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Encyclopedia > Klein model

In geometry, the Klein model, also called the projective model, the Beltrami-Klein model, the Klein-Beltrami model and the Cayley-Klein model, is a model of n-dimensional hyperbolic geometry in which the points of the geometry are in an n-dimensional disk, or ball, and the lines of the geometry are line segments contained in the disk; that is, with endpoints on the boundry of the disk. Along with the Poincaré half-plane model and the Poincaré disk model, it was proposed by Eugenio Beltrami who used these models to show hyperbolic geometry was equiconsistent with Euclidean geometry. A triangle immersed in a saddle-shape plane (an hyperbolic paraboloid), as well as two diverging parallel lines. ... In non-Euclidean geometry, the Poincaré model is a model of two-dimensional hyperbolic geometry as a homogeneous space for the group of Möbius transformations. ... In geometry, the Poincaré disk model, also called the conformal disk model, is a model of n-dimensional hyperbolic geometry in which the points of the geometry are in an n-dimensional disk, or unit ball, and the lines of the geometry are segments of circles contained in the disk... Eugenio Beltrami (16 November 1835 - 18 February 1900) was an Italian mathematician notable for his work on non-Euclidean geometry, electricity, and magnetism. ... In mathematics, specifically in mathematical logic, formal theories are studied as mathematical objects. ...

Contents


Relation to the hyperboloid model

The hyperboloid model is a model of hyperbolic geometry within Minkowski space. If [t, x1, ..., xn] is a vector in real (n+1)-space, we may define the Minkowski quadratic form to be In geometry, the hyperboloid model, also known as the Minkowski model or the Lorentz model, is a model of hyperbolic geometry in which the points are points on one sheet of a hyperboloid of two sheets. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ...

Q([t, x_1, cdots, x_n]) = t^2 - x_1^2 - cdots - x_n^2.

Corresponding to the Minkowski quadratic form Q there is a Minkowski bilinear form B, defined by

B(u,v) = (Q(u + v) − Q(u) − Q(v)) / 2.

If

u = [x_0, x_1, cdots, x_n], v = [y_0, y_1, cdots, y_n]

then we may write this as

B(u, v) = x_0 y_0-x_1 y_1 - cdots - x_n y_n = x_0 y_0 - mathbf{x} cdot mathbf{y}.

We may use this to put a hyperbolic metric on certain of the points of Minkowski projective space, which is to say, of lines through the origin which are rays defined by a vector u such that Q(u)>0. If u and v are two such vectors, then we may define a distance between them by In mathematics, a projective space is a fundamental construction from any vector space. ...

d(u, v) = operatorname{arccosh}(frac{B(u,v)}{sqrt{Q(u)Q(v)}}).

This is a homogenous function, and so defines a distance between projective points. We can obtain either the hyperboloid model or the Klein model by normalizing these projective points. If we normalize u and v by changing sign if needed to make the first coordinate positive, and then dividing u and v to obtain u' = frac{u}{sqrt{Q(u)}}, v' = frac{v}{sqrt{Q(v)}} respectively, so that the points satisfy Q(u') = Q(v') = 1, we obtain the hyperboloid model. If instead we normalize u and v by dividing through by the first coordinate, which since Q(u) and Q(v) are greater than zero cannot be zero, we obtain a subset of the projective plane, which are points in the interior of a unit disk. We may also view this as intersecting the lines through the origin with the hypersurface t=1. In geometry, the hyperboloid model, also known as the Minkowski model or the Lorentz model, is a model of hyperbolic geometry in which the points are points on one sheet of a hyperboloid of two sheets. ...


Distance formula

From the projective hyperbolic distance function we may derive a distance function for the points in the unit disk. If s and t are two vectors with norm less than one, then we may define u as the vector in Minkowski space whose t coordinate is 1 followed by the coordinates for s, and v as the same for t. Then

d(s, t) = operatorname{arccosh}(frac{B(u,v)}{sqrt{Q(u)Q(v)}})

defines a distance function on the unit disk; this is the distance function of the Klein model. In terms of the original vectors s and t, we may now rewrite this as

d(s, t) = operatorname{arccosh}(frac{1 - s cdot t}{sqrt{(1-s cdot s)(1-t cdot t)}}).

Relation to the Poincaré disk model

Both the Poincaré disk model and the Klein model are models of hyperbolic space on the unit n-disk. If u is a vector of norm less than one representing a point of the Poincaré disk model, then the corresponding point of the Klein model is given by In geometry, the Poincaré disk model, also called the conformal disk model, is a model of n-dimensional hyperbolic geometry in which the points of the geometry are in an n-dimensional disk, or unit ball, and the lines of the geometry are segments of circles contained in the disk...

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

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

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

Given two points on the boundry of the unit disk, the Klein model line is the chord between them, and the corresponding Poincaré model line is a circular arc on a subspace bisecting the disk, orthogonal to the boundry of the disk. The relationship between the two is simply 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 other model line.


Angles in the Klein model

Given two intersecting lines in the Klein model, which are intersecting chords in the unit disk, we can find the angle between the lines by mapping the chords, expressed as parametric equations for a line, to parametric functions in the Poincaré disk model, finding unit tangent vectors, and using this to determine the angle.


We may also compute the angle between the chord whose enpoints are u and v, and the chord whose endpoints are s and t, by means of a formula. Since the endpoints are the same in the Klein model and the Poincaré disk model, the formulas are identical for each model.


If both chords are diameters, so that v=-u and t=-s, then we are merely finding the angle between two unit vectors, and the formula for the angle θ is

cos(theta) = u cdot s.

If v=-u but not t=-s, the formula becomes

cos^2(theta) = frac{P^2}{QR},

where

P = u cdot (s-t),
Q = u cdot u,
R = (s-t) cdot (s-t) - (s_1t_2-s_2t_1)^2.

If both chords are not diameters, the general formula obtains

cos^2(theta) = frac{P^2}{QR},

where

P = (u-v) cdot (s-t) - (u_1v_2-u_2v_1)(s_1t_2-s_2t_1),
Q = (u-v) cdot (u-v) - (u_1v_2-u_2v_1)^2,
R = (s-t) cdot (s-t) - (s_1t_2-s_2t_1)^2.

Determining angles is greatly simplified when the question is to determine or construct right angles in the hyperbolic plane. A line in the Poincaré disk model corresponds to a circle orthogonal to the unit disk boundry, with the corresponding Klein model line being the chord between the two points where this intersects the boundry. The tangents to the intersection at the two endpoints intersect in a point called the pole of the chord. Any line drawn through the pole will intersect the Poincaré model circle orthogonally, and hence the line segments intersect the chord in the Klein model, which corresponds to the circle, as perpendicular lines.


Restating this, a chord B intersecting a given chord A of the Klein model, which when extended to a line passes through the pole of the chord A, is perpendicular to A. This fact can be used to give an easy proof of the ultraparallel theorem. In hyperbolic geometry, the Ultraparallel theorem states that every pair of ultraparallel lines in the hyperbolic plane has a unique common perpendicular hyperbolic line. ...


References

Eugenio Beltrami, Theoria fondamentale delgi spazil di curvatura constanta, Annali. di Mat., ser II 2 (1868), 232-255


Saul Stahl, The Poincaré Half-Plane, Jones and Bartlett, 1993


  Results from FactBites:
 
Lawrence Klein at AllExperts (835 words)
For his work in creating computer models to forecast economic trends in the field of econometrics at the Wharton School of the University of Pennsylvania, he was awarded the Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel in 1980.
Professor Klein is a graduate of Los Angeles City College, where he learned calculus; the University of California, Berkeley, where he began his computer modeling and earned a B.A. in Economics in 1942; he earned his PhD in Economics at the Massachusetts Institute of Technology (MIT) in 1944.
At the University of Michigan, Klein developed enhanced macroeconomic models, in particular the famous Klein-Goldberger model with Arthur Goldberger, which was based on foundations laid by Professor Jan Tinbergen of the Netherlands, later winner of the first economics prize in 1969.
  More results at FactBites »

 
 

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