| \mathbf{v} \right\| } \right) ,

which allows the vectors to be rescaled for convenience.

The Beltrami–Klein model is obtained from the hyperboloid model by rescaling all vectors so that the timelike component is 1, that is, by projecting the hyperboloid embedding through the origin onto the plane x0 = 1. The distance function, in its homogeneous form, is unchanged. Since the intrinsic lines (geodesics) of the hyperboloid model are the intersection of the embedding with planes through the Minkowski origin, the intrinsic lines of the Beltrami–Klein model are the chords of the sphere.

In the gyrovector space approach to hyperbolic geometry, vector algebra in the Beltrami–Klein model can be developed using relativistic 3-velocities as the vectors, analogously to the use of ordinary vectors in Euclidean geometry.

Relation to the Poincaré ball model[edit]

Both the Poincaré ball model and the Beltrami–Klein model are models of the n-dimensional hyperbolic space in the n-dimensional unit ball in Rn. If u is a vector of norm less than one representing a point of the Poincaré disk model, then the corresponding point of the Beltrami–Klein model is given by

s = \frac{2u}{1+u \cdot u}.

Conversely, from a vector s of norm less than one representing a point of the Beltrami–Klein model, the corresponding point of the Poincaré disk model is given by

u = \frac{s}{1+\sqrt{1-s \cdot s}} = 
\frac{\left(1-\sqrt{1-s \cdot s}\right)s}{s \cdot s}.

Given two points on the boundary of the unit disk, which are traditionally called ideal points, the straight line connecting them in the Beltrami–Klein model is the chord between them, while in the corresponding Poincaré model the line is a circular arc on the two-dimensional subspace generated by the two boundary point vectors, meeting the boundary of the ball at right angles. The two models are related through a projection from the center of the disk; a ray from the center passing through a point of one model line passes through the corresponding point of the line in the other model.

See also[edit]

Notes[edit]

  1. ^ Beltrami, Eugenio (1868). "Saggio di interpretazione della geometria non-euclidea". Giornale di Mathematiche VI: 285–315. 
  2. ^ Beltrami, Eugenio (1868). "Teoria fondamentale degli spazii di curvatura costante". Annali. di Mat., ser II 2: 232–255. doi:10.1007/BF02419615. 
  3. ^ Stillwell, John (1999). Sources of hyperbolic geometry (2. print. ed.). Providence: American mathematical society. pp. 7–62. ISBN 0821809229. 
  4. ^ Shafarevich, I. R.; A. O. Remizov (2012). Linear Algebra and Geometry. Springer. ISBN 978-3-642-30993-9. 
  5. ^ Cayley, Arthur (1859). "A Sixth Memoire upon Quantics". Philosophical Transactions of the Royal Society 159: 61–91. doi:10.1098/rstl.1859.0004. 
  6. ^ Klein, Felix (1871). "Ueber die sogenannte Nicht-Euklidische Geometrie". Mathematische Annalen 4: 573–625. doi:10.1007/BF02100583. 
  7. ^ Hyperbolic Geometry , J.W.Cannon, W. J. Floyd, R. Kenyon, W. R. Parry
  8. ^ answer from math stackexchange
  9. ^ hyperbolic toolbox
  10. ^ hyperbolic toolbox
  11. ^ Greenberg, Marvin Jay (2003). Euclidean and non-Euclidean geometries : development and history (3rd ed. ed.). New York: Freeman. pp. 272–273. ISBN 9780716724469. 

References[edit]