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.

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.

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.

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

– 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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