Prison Shape Models

Chapter Title : Prison Shape and Travel Distance

https://web.archive.org/web/20211029000320/https://aviation.stackexchange.com/questions/89939/is-there-a-place-where-altitude-while-flying-is-negative

https://web.archive.org/web/20221218201824/https://en.wikipedia.org/wiki/List_of_lowest_airports

I am not going to answer the question.  If you want me to give you an answer instead of thinking about it for yourself or observing things for yourself then skip this chapter

I intentionally never answer the question of the shape of the earth in this chapter because I want the viewer to develop their own thinking skills instead of me giving them an answer.  However I do compare different models about the shape of the earth and what certain results might end up being based on mathematical and physics assumptions that the reader should not really presume to be true, but instead contemplate and observe if they are true.  That is if they even care to bother because personally knowing the shape of the earth is not very important in terms of every day decisions for most people.

If you do not already know certain things then you should skip this chapter or learn these things then come back and read the chapter.  

Do not feel bad about skipping this chapter.  This is one of the least important chapters in the entire book in the opinion of the author because it does not matter what shape your slave pen is or if it is spinning, what matters is how to escape the soft "mind control" used to promote covert slavery

Before reading this chapter you should already be familiar with the following

Algebra

Trigonometry

Cartesian Coordinates

Polar Coordinates

How to convert degrees to radians

The connection between radians and the lengths of curved objects that are shaped like parts of a circle

How to use sine and cosine to measure the length of objects given an angle and another length

How to use inverse tangent to figure out an angle given two perpendicular lengths or a slope of a line 

You should understand how to take limits in calculus ( this is not necessary to understand most of the material but will be used in some of it )

What you do not need to understand this material is any high school or college geometry class material that is not covered in algebra, trigonometry, calculus or physics

Notation

* means multiply or times or multiplication sign

/ means divide by or division sign

x ^ N means x to the N power

x ^ 2 means x squared

x ^ 0,5 means the square root of x

Pi means pie or approximately 3.14

Perimeter arc length or path length of a section of a circle = Radius * Angle in Radians 

The great circle distance

The great circle distance is the distance of the shortest travel path between any two points on the surface of a sphere

If you are traveling the shortest travel route between two points of the same latitude that are not on the equator then you will change latitude while traveling from one point to the other in the globe earth model

Examples of traveling from one point to another at a constant latitude are not examples of the shortest travel routes in the globe earth model but such travel routes could be used for experimental purposes to compare results with those predicted by flat earth models.

Multiple Types of North and South

Celestial Poles and Astronomical North and South - It is claimed that for those in the Northern Hemisphere stars rotate around a celestial pole and a Star called Polaris is currently near that pole.  It is claimed that in the Southern Hemisphere stars rotate around a celestial pole and a star called Sigma Ontantis is currently near that pole.  North and South can be defined based on how traveling effects what star position you see in the sky.

"A pole star or polar star is a star, preferably bright, nearly aligned with the axis of a rotating astronomical body.

Currently, Earth's pole stars are Polaris (Alpha Ursae Minoris), a bright magnitude-2 star aligned approximately with its northern axis that serves as a pre-eminent star in celestial navigation, and a much dimmer magnitude-5.5 star on its southern axis, Polaris Australis (Sigma Octantis)."

https://en.wikipedia.org/wiki/Pole_star

"This makes Polaris, colloquially known as the "North Star", useful for navigation in the Northern Hemisphere: not only is it always above the north point of the horizon, but its altitude angle is always (nearly) equal to the observer's geographic latitude (though it can, of course, only be seen from locations in the Northern Hemisphere)."

https://web.archive.org/web/20221213155646/https://en.wikipedia.org/wiki/Celestial_pole 

"Altitude (alt.), sometimes referred to as elevation (el.), is the angle between the object and the observer's local horizon. For visible objects, it is an angle between 0° and 90°"

https://web.archive.org/web/20220722231628/https://en.wikipedia.org/wiki/Altitude_angle

Instantaneous Axis of Rotation of the earth - It has been claimed that the location for the axis of earth's rotation changes with time.  One end of the surface of the Earth that touches the axis can be labeled as North Pole and the other as the South Pole.  Going toward the North Pole while traveling on the surface of the earth at sea level is North and going toward the South Pole while traveling on the surface of the earth at sea level is South.

North and South as measured by compasses - If a compass needle is oriented perpendicular to up and down it is claimed that one end of the compass needle will point North and the other end will point South.  It is claimed that the magnetic North Pole can be found by following the end of the compass needle that point's North in the Northern hemisphere and the magnetic South pole can be found by following the opposite end of the compass needle that point's South in the Southern hemisphere.  It is claimed that the location of the magnetic North pole and South pole change over time and that therefor the magnetic directions of North and South change with time.

North and South measured as the directions that are perpendicular to East and West based on equator - If you measure East and West based on the direction you would move to travel on the equator, North and South would be perpendicular to both East and West and Up and Down at any location based on this convention.  This would create some difficulty because you could only measure it when standing on the equator and you would have to know where the equator is.  I believe that most flat earther's and globe earther's agree about the location of the equator based on various physical things that happen there.

North and South measured as the directions perpendicular to East and West based on sunrise and sunset is a problematic way to measure North and South - This is the idea that if the sun rises in the East then appears to move West to an observer standing on the earth and sets in the West then the North and South would be the two directions perpendicular to East and West.  This is problematic because the sun is said to not always rise and set purely East and West but actually sometimes rises North East or North West of an observer standing on the earth or sets South East or Southwest of an observer standing on the earth.  If you used a compass to measure North and South it would not always line up with being perpendicular to Sunrise and Sunset in all location at all times. 

Geodetic North - Geographic North

Geodetic North or geographic North points toward a geographic North Pole.  90 degrees latitude is set at the Geographic North Pole, not a North pole based on a instantaneous axis of rotation of the earth, stars in the sky or compass readings.

It is claimed that if people used stars in the sky, the instantaneous axis of rotation of the earth or a compass to decide a north pole location empirically then this location change over time.  This would make the latitude of other landmarks on earth change over time which would be problematic for mapmakers.

In order to have a location that does not change as much over time.  A committee decided a specific place for a geographic north pole as a physical location on earth they would use as a standard for 90 degrees Latitude when making maps.  Although this location is supposed to provide a more stable way to measure Latitude it still is claimed to change in position relative to other locations due to plate tectonics.

"True north (also called geodetic north or geographic north) is the direction along Earth's surface towards the geographic North Pole or True North Pole."

https://web.archive.org/web/20221118050652/https://en.wikipedia.org/wiki/True_north

"The North Pole is by definition the northernmost point on the Earth, lying antipodally to the South Pole. It defines geodetic latitude 90° North, as well as the direction of true north. "

https://web.archive.org/web/20230107011605/https://en.wikipedia.org/wiki/North_Pole

Precise Definition

"The Earth's axis of rotation – and hence the position of the North Pole – was commonly believed to be fixed (relative to the surface of the Earth) until, in the 18th century, the mathematician Leonhard Euler predicted that the axis might "wobble" slightly. Around the beginning of the 20th century astronomers noticed a small apparent "variation of latitude", as determined for a fixed point on Earth from the observation of stars. Part of this variation could be attributed to a wandering of the Pole across the Earth's surface, by a range of a few metres. The wandering has several periodic components and an irregular component. The component with a period of about 435 days is identified with the eight-month wandering predicted by Euler and is now called the Chandler wobble after its discoverer. The exact point of intersection of the Earth's axis and the Earth's surface, at any given moment, is called the "instantaneous pole", but because of the "wobble" this cannot be used as a definition of a fixed North Pole (or South Pole) when metre-scale precision is required.

It is desirable to tie the system of Earth coordinates (latitude, longitude, and elevations or orography) to fixed landforms. However, given plate tectonics and isostasy, there is no system in which all geographic features are fixed. Yet the International Earth Rotation and Reference Systems Service and the International Astronomical Union have defined a framework called the International Terrestrial Reference System."

https://web.archive.org/web/20230107011605/https://en.wikipedia.org/wiki/North_Pole

Wikipedia Article on "International Terrestrial Reference System" at the link below

https://en.wikipedia.org/wiki/International_Terrestrial_Reference_System_and_Frame

North Pole Centered Flat Model

The north pole in the globe is replaced with the center of the flat earth

The center of the flat earth is electromagnetic and effects compass needles

The south pole in the globe is replaced with a ice wall

The ice wall forms a circle shape

You can go around the world going purely east or west without going north or south by staying at a constant distance away from the north pole

South Pole Centered Flat Model

The south pole in the globe is replaced with the center of the flat earth

The center of the flat earth is electromagnetic and effects compass needles

The north pole in the globe is replaced with a ice wall

The ice wall forms a circle shape

You can go around the world going purely east or west without going north or south by staying at a constant distance away from the north pole

Simplified globe earth model

Modeling the earth as a perfect sphere and ignoring the difference between the polar and equatorial radius

Ignoring the difference between the axial and magnetic north pole and presuming they are in the same location on the globe model to simplify the model comparison

Ignoring the difference between the axial and magnetic south pole and presuming they are in the same location on the globe model to simplify the model comparison

Radius = Radius of the globe earth model in all calculations below including flat earth model calculations

All calculations in the sections below are based on travel at sea level

Traveling at a constant altitude above sea level would give a longer travel distance on a globe earth model than traveling at sea level but would not do so on a flat earth model

Traveling at a constant altitude below sea level would give a shorter travel distance on a globe earth model than traveling at sea level but would not do so on a flat earth model

How would altitude effect travel distance on a a globe earth model

As long as someone travels at a constant altitude within a certain range there will be little percent difference in the measurements if they traveled at a difference altitude than sea level based on the official equator and polar radius of the standard oblate spheroid earth model.  If someone flied at an altitude as high as 12 miles above sea level or traveled as low as 1 mile below sea level as long as they traveled at constant altitude it should not make much differences in the travel distance.

3,963 mile equatorial radius + 12 miles = 3975 miles

3,950 mile polar radius - 1 miles = 3949 miles

3975 - 3949 / 3949 = 0.66% rounded

See references to understand choice of altitudes which are rounded to a larger range than those chosen from references based on concord jet travel height and the elevation of the dead sea.  Note that commercial planes  can fly at a higher altitude than the elevation of the highest mountains in the world.  Although they might avoid flying over certain mountains for safety and legal reasons.

https://web.archive.org/web/20221219104515/https://en.wikipedia.org/wiki/Earth_radius

https://web.archive.org/web/20221027144331/https://www.worldatlas.com/geography/the-10-lowest-places-on-earth.html

https://web.archive.org/web/20221211032933/https://travelonthefly.com/how-high-do-planes-fly/

https://web.archive.org/web/20221222161437/https://en.wikipedia.org/wiki/List_of_highest_mountains_on_Earth

Travel Distance from pole to pole on a globe model or pole to ice wall on flat models

The distance traveling from the north pole to the south pole on the globe model at a constant longitude is equal to

Pi * radius

Therefore the distance from the north pole to the ice wall on the north pole centered flat model is equal to

Pi * radius 

Therefore the distance from the south pole to the ice wall on the south pole centered flat model is equal to

Pi * radius 

Why?

This is assuming the pure north south travel distances between any two points on a flat earth model are the same as on the globe earth model.  Because if people were wrong about the earth being a globe as flat earth believers claim then people would know the distance traveling purely north or south between any two locations purely north and south from each other by measuring how far they would have to walk going purely north or south from one point to the other without traveling east or west.  This assumption might not necessarily be correct but it gives me something to work with to mathematically compare the models.

Travel distances when traveling at a constant longitude for the different models

I will presume the north south travel distance when traveling at a constant longitude will be the same on the two flat earth models and the globe earth model.  This is for reasons I described above

Traveling at a constant longitude between two latitudes on the globe model

( First Latitude - Second Latitude ) * ( 2 * Pi Radians / 360 degrees ) * ( Radius )

Traveling at a constant longitude between two latitudes on the north pole centered flat model

( First Latitude - Second Latitude ) * ( 2 * Pi Radians / 360 degrees )  * ( Radius )

Traveling at a constant longitude between two latitudes on the south pole centered flat model

( First Latitude - Second Latitude ) * ( 2 * Pi Radians / 360 degrees )  * ( Radius )

Travel distances when traveling at a constant latitude for the different models

The east west travel distances between places would not be the same on the flat earth and globe earth models when traveling at a constant latitude

Traveling at a constant latitude between two longitudes on globe model

( First longitude - Second Longitude ) * ( 2 * Pi Radians / 360 degrees ) * Radius * cosine ( latitude )

Traveling from  the north pole at positive 90 degrees latitude to another Latitude at a constant longitude on the north pole centered flat earth model

 Pi * Radius * (  90 degrees - Latitude ) / 180 degrees

Traveling at a constant latitude between two longitudes on the north pole centered flat model

Pi * Radius * ( ( 90 Degrees - Latitude ) / 180 degrees ) * ( First longitude - Second Longitude ) * ( 2 * Pi Radians / 360 degrees ) 

Travel distance for the north pole centered flat  model divided by travel distance for the globe model when traveling at a constant latitude between two longitudes

( 90 degrees - Latitude ) / ( 180 degrees * cosine ( latitude ) )

f (x) = (  0.5 pi - x ) / ( pi * cosine ( x ) )

Limit as x approaches + 0.5 pi of f ( x )  = 1 / pi    approximately or exactly

The limit was estimated graphically using geogebra

Limit as x approaches - 0.5 pi of f ( x ) = Does not exist ( positive infinity )

Traveling from  the south pole at negative 90 degrees latitude to another Latitude at a constant longitude on the south pole centered flat earth model

 Pi * Radius * (  Latitude + 90 degrees ) / 180 degrees

Traveling at a constant latitude between two longitudes on the south pole centered flat model

Pi * Radius * ( ( Latitude + 90 degrees ) / 180 degrees ) * ( First longitude - Second Longitude ) * ( 2 * Pi Radians / 360 degrees ) 

Travel distance for the south pole centered flat  model divided by travel distance for the globe model when traveling at a constant latitude between two longitudes

( Latitude + 90 degrees ) / ( 180 degrees * cosine ( latitude ) )

f (x) = ( x + 0.5 pi ) / ( pi * cosine ( x ) )

Limit as x approaches - 0.5 pi of f ( x )  = 1 / pi    approximately or exactly

The limit was estimated graphically using geogebra

Limit as x approaches + 0.5 pi of f ( x ) = Does not exist ( positive infinity )

North Pole Centered Flat Model travel distance divided by Globe Model travel distance

Rounded values are truncated

Latitude in degrees, ratio

approaching + 90 degrees, 0.31830988 rounded approaches approximately or exactly equal to 1 / pi 

+ 89.9 degrees, 0.31831004 rounded

+ 80 degrees, 0.31993169 rounded

+ 70 degrees, 0.32486715 rounded

+ 60 degrees, 0.33333333 rounded 1/3

+ 50 degrees, 0.34571640 rounded

+ 40 degrees, 0.36261313 rounded

+ 30 degrees, 0.38490017 rounded

+ 20 degrees, 0.41384691 rounded

+ 10 degrees, 0.45130071 rounded

0 degrees. 0.5 exact

- 10 degrees, 0.56412589 rounded

- 20 degrees, 0.65033086 rounded

- 30 degrees, 0.76980035 rounded

- 40 degrees, 0.94279415 rounded

- 42.5672 degrees, 0.99999998 rounded

- 42.5672006834, duckduckgo search engine calculator gave value of exactly 1 but this should not be exact

- 42.5673 degrees, 1.00000234 rounded

- 45 degrees, 1.06066017 rounded

- 50 degrees, 1.21000742 rounded

- 60 degrees, 1.66666666 rounded 1 + 2 / 3

- 65 degrees, 2.03756247438 rounded

- 70 degrees, 2.59893724 rounded

- 75 degrees, 3.54172802 rounded

- 80 degrees, 5.43883878 rounded

- 85 degrees, 11.15499898 rounded

-87.5 degrees, 22.60717471 rounded

- 88.75 degrees, 45.52192478 rounded

-89.375 degrees, 91.35674913 rounded

-89.6875 degrees, 183.02909204 rounded

approaches - 90 degrees limit does not exist ( positive infinity )

Table excluded for South Pole Centered Flat Model travel distance divided by Globe Model travel distance

South pole centered flat model travel distances can be got from the north centered travel distances inputting the latitude you want in degrees but with the opposite sign

 

https://www.geogebra.org/

https://duckduckgo.com/?q=%3D+(+90-89.9+)+%2F+(+180*cos+(+89.9*pi+%2F+180+)+)&ia=calculator

https://duckduckgo.com/?q=%3D+(+90%2B42.5672+)+%2F+(+180*cos+(+42.5672*pi+%2F+180+)+)&ia=calculator

https://duckduckgo.com/?q=%3D+(+90%2B42.5673+)+%2F+(+180*cos+(+42.5673*pi+%2F+180+)+)&ia=calculator

https://duckduckgo.com/?q=%3D+(90+%2B+89.375)+%2F2&t=&ia=calculator

https://duckduckgo.com/?q=0.7429378052822*180%2Fpi&ia=calculator

gives pi for limit of the reciprocal of the function implying 1/pi is the limit for the function

http://web.archive.org/web/20221229072533/https://www.symbolab.com/solver/limit-substitution-calculator/%5Clim_%7Bx%5Cto%200.5%5Ccdot%20pi%20%7D%5Cfrac%7B%5Cleft%28%20pi%5Ccdot%20cos%5Cleft%28x%5Cright%29%5Cright%29%7D%7B%5Cleft%280.5%5Ccdot%20pi-x%5Cright%29%7D?or=input

Output accessed online at the date of 2022 December 24 for result of exactly 1

https://duckduckgo.com/?q=%3D+(+90+%2B+42.5672006834+)+%2F+(+180+*+cos+(+42.5672006834+*+pi+%2F+180+)+)&ia=calculator

How far can people easily travel without special legal clearance travel North and South?

https://web.archive.org/web/20220606172713/https://en.wikipedia.org/wiki/List_of_countries_by_northernmost_point

https://web.archive.org/web/20220921195423/https://en.wikipedia.org/wiki/List_of_countries_by_southernmost_point

Until 1995, scientific research staff were stationed permanently at a meteorological station on Campbell Island. Since then, the islands have been uninhabited, though they are periodically visited by researchers and tourists. Protection of reserves was strengthened in 2014, becoming the largest natural sanctuary in the nation.

[3]https://web.archive.org/web/20221016042902/https://en.wikipedia.org/wiki/New_Zealand_Subantarctic_Islands

The island group is barren, windswept, bitterly cold, and uninhabited.

https://web.archive.org/web/20221111221615/https://en.wikipedia.org/wiki/Southern_Thule

Copyright Carl Janssen 2022