Phenomenon of hyperbolic refraction in space with Manhattan metric

Phenomenon of hyperbolic refraction in space with
Manhattan metric
March 10, 2019
Abstract
Description of the law of hyperbolic refraction of a distributed straight line in space with Manhattan metric when it passes through a boundary of two
areas with ”city-blocks” of different sizes, that the ratio of the hyperbolic sine of the incident and refracted hyperbolic angles equals to the ratio of the sizes of the underlying blocks.The work contains the description of the computer-based experiment to allow independent researcher to verify the conclusions.
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Basic Concepts

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Distributed straight line

Since in Manhattan metrics there are multiple straight (shortest distance) lines connecting any pair of points, I will introduce a concept of a distributed straight line connecting a pair of points {P1, P2} as a functional, value of which in any given point is a probability that a straight line {P1, P2} will pass through that point.The probability is calculated as the ratio of a number of all straight lines connecting points {P1, P2} and passing through a given point to a number of all straight lines connecting {P1, P2}.

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Computer-generated Distributed straight line
Figure 3: Computer-generated distributed straight line connecting points {P1, P2}
on 700 X 540 blocks(pixels). Gray color intensity corresponds probability. White-colored pixels also have non-zero probability, but it is too small to be displayed.

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Long Gravity Axis of a distributed straight line
Long gravity axis is a symmetry axis of the distributed straight line
along which the line is perfectly balanced in physical sense of the word, if we
consider probability represents a unit weight. It has been shown with the
help of computer algorithm that a long gravity axis will be just a straight
classical line connecting P1 and P2.
Figure 4: Computer-generated distributed straight line with long gravity axis
– a solid black line in the middle

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Distributed straight line passing through a boundary of two areas with different size blocks
Figure 5: Distributed straight line passing through a boundary of two areas with different size blocks

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Computer-generated example of distributed straight line passing through a boundary of two areas with block sizes 3 and 5
Figure 6: Computer-generated example of distributed straight line passing through a boundary of two areas with block sizes 3 and 5

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Computer-generated example of distributed straight line passing through a boundary of two areas with block sizes 5 and 2
Figure 7: Computer-generated example of distributed straight line passing through a boundary of two areas with block sizes 5 and 2

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