Poincaré disk model

Poincaré disk model
Poincaré disc model of great rhombitruncated {3,7} tiling.
Poincaré 'ball' model view of the icosahedral honeycomb in hyperbolic 3-space

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 straight lines of the hyperbolic geometry are segments of circles contained in the disk orthogonal to the boundary of the disk, or else diameters of the disk. Along with the Klein model and the Poincaré half-space model, it was proposed by Eugenio Beltrami who used these models to show hyperbolic geometry was equiconsistent with Euclidean geometry.

Contents

Metric

If u and v are two vectors in real n-dimensional vector space Rn with the usual Euclidean norm, both of which have norm less than 1, then we may define an isometric invariant by

\delta (u, v) = 2 \frac{\lVert u-v \rVert^2}{(1-\lVert u \rVert^2)(1-\lVert v \rVert^2)},\,

where \lVert \cdot \rVert denotes the usual Euclidean norm. Then the distance function is

d(u, v) = \operatorname{arccosh} (1+\delta (u,v)).\,

Such a distance function is defined for any two vectors of norm less than one, and makes the set of such vectors into a metric space which is a model of hyperbolic space of constant curvature −1. The model has the conformal property that the angle between two intersecting curves in hyperbolic space is the same as the angle in the model.

The associated metric tensor of the Poincaré disk model is given by

ds^2 = 4 \frac{\sum_i dx_i^2}{(1-\sum_i x_i^2)^2}

where the xi are the Cartesian coordinates of the ambient Euclidean space. The geodesics of the disk model are circles perpendicular to the boundary sphere Sn−1.

Relation to the hyperboloid model

The Poincaré disk model, as well as the Klein model, are related to the hyperboloid model projectively. If we have a point [tx1, ..., xn] on the upper sheet of the hyperboloid of the hyperboloid model, thereby defining a point in the hyperboloid model, we may project it onto the hypersurface t = 0 by intersecting it with a line drawn through [−1, 0, ..., 0]. The result is the corresponding point of the Poincaré disk model.

For Cartesian coordinates (txi) on the hyperboloid and (yi) on the plane, the conversion formulae are:

y_i = \frac{x_i}{1 + t}
(t, x_i) = \frac {\left( 1+\sum{y_i^2},\, 2 y_i \right)} {1-\sum{y_i^2}}

Compare the formulae for stereographic projection between a sphere and a plane.

Analytic geometry constructions in the hyperbolic plane

A basic construction of analytic geometry is to find a line through two given points. In the Poincaré disk model, lines in the plane are defined by portions of circles having equations of the form

x^2 + y^2 + a x + b y + 1 = 0,\,

which is the general form of a circle orthogonal to the unit circle, or else by diameters. Given two points u and v in the disk which do not lie on a diameter, we can solve for the circle of this form passing through both points, and obtain


\begin{align}
& {} \quad x^2 + y^2 + \frac{u_2(v_1^2+v_2^2)-v_2(u_1^2+u_2^2)+u_2-v_2}{u_1v_2-u_2v_1}x \\[8pt]
& {} + \frac{v_1(u_1^2+u_2^2)-u_1(v_1^2+v_2^2)+v_1-u_1}{u_1v_2-u_2v_1}y + 1 = 0.
\end{align}

If the points u and v are points on the boundary of the disk not lying at the endpoints of a diameter, the above simplifies to

x^2+y^2+\frac{2(u_2-v_2)}{u_1v_2-u_2v_1}x - \frac{2(u_1-v_1)}{u_1v_2-u_2v_1}y + 1 = 0.

Angles

We may compute the angle between the circular arc whose endpoints (ideal points) are given by unit vectors u and v, and the arc whose endpoints are s and t, by means of a formula. Since the ideal points are the same in the Klein model and the Poincaré disk model, the formulas are identical for each model.

If both models' lines 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, in terms of the wedge product,

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

where

P = u \cdot (s-t),\,
Q = u \cdot u,\,
R = (s-t) \cdot (s-t) - (s \wedge t) \cdot (s \wedge t).\,

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 \wedge v) \cdot (s \wedge t),\,
Q = (u-v) \cdot (u-v) - (u \wedge v) \cdot (u \wedge v),\,
R = (s-t) \cdot (s-t) - (s \wedge t) \cdot (s \wedge t).\,

Using the Binet–Cauchy identity and the fact that these are unit vectors we may rewrite the above expressions purely in terms of the dot product, as

P = (u-v) \cdot (s-t) + (u \cdot t)(v \cdot s) - (u \cdot s)(v \cdot t),\,
Q = (1 - u \cdot v)^2,\,
R = (1 - s \cdot t)^2.\,

Artistic realizations

The M.C. Escher print Circle Limit IV is an artistic visualization of the Poincaré disk.

See also

References

  • James W. Anderson, Hyperbolic Geometry, second edition, Springer, 2005
  • Eugenio Beltrami, Teoria fondamentale degli spazii di curvatura costante, Annali. di Mat., ser II 2 (1868), 232-255
  • Saul Stahl, The Poincaré Half-Plane, Jones and Bartlett, 1993

Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • Poincaré model — can refer to:*Poincaré disk model, a model of n dimensional hyperbolic geometry *Poincaré half plane model, a model of two dimensional hyperbolic geometry …   Wikipedia

  • Poincaré metric — In mathematics, the Poincaré metric, named after Henri Poincaré, is the metric tensor describing a two dimensional surface of constant negative curvature. It is the natural metric commonly used in a variety of calculations in hyperbolic geometry… …   Wikipedia

  • Poincaré half-plane model — Stellated regular heptagonal tiling of the model.In non Euclidean geometry, the Poincaré half plane model is the upper half plane, together with a metric, the Poincaré metric, that makes it a model of two dimensional hyperbolic geometry.It is… …   Wikipedia

  • 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… …   Wikipedia

  • Hyperboloid model — In geometry, the hyperboloid model, also known as the Minkowski model or the Lorentz model (after Hermann Minkowski and Hendrik Lorentz), is a model of n dimensional hyperbolic geometry in which points are represented by the points on the forward …   Wikipedia

  • Unit disk — For other uses, see Disc (disambiguation). An open Euclidean unit disk In mathematics, the open unit disk (or disc) around P (where P is a given point in the plane), is the set of points whose distance from P is less than 1 …   Wikipedia

  • Hyperbolic geometry — Lines through a given point P and asymptotic to line R. A triangle immersed in a saddle shape plane (a hyperbolic paraboloid), as well as two diverging ultraparall …   Wikipedia

  • Differential geometry of surfaces — Carl Friedrich Gauss in 1828 In mathematics, the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives:… …   Wikipedia

  • Möbius transformation — Not to be confused with Möbius transform or Möbius function. In geometry, a Möbius transformation of the plane is a rational function of the form of one complex variable z; here the coefficients a, b, c, d are complex numbers satisfying ad − …   Wikipedia

  • non-Euclidean geometry — geometry based upon one or more postulates that differ from those of Euclid, esp. from the postulate that only one line may be drawn through a given point parallel to a given line. [1870 75; NON + EUCLIDEAN] * * * Any theory of the nature of… …   Universalium

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”