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In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or diameters of the unit circle.
The Poincaré disk model for hyperbolic geometry is the pair (D,H) where D consists of all points z in C such that |z|<1, and H consists of all Möbius transformations T for which T (D)=D.….
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9.2. THE POINCARE DISK MODEL´ 101 9.2.1 Construction of Lines This sounds nice, but how do you draw them? 1. Start with a circle Γ centered at O and consider the ray −→ OA, where A lies on the circle, Γ. 2. Construct the line perpendicular to −→ OA at A. 3. Choose a point P on this perpendicular line for the center of the second ...
May 16, 2024 · The Poincaré disk is a model for hyperbolic geometry in which a line is represented as an arc of a circle whose ends are perpendicular to the disk's boundary (and diameters are also permitted). Two arcs which do not meet correspond to parallel rays, arcs which meet orthogonally correspond to perpendicular lines, and arcs which meet on the ...
One useful feature of the Poincaré disk model is that there does exist such an easily expressed relationship between the Euclidean and hyperbolic distance. Property 1 For each pair and of points in we have that: where is the hyperbolic distance between the two points.
4 The Poincare Disc Model and Metric 11 5 Hyperbolic Geodesics 14 6 The Hyperbolic Pythagorean Theorem 16 The proofs and ideas in the rst four sections are accredited to Anderson [1] unless otherwise stated. The material in the fth section is accredited to Gamelin [2] and the proofs and ideas in the sixth section are due to Unger [3].
Nov 20, 2013 · The Poincare disk model for hyperbolic geometry is the ge-ometry (D, H) where D is the open unit disk. Note. The unit circle is not part of the Poincare disk, but it still plays an important role. It is called the circle at infinity, denoted S1 ∞. Recall.
