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Model Disc
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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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TWIN DISC MODEL EC200 BOAT CONTROL BOX SM LEVER |  |
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Denon Remote Control Model RC-246 for 5-Disc CD Changer | |
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Role Models (Blu-ray Disc, 2009) | |
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SINGER MODEL 603 MANUAL/ATTACHMENTS/SPECIAL DISCS | |
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Role Models (Blu-ray Disc, 2009) | |
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JVC Model XL-F108 5 Disc CD Changer Compact Disk Player | |
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Sony 300 Disc CD Player Model CDP-CX355 | |
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PHILIPS BLU-RAY DISC PLAYER MODEL # BDP 5005 N.I.O.B. | |
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Magnavox Blu-Ray DVD Disc Player, Model NB500MG1F | |
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Kenwood Compact Disc Player Model# DP-2030 | |
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Sony 1080p Blu-ray Disc Player - Model BDP-S350 | |
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Role Models (Blu-ray Disc, 2009) | |
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Magnavox Blu-Ray DVD Disc Player, Model NB500MG1F | |
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How the Blu-ray Player Reviews Can Help You Buying a Blu-ray Model
Blu-ray is a new technology in optical disc formats. The format allows you to record rewrite and play High-Definition video apart from storing a large volume of data. Hence, the Blu-ray players are getting popular among the viewers, who look for high definition picture quality while watching movies. With the increasing popularity of the Blu-ray players numerous companies are introducing various new models that are hitting the market time to time.
If you are looking forward to invest in one of the Blu-ray models, I must suggest you to go through the blu-ray player reviews to make your decision more informed. Here are the reasons that make reviews worth reading.
Detail about the features and specification
Any buyer of electronic products should know the features and specifications of the products before purchasing it. As Blu-ray disc player is a new product many people don’t have idea about its features and technologies it is built with. Hence, before you purchase the device, you need to read the Blu-ray disc player reviews to find details about the specifications of the models. Often, the reviews published by the companies of the models are best source of information.
Performance of the Blu-ray Models
It is worth knowing the performance of any electronic product before you buy one for yourself. When it comes to performance, views may differ. But if a product is worth praising there might be many thumbs up for it. bluray player reviews can inform you about the performance of the models based on the technologies they are built with. Sometimes, user reviews can provide you more information as they are based on personal experiences.
Price Details of the Model for Comparison
One of the crucial aspects that determine the sales of an electronic product is price. it is the factor that influences buying decisions of the viewers. Although performance of the product is right judge, one should consider the budget factor. Blu-ray player reviews can provide you perfect information about the price of the models. You can go through different reviews and compare prices of the models to zero in on the one that best suits your budget and satisfy maximum of your need from a Blu-ray player.
Users’ Comment on the Models
Often, users’ remark on a product makes a lot of difference. As most of the users publish their reviews based on their experiences, you can get good idea about the performance of the Blu-ray models from the Blu-ray disc reviews. However, sometimes the reviews are biased and only personal opinion. So, be careful while taking your decision based on personal reviews. You can also consider the opinions of the bluray users that are provided in the blogs.
New Blu-ray Releases
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Don’t take your decision about purchasing a blu-ray player hastily. Go through the blu-ray disc reviews to make an informed decision and make the best buy for your home entertainment.
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.
Does anybody know about the Lincoln LS 2003 model. I want to know how loud the factory system with 6 disc cd?
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DVD Review: Toy Story 2 (Two-Disc Special Edition Blu-ray/DVD Combo)
The toys are back in town with this sequel that is just as good as the original. Once again, Woody the cowboy (Tom Hanks) is taken from home, and this time it's up to Buzz Lightyear (Tim Allen) to organize a rescue team.
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