The Pale Blue Dot? Chapter Seven: The Global Positioning System (GPS)

Originally Written By Thomas Perez. October 16, 2019 at 10:37pm. Copyright 2019. Updated 2020.

I am often asked the question that if the Earth is flat, how does one explain the Global Positioning System? Better known as GPS. To answer that question I will quote in full the following short article taken from Physics.Org…

“The Global Positioning System (GPS) is a network of about 30 satellites orbiting the Earth at an altitude of 20,000 km. The system was originally developed by the US government for military navigation but now anyone with a GPS device, be it a SatNav, mobile phone or handheld GPS unit, can receive the radio signals that the satellites broadcast.

Wherever you are on the planet, at least four GPS satellites are ‘visible’ at any time. Each one transmits information about its position and the current time at regular intervals. These signals, travelling at the speed of light, are intercepted by your GPS receiver, which calculates how far away each satellite is based on how long it took for the messages to arrive.

Once it has information on how far away at least three satellites are, your GPS receiver can pinpoint your location using a process called trilateration.


The red dot is supposed to be where you are located (T. Perez).

Imagine you are standing somewhere on Earth with three satellites in the sky above you. If you know how far away you are from satellite A, then you know you must be located somewhere on the red circle. If you do the same for satellites B and C, you can work out your location by seeing where the three circles intersect. This is just what your GPS receiver does, although it uses overlapping spheres rather than circles.

The more satellites there are above the horizon the more accurately your GPS unit can determine where you are.

GPS and Relativity

GPS satellites have atomic clocks on board to keep accurate time. General and Special Relativity however predict that differences will appear between these clocks and an identical clock on Earth.

General Relativity predicts that time will appear to run slower under stronger gravitational pull – the clocks on board the satellites will therefore seem to run faster than a clock on Earth.

Furthermore, Special Relativity predicts that because the satellites’ clocks are moving relative to a clock on Earth, they will appear to run slower.

The whole GPS network has to make allowances for these effects – proof that Relativity has a real impact.”

Notation: I do not deny the existence of orbiting GPS’s, I merely question and deny their applications upon an alleged ball Earth. However, according to Newsweek, “99% of our transmissions come from under the water (oceans) not from satellites in space.” So either way you put it, floating satellites in space or under water transmissions are fine for a flat Earther.

With that said, let us continue.

According to the source above, all GPS systems use trilateration as opposed to triangulation. The difference between the two is that trilateration involves the measuring of distances. Where as triangulation involves the measurement of angles.

“As GPS satellites broadcast their location and time, trilateration measure distances to pinpoint their exact position on Earth. While surveyors use triangulation to measure distant points, GPS positioning does not involve any angles whatsoever.” Moreover, “Through the measurement of distances, your precise GPS location can be determined. Yet several factors such as HDOP, PDOP, GDOP and the atmosphere can affect GPS accuracy and error.”

“HDOP (geodensy) is an Acronym for horizontal dilution of precision. A measure of the geometric quality of a GPS satellite configuration in the sky. HDOP is a factor in determining the relative accuracy of a horizontal position. The smaller the DOP number, the better the geometry.”

“PDOP. Dilution of precision (DOP), or geometric dilution of precision (GDOP), is a term used in satellite navigation and geomatics engineering to specify the additional multiplicative effect of navigation satellite geometry on positional measurement precision.”

“Thus, PDOP is Position of DOP and can be thought of as 3D positioning or the mean of DOP, and most often referred to in GPS; HDOP is Horizontal of DOP; VDOP is Vertical of DOP.” (Emphasis T. Perez).

“GDOP: One Satellite directly overhead w/an abundance of additional satellites spaced evenly around the sky.”

According to gisgeography, “The Global Positioning System uses the World Geodetic System (WGS84) as its reference coordinate system. It comprises of a reference ellipsoid, a standard coordinate system, altitude data and a geoid.”

Let us now consider how the GPS is used within the noted frame work above; ellipsoid, standard coordinate system, altitude and data.


An ellipsoid is “A spheroid, or ellipsoid of revolution…a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. …If the ellipse is rotated about its minor axis, the result is an oblate (flattened) spheroid, shaped like a lentil.”

Similarly, “An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scaling, or more generally, of an affine transformation. An ellipsoid is a quadric surface; that is, a surface that may be defined as the zero set of a polynomial of degree two in three variables.” (Wiki). (Emphasis T. Perez).

There are a few things here that need to be pointed out. The first is the “lentil.” The shape of a lentil can be seen in the following two pictures. Obviously, lentils are not spheres…

In the case of GPS’s, ellipsoids must be achieved. Ellipsoids are shapes that are transformed into that of a flat or oblate surface area. Note the following citation and illustration…

Figure 1 shows a red circle transformed in this way into a blue ellipse. Notice that the blue image has the same height as the original red circle, but is only half as wide.”

Figure 1. Scale x by 1/2.

This principle, in short, is taken from the guidelines of the Karman Line, of which is covered in chapter sixteen. The Karman Line works its way up. Moreover, “The main difference between an ellipsoid and ellipse is that the ellipsoid is a closed quadric surface that is a three dimensional analogue of an ellipse and ellipse is a type of curve on a plane.”

Directional scaling is “In Euclidean geometry, uniform scaling (or isotropic scaling)…a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale factor that is the same in all directions.” (Wiki) (Emphasis T. Perez).

Durand; Cutler. “Transformations” (PowerPoint). Massachusetts Institute of Technology. Retrieved 12 September 2008.

The key words here that one should pay close attention to are the words; “transformation,” “directional scaling,” “increases, “diminishes,” “shrinks,” “transformed,” and “deform” or “deforming.” It would seem that anyway you want to slice it, pun intended, the shape of a sphere must be altered in some fashion to the conform to a flat or oblate plane (surface).

But exactly what do they mean by “affine transformations?” In short, affine transformations “is a linear mapping method that preserves points, straight lines, and planes. Sets of parallel lines remain parallel after an affine transformation.

“The affine transformation technique is typically used to correct for geometric distortions or deformations that occur with non-ideal camera angles. For example, satellite imagery uses affine transformations to correct for wide angle lens distortion, panorama stitching, and image registration. Transforming and fusing the images to a large, flat coordinate system is desirable to eliminate distortion. This enables easier interactions and calculations that don’t require accounting for image distortion.”

Standard Coordinate System

“A coordinate system is used to define a location on the Earth. It is created in association with a map projection, datumand reference ellipsoid and describes locations in terms of distances or angles from a fixed reference point.” A datum is “a fixed starting point of a scale or operation. An accurate datum is formed by which other machining operations can be carried out.” (Emphasis T. Perez)

Oxford Dictionary.

Altitude and Data

“Altitude is a distance measurement, usually in the vertical or “up” direction, between a reference datum and a point or object. …Although the term altitude is commonly used to mean the height above sea level of a location, in geography the term elevation is often preferred for this usage.” Starting at a reference datum to a point or object – hence starting above the Karman Line.

In Reference to Inertial Forces

Is the Earth inertial, or a non-inertial frame of reference? According to the Britannia, “A coordinate system attached to the Earth is not an inertial reference frame because the Earth rotates and is accelerated with respect to the Sun.”

However, “A non-inertial reference frame is a frame of reference that is undergoing acceleration with respect to an inertial frame. An accelerometer at rest in a non-inertial frame will, in general, detect a non-zero acceleration. While the laws of motion are the same in all inertial frames, in non-inertial frames, they vary from frame to frame depending on the acceleration.” However, inertial forces are used in a fictitious manner.

Emil Tocaci, Clive William Kilmister (1984). Relativistic Mechanics, Time, and Inertia. Springer. p. 251. ISBN 90-277-1769-9.

Wolfgang Rindler (1977). Essential Relativity. Birkhäuser. p. 25. ISBN 3-540-07970-X.

Ludwik Marian Celnikier (1993). Basics of Space Flight. Atlantica Séguier Frontières. p. 286. ISBN 2-86332-132-3.

“In classical mechanics, is often possible to explain the motion of bodies in non-inertial reference frames by introducing additional fictitious forces (also called inertial forces, pseudo-forces and d’Alembert forces) to Newton’s second law. Common examples of this include the Coriolis force and the centrifugal force. In general, the expression for any fictitious force can can be derived from the acceleration of the non-inertial frame. As stated by Goodman and Warner, “One might say that F = ma holds in any coordinate system provided the term ‘force’ is redefined to include the so-called ‘reversed effective forces’ ‘inertia forces’.”

Harald Iro (2002). A Modern Approach to Classical MechanicsWorld Scientific. p. 180. ISBN 981-238-213-5.

Albert Shadowitz (1988). Special relativity(Reprint of 1968 ed.). Courier Dover Publications. p. 4. ISBN 0-486-65743-4.

Lawrence E. Goodman & William H. Warner (2001). Dynamics (Reprint of 1963 ed.). Courier Dover Publications. p. 358. ISBN 0-486-42006-X.

“In the theory of general relativity, the curvature of spacetime causes frames to be locally inertial, but globally non-inertial. Due to the non-Euclidean geometry of curved space-time, there are no global inertial reference frames in general relativity. More specifically, the fictitious force which appears in general relativity is the force of gravity.”

That a given frame is non-inertial can be detected by its need for fictitious forces to explain observed motions. For example, the rotation of the Earth can be observed using a Foucault pendulum. The rotation of the Earth seemingly causes the pendulum to change its plane of oscillation because the surroundings of the pendulum move with the Earth. As seen from an Earth-bound (non-inertial) frame of reference, the explanation of this apparent change in orientation requires the introduction of the fictitious Coriolis force.

Raymond A. Serway (1990). Physics for scientists & engineers (3rd ed.). Saunders College Publishing. p. 135. ISBN 0-03-031358-9.

V. I. Arnol’d (1989). Mathematical Methods of Classical Mechanics. Springer. p. 129. ISBN 978-0-387-96890-2.

Milton A. Rothman (1989). Discovering the Natural Laws: The Experimental Basis of Physics. Courier Dover Publications. p. 23. ISBN 0-486-26178-6.

Sidney Borowitz & Lawrence A. Bornstein (1968). A Contemporary View of Elementary Physics. McGraw-Hill. p. 138. ASIN B000GQB02A.

Leonard Meirovitch (2004). Methods of analytical Dynamics (Reprint of 1970 ed.). Courier Dover Publications. p. 4. ISBN 0-486-43239-4.

Giuliano Toraldo di Francia (1981). The Investigation of the Physical WorldCUP Archive. p. 115. ISBN 0-521-29925-X.

As pointed out by Ryder for the case of rotating frames as used in meteorology…

“A simple way of dealing with this problem is, of course, to transform all coordinates to an inertial system. This is, however, sometimes inconvenient. Suppose, for example, we wish to calculate the movement of air masses in the earth’s atmosphere due to pressure gradients. We need the results relative to the rotating frame, the earth, so it is better to stay within this coordinate system if possible. This can be achieved by introducing fictitious (or “non-existent”) forces which enable us to apply Newton’s Laws of Motion in the same way as in an inertial frame.”

Peter Ryder, Classical Mechanics, pp. 78-79

Peter Ryder (2007). Classical Mechanics. Aachen Shaker. pp. 78–79. ISBN 978-3-8322-6003-3.

NASA, for example, has done just that. According to a PDF online document entitled ‘Algorithm Theoretical Basis Document for ASTER Level-1 Data Processing (Ver. 3.0),’ the following is cited; “Orbital Reference Frame to Earth Inertial Frame: The line of sight vectors in the Orbital Reference Coordinate Frame can be converted to the expression in the Earth Inertial Coordinate Frame as follows.” “Earth Inertial Frame to Earth Fixed Frame: The line of sight vectors in the Earth Inertial Coordinate Frame can be converted to the expression in the Earth Fixed Coordinate Frame as follows.”

Aboard the Terra, an Earth observing system and multi-national NASA scientific research satellite in a Sun-synchronous orbit around the Earth, the “ASTER is an advanced multispectral imager which is selected by NASA to fly on EOS-AM1 polar orbiting spacecraft with other 4 sensors in June 1998, and covers a wide spectral region from visible to thermal infrared by 14 spectral bands with high spatial and spectral and radiometric resolutions. ASTER stands for the Advanced Spaceborne Thermal Emission and Reflection radiometer. The EOS-AM1 spacecraft will operate in a circular, near polar orbit at 705 km altitude. The orbit is sun-synchronous with a local time of 10:30 a.m. The recurrent cycle is 16 days.”

“The basic concept of ASTER is to acquire quantitative spectral data of reflected and emitted radiation from the earth’s surface in the 0.5-2.5 and 8-12 mm atmospheric windows at spatial and spectral resolutions appropriate for various science objectives. The general purpose of the science investigation by ASTER is to study the interaction among the geosphere, hydrosphere, cryosphere, and atmosphere of the Earth from geophysical point of view.”

“The pointing performance of the instrument and the spacecraft is vital for band-to-band registration of ASTER, since ASTER optics consists of three telescopes. Boresights of each telescope must be stable enough to satisfy the required registration accuracies which are ±0.2 pixels for intra-telescopes and ±0.3 pixels for inter-telescopes.”

NASA uses a theoretical system of coordinations.

“Theoretical Basis of Geometric System Correction:
The geometric system correction is divided into several parts as follows:

(1) The pointing correction
(2) The coordinates transformation from Navigation Base Reference of the spacecraft to the
Orbital Reference Frame
(3) The coordinates transformation from the Orbital Reference Coordinate Frame to the Earth
Inertial coordinate Frame
(4) The coordinates transformation from the Earth Inertial Coordinate Frame to the Earth
Greenwich Coordinate Frame
(5) Identification of a cross-point between the earth surface and an extended line of the vector.”

(1) NASA (.gov) › eospso › files PDF

In another PDF document found online, we learn that the only system used to designate ones position on Earth is the “Earth centered, Earth fixed coordinates.” According to a paper entitled ‘Coordinates’ written by James R. Clynch – Naval Postgraduate School, 2002, “These Earth Centered, Earth Fixed ECEF coordinates are the ones used by most satellites systems to designate an Earth position. This is done because it gives precise values. without having to choose a specific ellipsoid. Only the center of the earth and the orientation of the axis is needed.”

(2). Naval Postgraduate School › oc › …PDF

In this case Clynch is referring to “The intersection of a plane and a sphere is a circle (or is reduced to a single point, or is empty). Any ellipsoid is the image of the unit sphere under some affine transformation, and any plane is the image of some other plane under the same transformation. So, because affine transformations map circles to ellipses, the intersection of a plane with an ellipsoid is an ellipse or a single point, or is empty. Obviously, spheroids contain circles. This is also true, but less obvious, for triaxial ellipsoids.” 

Albert, Abraham Adrian (2016) [1949], Solid Analytic Geometry, Dover, p. 117, ISBN 978-0-486-81026-3.

See picture below…

According to Clynch, “There are two generic types of coordinates: Cartesian, and Curvilinear of Angular. Those that provide x-y-z type values in meters, kilometers or other distance units are called Cartesian. Those that provide latitude, longitude, and height are called curvilinear or angular. The Cartesian and angular coordinates are equivalent, but only after supplying some extra information. For the spherical earth model only the earth radius is needed. For the ellipsoidal Earth, two parameters of the ellipsoid are needed. (These can be any of several sets. The most common is the semi-major axis, called “a” and the flattening, called “f”).


Similarly NASA also cites the following from the same PDF document, under the category ‘Panoramic Correction.’ “As mentioned above, the altitude of the sensor (or the distance between the earth surface and the sensor) has some geometric impact to the data, This altitude changes predominantly by following three factors.

“1) Change of the spacecraft orbit: Nominal major axis of the spacecraft orbit is 7078 km as shown in the table 2-1. This gives an altitude of 700 km at the equator, assuming the WGS 84 Earth parameter: Considering the small flattening factor f of the orbit (1/f = 1.389 x 10-6 ), the radius of the orbit may change only slightly. So, the radius can be considered constant for few cycles. However, long term change of the radius may occur so that the change is of magnitude of 11km.”

“2) Change of the earth radius. By the WGS 84(World Geodetic System 84), the earth major axis a is 6378.137km and 1/f ( f : flattening factor ) = 298.25723563 ( the earth minor axis b is 6356.752 km ) So, if the orbit of the spacecraft is close to a circle, the distance from the ellipsoid changes with
magnitude of 21 km.”



Besides the academic explanations concerning the GPS’s that most people may have already known about, we or at least some, have found out that certain unknown variables are at play here with reference to global positioning networks. Mathematical formulas and equations are used interchangeably at will according to the need at hand. Formulas and equations are based upon fictitious forces due to “no global inertial reference frames in general relativity.” Hence transforming all coordinate systems into inertial frames of reference. This juxtaposition of elaborate mathematical expressions is nothing new. We have read an example of this type of “inversion,” if you will, before in chapter one, when the ancient Chinese astronomers book, ‘Huainanzi,’ inverted Eratosthenes’ calculation pertaining to the curvature of the Earth; while another text (Zhoubi SuanJing) revealed a method of determining distances (like the Sun) by measuring noon-time shadows from various latitudes, similar to Eratosthenes.

All GPS’s use lateral (trilateration) directional functions. In order for this to work on a prescribed global Earth they have to perform transformations. These transformations are done through what they call the “flattening factor.” The need to change from a spherical Earth to an “lentil” (flat) Earth is absolutely necessary when performing Euclidean geometry. “In Euclidean geometry, given a point and a line, there is exactly one line through the point that is in the same plane as the given line and never intersects it.” Upon the concept of Euclidean geometry precision “datums” can be performed – starting above the Karman Line (62 mi high), which is the boarder between Earth’s atmosphere and space.

Hence a flat Earth provides a liner transformation that increases or shrinks the size within a scale factor in all directions. This is done through “directional scaling.” Once the geodesy (circles within a flat circle, as shown above) and transformation is worked out mathematically the GPS, or in this case as I prefer to call it: ‘The FPS’ (Flat Positioning System) is achieved flawlessly. Hence all transformations are done systematically through linear, horizontal, up, down, vertical, trilateral directional datums; from point A to B (from satellite to tower), within a scale upon a flat Euclidean non-inertial geometrical Earth plane (surface). When you think of it, the GPS simply does not work on a ball Earth model because signals will always get obstructed due to a receiver, or tower, being on the “other side” of the Earth. However, signal obstructions are simply not the case; given the fact that many people can text and even video chat from Germany, Egypt, Russia and even Asia to folks located in the United States. A feat impossible upon a ball Earth; given the fact that towers are placed upon a surface that is allegedly curving every 8 inches per mile upon a fictitious globe/ball Earth.

We are told that 4 GPS Satellites are in Line-of-Sight. Using a constellation of 24 GPS satellites ensures that at least 4 satellites are within line-of-sight of any location on Earth at all times. The magic number is 4 because of the way that GPS calculates your exact position.” “GPS satellites fly in medium Earth orbit (MEO) at an altitude of approximately 20,200 km (12,550 miles). Each satellite circles the Earth twice a day.”

“Using a constellation of 24 GPS satellites ensures that at least 4 satellites are within line-of-sight of any location on Earth at all times. GPS satellites only last about 10 years, and often need servicing during their lifetime, which is why there are currently 32 GPS satellites in orbit.” The following GIF illustrates this…

But from what we have learned thus far, this is how it looks upon a flat Earth. Using the Peirce quincuncial projection of the world we can see the four corners of the Earth (Isaiah 11:12, Ezekiel 7:2, Revelation 7:1, etc) with the square. As they are angular. Now picture the 4 major GPS (or as I prefer to call it, FPS) satellites (large orange dots) at each corner pinging signals to other neighboring small non-orbiting satellites. The larger ones do not rotate. And since they do not rotate, the smaller ones do not have too either; since acceleration, as we have learned above, is considered a fictitious concept here. Also note the non-orange circles and circles within…

Following picture for color display only…

Non-rotating, or if you insist, rotating satellites work fine upon a flat Earth. The second picture above illustrates this by the curving line drawn across the middle going through the pole.