Is the Poincare disk a model for hyperbolic geometry?

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Is the Poincare disk a model for hyperbolic geometry?

The Poincare disk is a model for hyperbolic geometry. Proving this assertion´ meansprovingthat,withthetermspoint,line,distance,etc.interpretedasabove,all the axioms of hyperbolic geometry are satisﬁed. Since the model is described within Euclideangeometry,thoseproofsareallEuclideanproofs.

What kind of line is in the Poincare disk?

A ‘‘point’’ in the Poincare disk is a Euclidean point´ that is inside . There are two kinds of ‘‘lines’’ in the Poincare disk. The ﬁrst kind´ of line is a diameter of ; more speciﬁcally, a Poincare line of the ﬁrst kind consists´ of all the points on a diameter of  that lie inside .

Which is a valid Poincare disk model in H2?

Thus we have circles of arbitrary radius a orthogonally cutting T radius as valid Poincaré disk model in H2 on either side of the boundary circle which is hence no truncated geodesic arc in this sense, considered from Poincaré’s defining metric. Either side/segment is equally valid.

How are the locus points of the disc related?

Points on the present new outer hyperbolic locus points (B, Q) correspond to the conventional inner hyperbolic locus points (A, P) of disc model obtained by mutual inversions. A diameter of the boundary circle maps to line at infinity in inversion.

How is a triangle determined in the Poincare disk?

Given any two points in the Poincaré disk, there is a unique geodesic that passes through both of them. Hence, if two points are given and a third point is chosen that is not contained in the geodesic determined by the first two, these three points determine a unique hyperbolic triangle.

How are angles preserved on a Poincare disk?

If you tile the Poincaré circle with whatever, the symmetry group is made up of Möbius transformations. On the Poincaré disk, a straight line is a circle that intersects the edge of the disk at right angles. Also, angles are preserved – if two “lines” are at 60 degrees, then they stay at 60 degrees even if you move the focus around.

How to calculate the angle of parallelism in hyperbolic geometry?

The metric of the Poincaré half-plane model of hyperbolic geometry parametrizes distance on the ray { (0, y ) : y > 0 } with logarithmic measure. Let log y = a, so y = e a where e is the base of the natural logarithm.

How is the angle of parallelism of a line determined?

The angle depends on the segment length a between the right angle and the vertex of the angle of parallelism. Given a point not on a line, drop a perpendicular to the line from the point. Let a be the length of this perpendicular segment, and be the least angle such that the line drawn through the point does not intersect the given line.

Which is the best description of the Poincare disk model?

In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which the points of the geometry are inside the unit disk, and the straight lines consist of all circular arcs contained within that disk that are orthogonal to the boundary of the disk, plus all diameters of the disk.

What is the Poincare model of triheptagonal tiling?

Poincaré disk model of the truncated triheptagonal tiling. In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which the points of the geometry are inside the unit disk, and the straight lines consist of all circular arcs contained within that disk

Which is the best description of the Poincare model?

In geometry, the Poincaré disk model also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which the points of the geometry are inside the unit disk, and the straight lines consist of all segments of circles contained within that disk that are orthogonal to the boundary of the disk, plus all diameters of the disk.

Which is the largest possible triangle in hyperbolic geometry?

The interior angles of an ideal triangle are all zero. An ideal triangle has infinite perimeter. An ideal triangle is the largest possible triangle in hyperbolic geometry. In the standard hyperbolic plane (a surface where the constant Gaussian curvature is −1) we also have the following properties:

What do geodesics look like in hyperbolic geometry?

Geodesics are arcs of circles which meet the edge of the disk at 90°. Geodesics which pass through the center of the disk appear straight. Practice drawing geodesics in the Poincaré disk with Hyperbolic Geometry Exploration . Ideal hyperbolic triangle in the upper half space model.

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Is the Poincare disk a model for hyperbolic geometry?

What kind of line is in the Poincare disk?

How is the angle between two Poincare lines measured?

The angle between two Poincare lines is measured by´ measuring the Euclidean angle between their tangent lines. The Poincare disk is a model for hyperbolic geometry. Proving this assertion´ meansprovingthat,withthetermspoint,line,distance,etc.interpretedasabove,all the axioms of hyperbolic geometry are satisﬁed.

How are angles measured in the Poincare disc?

Given a hyperbolic line and a point not on it, there are infinitely many hyperbolic lines through the point that are parallel to the given hyperbolic line. When doing hyperbolic geometry using the Poincaré disc model, angles are measured the same way as in Euclidean geometry, but distances are not.

Is the disk of the Poincare model equivalent to the pseudosphere?

The disk of the Poincaré model is geometrically equivalent to the pseudosphere. In-other-words, any theorem proved in the Poincaré model, is a theorem in the pseudosphere.

Which is the natural logarithm of the Poincare disk model?

The vertical bars indicate Euclidean length of the line segment connecting the points between them in the model (not along the circle arc), ln is the natural logarithm . is the inverse hyperbolic function of hyperbolic cosine . is the inverse hyperbolic function of the hyperbolic tangent . . This reduces to the previous formula if .

How are lines defined in the Poincare disk model?

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 which is the general form of a circle orthogonal to the unit circle, or else by diameters.