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How do you find the hyperbolic distance in the Poincare Disc Model between i/2 and 1/2?
You do a line integral, along the shortest path, from one point to the other.
In the Poincare Disk model, an infinitesimal change (dx,dy) in the coordinates (x,y) corresponds to an infinitesimal distance ds, where:
(ds)^2 = [(dx)^2+(dy)^2] / [1-x^2-y^2]
This is the hyperbolic version of the Pythagorean Theorem.
In Euclidean goemetry: (ds)^2=(dx)^2+(dy)^2
You want to find the smallest possible value for the integral of (ds) along any path from (0,1/2) to (1/2,0).
If you already know that the "shortest" path is a circle perpendicular to the unit circle bounding the Poincare disc, that simplifies things a bit. A little bit of algebra gets the equation of the path from (0,1/2) to (1/2,0) and a rather nasty looking integral. (I probably messed up the arithmetic, so I'll let you do it yourself instead of showing you my answer.)
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Probably a better way is to use a "conformal" transformation (which preserves the metric). Translate (0,1/2) to the origin and then rotate the image of (1/2,0) back to the positive x-axis: (X,0). Then shortest path is the x-axis (y=0 and dy/dx=0) so you just have to integrate dx/(1-x^2) = Ln(Sqrt([1+X]/[1-X])).
Those conformal maps are fractional linear transformations of the complex plane:
w = (az+b)(cz+d)
which preserve the unit disk: |z|=1 implies |w|=1.
Those are rotations w=az (complex a with |a|=1)
and "translations": w = [z+(a+ib)] / [(a-ib)z+1].
Edit: That second approach is MUCH easier. I get Ln((25+4Sqrt(34))/9, or about 1.68, with some confidence that it might be right!
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Invisible English Plate Hanger Disc--2 Inch (For Plates 4 to 6 Inches in Diameter) Sale Price: $2.25 Average Rating:   |
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These plate hangers are wonderful. They work by solidly adhering to the surface of the back of the plate and are easily removed with warm water. The disc plate hanger also has excellent adhesion to unvarnished wood and hardboard... |
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Tracks: 1. Trouble Is - One Man Down; 2. Trouble Is - Chorus for Destruction; 3. Fall Out Boy - Dead on Arrival; 4. Fall Out Boy - Chicago is So Two Years Ago; 5. Acceptance - Permanent; 6. Acceptance - Seeing is Believing; 7... |
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2 Lumbar Vertebrae Model with prolapsed disc, flexibly mounted 2 Lumbar Vertebrae Model with prolapsed disc, flexibly mountedWith spinal nerves and dura mater of spinal cord.Measurements:6x40x40 cm Weight: 0,15 kg Product photo may not exactly match the product offered for sale... |
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E.c. Moore Inc. Mandrel And Snap-on Sanding Discs Mandrell - Model 7HPSS - Each Sale Price: $11.68 |
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Mandrel is for use with snap-on, brass center discs. Disc diameter is 7/8". |
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Maxell DVD-R & DVD-RAM Round Cartridges for Panasonic & Late Model Hitachi DVD Camcorders List Price: $24.95 Sale Price: $12.99 Average Rating:  |
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A specially designed round cartridge for the HITACHI DZMV350 - DZMV380 and specific PANASONIC model camcorders |
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About the Author
James Poole writes Blu-ray disc player reviews on different kinds of Blu-ray models. The blu-ray player reviews by the author are based on his personal experiences.
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