Talk:Global Positioning System/Archive 8: Difference between revisions

Revision as of 00:03, 2 August 2015

Revision as of 00:02, 14 August 2015

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::::So, instead of moving [[Triangulation]] over [[Triangulation (surveying)]] (which remains the primary topic), I propose to create a new [[Triangulation (mobile technology)]] which redirects to [[Mobile phone tracking]], and make a note in [[Triangulation (disambiguation)]]. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 18:28, 2 July 2015 (UTC)

== Edit war ==

Now that the article is locked, I think we should try to reach consensus as to the content dispute. I've gone back over the three previous discussions of this "equations of spheres" dispute and don't see anything new here, so I would argue in favor of leaving out the "equations of spheres" material. But I'm open to persuasion if someone can provide supporting quotes from the source material (quotes, not your own interpretation). [[User:Kendall-K1|Kendall-K1]] ([[User talk:Kendall-K1|talk]]) 13:59, 23 June 2015 (UTC)

Yes this is a good opportunity to discuss editing changes. There have been a lot of complaints that the fact that the equations in the Problem description section describe spheres is not documented. I think that these complaints are just excuses since it is obvious to me that they are the equations of spheres. However to call the bluff of these people doing the complaining, I have provided a reference along with explanation to show that they are the equations of spheres. This explanation is shown below. This explanation will aid the understanding of GPS so if you are a supporter of improving the GPS document making it more readable and understandable, you will support including the explanation below as a part of the GPS document. On the other hand if you want to degrade the GPS document make it less understandable, you may oppose the inclusion of this explanatory material. So let's find out who the good people are and who the enemies of Wikipedia are or otherwise explain your position. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 18:27, 23 June 2015 (UTC)

Problem description

The receiver uses messages received from satellites to determine the satellite positions and time sent. The ''x, y,'' and ''z'' components of satellite position and the time sent are designated as [''x_i, y_i, z_i, s_i''] where the subscript ''i'' denotes the satellite and has the value 1, 2, ..., ''n'', where ''n''&nbsp;&ge;&nbsp;4. When the time of message reception indicated by the on-board receiver clock is ''t&#771;'', the true reception time is {{nobreak|1=''t'' = ''t&#771;'' - ''b''}}, where ''b'' is the receiver's clock offset from the much more accurate GPS system clocks employed by the satellites. The receiver clock offset is the same for all received satellite signals (assuming the satellite clocks are all perfectly synchronized). The message's transit time is {{nobreak|1=''t&#771;'' - ''b'' - ''s_i''}}<!--, where ''s_i'' is the satellite time-->. Assuming the message traveled at [[Speed of light|the speed of light]], ''c'', the distance traveled is {{nobreak|1=(''t&#771;'' - ''b'' - ''s_i'') ''c''}}. <!--(''t~_i - b − t_i'')''c''.-->

For n satellites, the equations to satisfy are:

:<math>(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2 = \bigl([ \tilde{t} - b - s_i]c\bigr)^2, \; i=1,2,\dots,n</math>

or in terms of ''pseudoranges'', <math> p_i = \left ( \tilde{t} - s_i \right )c</math>, as

:<math>\sqrt{(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2} + bc = p_i, \;i=1,2,...,n</math> .<ref name=GPS_BASICS_Blewitt>section 4 beginning on page 15 [http://www.nbmg.unr.edu/staff/pdfs/Blewitt%20Basics%20of%20gps.pdf GEOFFREY BLEWITT: BASICS OF THE GPS TECHNIQUE]</ref><ref name=Bancroft>{{cite web|url=http://www.macalester.edu/~halverson/math36/GPS.pdf|archiveurl=http://web.archive.org/web/20110719232148/http://www.macalester.edu/~halverson/math36/GPS.pdf|archivedate=July 19, 2011|title=Global Positioning Systems|format=PDF|accessdate=October 15, 2010}}</ref>

Comparison of these equations with the Equations in R3 section of [[Sphere]] in which <math>(x-x_i)</math> corresponds to <math>(x-x_0)</math>, <math>(y-y_i)</math> corresponds to <math>(y-y_0)</math>, <math>(z-z_i)</math> corresponds to <math>(z-z_0)</math>, and <math>\bigl([ \tilde{t} - b - s_i]c\bigr)</math> corresponds to <math>r</math> shows that these equations are spheres as documented in [[Sphere]].

Since the equations have four unknowns [''x, y, z, b'']&mdash;the three components of GPS receiver position and the clock bias&mdash;signals from at least four satellites are necessary to attempt solving these equations. They can be solved by algebraic or numerical methods. Existence and uniqueness of GPS solutions are discussed by Abell and Chaffee.<ref name="Abel1"/> When ''n'' is greater than 4 this system is overdetermined and a fitting method must be used.

With each combination of satellites, GDOP quantities can be calculated based on the relative sky directions of the satellites used.<ref>{{cite web|url=http://www.colorado.edu/geography/gcraft/notes/gps/gps.html#Gdop|title=Geometric Dilution of Precision (GDOP) and Visibility|first=Peter H.|last=Dana|publisher=University of Colorado at Boulder|accessdate=July 7, 2008}}</ref> The receiver ___location is expressed in a specific coordinate system, such as latitude and longitude using the [[WGS 84]] [[datum (geodesy)|geodetic datum]] or a country-specific system.<ref>{{cite web|url=http://www.colorado.edu/geography/gcraft/notes/gps/gps.html#PosVelTime|title=Receiver Position, Velocity, and Time|author=Peter H. Dana|publisher=University of Colorado at Boulder|accessdate=July 7, 2008}}</ref> [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 18:27, 23 June 2015 (UTC)

{{reflist-talk}}

:This is essentially the exact same argumentation used before, and as before not only do the equations not, in fact, represent spheres, the sources you have cited also do not, in fact, claim that they do. [[User:Siafu|siafu]] ([[User talk:Siafu|talk]]) 22:17, 23 June 2015 (UTC)

State what you are talking about, [[User:Siafu|siafu]], what you say makes no sense. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 05:17, 24 June 2015 (UTC)

:I'll only take direct quotation from reliable sources as a valid argument; explaining using your own words has no value in dispute resolution. The relationship between spheres and synchronized ranges is well sourced; it deserves mention in the article because it's a useful stepping stone for the more realistic relationship between pseudoranges and hyperboloids. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 03:05, 24 June 2015 (UTC)

What you say, [[User:Fgnievinski|Fgnievinski]], is idiotic nonsense. Words are necessary to form a bridge between a document and references. Saying that words cannot be used as a bridge between a document and a reference shows a failure to understand both Wikipedia and GPS. You don't have the competence to decide what will be taken and what will not. I don't believe you even possess a license to practice engineering. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 05:17, 24 June 2015 (UTC)

:[[User:RHB100|RHB100]], you have been shown an incredible amount of forebearance, considering that this "dispute" of you vs. the world has been going on literally ''for years'' without administrative involvement having been previously invoked. Petty nonsense like this, and your comments above, is not going to be tolerated any further-- I am certainly fed up with your constant insults and nonsensical derision. If you want to be taken seriously at wikipedia, you need to familiarize yourself with its policies, starting with [[WP:3RR]] (which if you think I'm in violation of it, you clearly haven't read it), [[WP:NPA]], [[WP:V]], and [[WP:CRED]]. Your supposed credentials, or anyone else's mean nothing here, and frankly we're not impressed, since many of us have more advanced degrees than you claim, and plenty of experience in the field. Furthermore, if all you are going to do is repeat the same things you have said before-- which are you doing now-- you can only expect the exact same response, which is rejection of your proposals. If you continue to insult the competence of your fellow editors, you will be again reported to administrator's noticeboard; do not expect your promise to seek consensus to be sufficient in that case. [[User:Siafu|siafu]] ([[User talk:Siafu|talk]]) 05:27, 24 June 2015 (UTC)

:{{reply|RHB100}} I don't trust ''your'' words; I'd gladly take a published author's words, of course. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 13:58, 24 June 2015 (UTC)

Siafu, if you want to be taken seriously on Global Positioning Systems, you need to go back and review the equations for a sphere in Analytic Geometry or elsewhere. Your comments indicate that you do not understand the equations of a sphere. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 17:13, 24 June 2015 (UTC)

[[User:Fgnievinski|Fgnievinski]], state the specific words you do not trust without regard of who wrote the words. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 17:13, 24 June 2015 (UTC)

:"who wrote the words" is essential to [[WP:V]] and [[WP:RS]], which are central policy on WP. [[User:Kendall-K1|Kendall-K1]] ([[User talk:Kendall-K1|talk]]) 18:07, 24 June 2015 (UTC)

[[User:Siafu|siafu]], I find your comment that the equations above do not represent spheres completely ridiculous. Do you actually believe that? Even more mind boggling is that you seem to be saying that the equation in the Wikipedia article [[Sphere]]s does not represent a sphere. It is absolutely mind boggling that you would make such a statement. Haven't you studied Analytic Geometry and Calculus? [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 17:44, 24 June 2015 (UTC)

:I do not trust any of your words, since what is obvious to you is erroneous to me. So unless you can find someone else saying what you say, there's no point in further discussions. (And here's some explanation: You started well in comparing the pseudorange equation to [[Sphere#Equations in three-dimensional space]], but then made an error in concluding that, because "<math>\bigl([ \tilde{t} - b - s_i]c\bigr)</math> corresponds to <math>r</math>", this correspondence "shows that these [pseudorange] equations are spheres". The spherical radius is a geometrical quantity, as you'd measure with a ruler; there's no allowance for lack of synchronization in the equation for a sphere.) [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 18:16, 24 June 2015 (UTC)

Well much of the entire Problem description section as well as several other sections of the article and the Error Analysis of GPS are my words. So if you rely on very much of anything in the GPS article, you are trusting my words. <math>\bigl([ \tilde{t} - b - s_i]c\bigr)</math> is not a pseudorange. It is the computed distance from satellite i to the receiver. <math> p_i = \left ( \tilde{t} - s_i \right )c</math> is the pseudorange. The equation,

:<math>(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2 = \bigl([ \tilde{t} - b - s_i]c\bigr)^2, \; i=1,2,\dots,n</math> are not pseudorange equations. They are equations equating the sum of the squares of the three components of the distance from receiver to satellite i for i=1,2,3,4, ... to the square of the computed, i.e. approximately correct total distance from receiver to satellite i. [[User:Fgnievinski|Fgnievinski]], you are not using clear and precise language. You are calling equations using the computed, , i.e. approximately correct total distance from receiver to satellite i, pseudorange equations. Your failure to think in terms of clear and precise mathematics is misleading you. I urge you to try and think this through again spending considerably more time on clear and precise mathematical thinking. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 19:09, 24 June 2015 (UTC)

[[User:Siafu|siafu]], let's talk a little more on impersonal issues such as mathematics. Let's forget about things like my degrees and me against the world. Let's try to find out why our understanding of GPS differs. The problem may be due to not using sufficiently clear and precise language. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 22:43, 24 June 2015 (UTC)

===Advice to Go Forward===

Everyone who has been edit-warring should read [[WP:DR|the dispute resolution policy]], [[WP:CIVIL|the civility policy]], and [[WP:NPA|the policy against personal attacks]]. Then follow one of the procedures described in [[WP:DR|the dispute resolution policy]]. The article is currently locked so as to prevent further edit-warring. Decide which of the content dispute resolution procedures to follow while the article is locked. Any more civility violations (and I already see several) are likely to result in blocks. Comment on content, not on contributors. Nobody "wins" if the dispute is taken to [[WP:ANI]], only losers (blocked editors) and non-winners. Some content dispute resolutions can be win-win. [[User:Robert McClenon|Robert McClenon]] ([[User talk:Robert McClenon|talk]]) 18:29, 24 June 2015 (UTC)

:[[User:Robert McClenon|Robert McClenon]], I will read that which you recommend. I think we are beginning to make progress. I now see that the problem is imprecise mathematical thinking. This is difficult to communicate but I am trying. For the first time, I have gotten a meaningful though incorrect response to my disagreement. I have been angered by the robot like response I received earlier and I expressed my irritation at the robot like responder. But now we may be able to get a resolution. Because of the difficulty of communicating mathematically with those lacking an adequate mathematical background, I am not sure we can come to a meeting of the minds but it is worth trying. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 19:21, 24 June 2015 (UTC)

::In view of the mathematical complexity of the subject, if the editors remain civil, one possible option would be [[WP:RFM|formal mediation]] by a member of the [[WP:MEDCOM|Mediation Committee]]. [[User:Robert McClenon|Robert McClenon]] ([[User talk:Robert McClenon|talk]]) 19:28, 24 June 2015 (UTC)

[[User:Robert McClenon|Robert McClenon]], This sounds like a good idea. First, I would like to discuss the issues a little more and see if we can make progress. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 22:39, 24 June 2015 (UTC)

=== Pseudorange forms not a sphere but the region between two spheres? ===

{{reply|RHB100}} How about this interpretation as a middle ground (assuming it can be externally sourced): while synchronized ranges describe circles on a plane and spheres in space (only their boundaries, i.e., not a [[Disk (mathematics)|disk]]/[[Ball (mathematics)|ball]]), pseudoranges can be interpreted geometrically as specifying a [[locus (mathematics)|locus]] (i.e., describing the positions lying inside a region) given by a ring ([[Annulus (mathematics)|annulus]]) on a plane and a [[spherical shell]] (non-infinitesimal or "thick") in space. In other words, the spherical radius (a constant) is replaced for a radial [[Interval (mathematics)|interval]], given by the geometrical distance +/- the receiver clock bias (in units of length, i.e., multiplied by the speed of light). [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 16:12, 25 June 2015 (UTC)

:A pseudorange is just a distance, not a volume. So the above interpretation seems invalid. A better stepping stone might be to state that the basic equation for a satellite as given above describes a sphere (=spherical surface) for each fixed value of the (unknown) clock bias ''b''. If ''b'' is equal to the pseudorange (divided by ''c'') for the satellite, the radius of the sphere is 0. When ''b'' is seen as an independent variable, the equation describes a sheaf of spheres, all centered on te satellite (more technically called a spherical cone). When ''b'' reaches the factual bias of the receiver clock, the sphere passes through the receiver. For further values of ''b'', the receiver lies inside the sphere. Combining equations for 4 satellites, there will be a value for ''b'', where all 4 spheres will have a common point, which will be the ___location of the receiver (if no other errors are present). &minus;[[User:Woodstone|Woodstone]] ([[User talk:Woodstone|talk]]) 17:28, 25 June 2015 (UTC)

:*{{reply|Woodstone}} Do you accept that a distance may be taken as a [[radius]] defining a spherical surface? Then it's not a far thought to consider how the volume bounded by two concentric spheres (a [[spherical shell]]) might follow from the lower and upper limits of a radial interval. And let's keep the discussion restricted to the geometrical [[three-dimensional space]] -- no 4D [[space-time]], in which the [[spherical cone]] interpretation applies, please. Then would you agree that, in 3D space, as value of the clock bias sweeps from zero to its true value, the concentric spherical surface shrinks or grows? Finally, you might want to check your definitions: pseudorange equals range (geometrical distance) plus receiver clock bias/error: p = d + b; you seem to be suggesting that p = b, which is mistaken -- the clock bias excludes the bulk of the [[travel time]], and the spheres never collapse into a point, unless the receiver is sitting a the GPS satellite. To restate the gist of it: radii are to ranges as spherical shells are to pseudoranges. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 18:55, 25 June 2015 (UTC)

::*Also, your point is already well sourced and described in section [[Global Positioning System#Spherical cones]], isn't? [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 19:31, 25 June 2015 (UTC)

::You are confused by the word pseudorange, which although containing the word range is not in this case a numberrange. It is a single number per satellite (the measured distance without clock correction). The equation has (radius of sphere around satellite) + ''bc'' = pseudorange, so for the value of ''b'' where bc=p, the sphere has radius 0. The ''b'' is a free variable, not the solution of the equations. I am here just trying to explain the situation in terms of spheres without saying that the equations describe spheres, which would be incorrect. &minus;[[User:Woodstone|Woodstone]] ([[User talk:Woodstone|talk]]) 21:30, 25 June 2015 (UTC)

::: {{reply|Woodstone}} 0) Let's try and discuss each issue separately.

::: 1) Above you called ''b'' the clock bias, so your suggestion that ''p'' = ''c'' ''b'' might be reasonably achieved reflects a misunderstanding of the theory. Let me source it:

:::* "Note that pseudorange is almost like range, except that it includes clock errors because the receiver clocks are far from perfect." [http://www.nbmg.unr.edu/staff/pdfs/Blewitt%20Basics%20of%20gps.pdf]

:::* "The pseudorange would equal the geometric distance from the satellite ... to the receiver if the propagation medium were a vacuum and if there were no clock errors..." [https://books.google.com.br/books?id=4qE6xYjYSHgC&lpg=PA173&ots=fr8Iz1G4RG&dq=gps%20pseudorange%20equals%20range%20plus%20receiver%20clock&pg=PA173#v=onepage&q=gps%20pseudorange%20equals%20range%20plus%20receiver%20clock&f=false]

::: So ''p'' = ''d'' + ''b'' ''c'', i.e., pseudorange equals range plus clock bias (the latter in units of length). Let me know if there remains any outstanding issue in the definition of pseudorange (considering the simplifications in the idealized scenario of vacuum propagation, no relativistic effects, and no other error sources.). [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 19:07, 26 June 2015 (UTC)

:::::You have it backwards. The pseudorange is the measured result, being the clock value in the receiver at reception minus the time stamp on the received message (times c). In the equation we have unknowns (x, y, z, b), that will eventually be resolved into the position and clock bias. The ''b'' is no more the actual clock bias than x is the actual first coordinate of the receiver. They are free variables, that only take specific value after solving a set of equations.&minus;[[User:Woodstone|Woodstone]] ([[User talk:Woodstone|talk]]) 10:19, 27 June 2015 (UTC)

::::::{{reply|Woodstone}} Receiver clock '''bias''' or '''error''' ''b'' is not to be confused with the receiver clock state or the reception time; more correctly, the former is only the deviation in the latter from the true reception time; this is the standard nomenclature as per cited sources above. Inventing your own nomenclature is not the way to go. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 03:20, 28 June 2015 (UTC)

::::::Some basic concepts for your understanding: measured pseudoranges ''p'' are modeled to first order as the sum of a geometric range or distance ''d'' and a receiver clock bias ''b'' ''c'' (times speed of light, ''c''); in its turn, the range ''d'' can be modeled as the signal travel time ''T'' / ''c'' (elapsed from transmission to reception -- based on idealized synchronized clocks -- divided by speed of light, ''c'') or, equivalently, as the norm of the vector difference between satellite and receiver positions. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 03:20, 28 June 2015 (UTC)

:::::::My statements above are completely consistent with your explanation here. However we are not discussing the abstract model, but the way to represent, interpret and solve the equations in a concrete case. The only known data are the timestamps on the messages, the reception times according to the receiver's clock and the orbital data. From this the positions of the satellites and the pseudoranges can be computed (ignoring for argument's sake any relativistic effects, inhomogeneous media, movement of the receiver and technical errors). The true reception times are unknown (effectively only the differences between the reception times are known). So what we do is postulate a position (''x, y, z'') and clock bias ''b'' and setup equations coupling these unknown quantities with the known data. Finally we solve the equations to know the actual position and clock bias (and thus the true clock and true ranges). &minus;[[User:Woodstone|Woodstone]] ([[User talk:Woodstone|talk]]) 10:17, 28 June 2015 (UTC)

::: 2) Your distinction between "trying to explain the situation in terms of spheres" on the one hand and, on the other hand, "saying that the equations describe spheres" is unyielding; of course we're not trying to prescribe causality, only to fit idealized models (sphres) to reality (GPS measurements) -- we're just trying to find a proper and useful analogy. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 19:07, 26 June 2015 (UTC)

:::::As stated many times each equation does not describe a sphere, because ''b'' is not a constant, but an additional variable. However for intuitive inderstanding of the situation it may be useful to consider the effect of holding ''b'' at a certain value.&minus;[[User:Woodstone|Woodstone]] ([[User talk:Woodstone|talk]]) 10:19, 27 June 2015 (UTC)

::::::I have shown many sources advancing the spherical surface interpretation for synchronized ranges; which one are your disputing? [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 03:20, 28 June 2015 (UTC)

:::::::The situation of synchronized clocks corresponds to a fixed value of ''b''=0, which I state above leaves the equation representing a sphere. No dispute here. &minus;[[User:Woodstone|Woodstone]] ([[User talk:Woodstone|talk]]) 10:17, 28 June 2015 (UTC)

::: 3) A pseudorange ''p'' is a scalar value, of course, and as such it cannot fully specify the lower and upper limits of an [[Interval (mathematics)|interval]] (your "numberrange", I assume). The radial interval that I introduced would result from correcting pseudoranges ''p'' of it's clock bias, i.e., ''p'' - ''b'' ''c''. Let's consider the following toy exercise. I'd give you two pseudorange measurement values and you'd try to locate the receiver on a planar board, by drawing circles centered at satellites located at known positions. If clock bias is uncorrected for, you'd draw the circles too big or too small (depending on the algebraic sign of ''b'') than it ought to be. If later I reveal the clock bias value, you could subtract it from each pseudorange measurement, and finally draw the circles with the correct radius. The volume between biased and unbiased circles (spheres) is the annulus (shell) that I introduced above. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 19:07, 26 June 2015 (UTC)

:::::I really don't see how this could help in understanding the picture. It complicates instead of simplifying. &minus;[[User:Woodstone|Woodstone]] ([[User talk:Woodstone|talk]]) 10:19, 27 June 2015 (UTC)

::::::Well, published authors seem to disagree with you, as per, e.g., Fig.2 in Langley (1991) [http://gauss.gge.unb.ca/gpsworld/EarlyInnovationColumns/Innov.1991.07-08.pdf].

<ref name=Langley>Richard Langley, The Mathematics of GPS, [http://gauss.gge.unb.ca/gpsworld/EarlyInnovationColumns/Innov.1991.07-08.pdf], 1991</ref>[[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 03:20, 28 June 2015 (UTC)

:::::::I'm not saying it's wrong, it's just not helpful. &minus;[[User:Woodstone|Woodstone]] ([[User talk:Woodstone|talk]]) 10:17, 28 June 2015 (UTC)

::::{{reply|Fgnievinski}}A previous objection that is worth noting is that implying that these equations describe spheres also implies that the surfaces are in some fashion isotropic, which they certainly are not. Among other things, the value of c, treated as a scalar, is dependent on the medium, and moreover the effects of the media being traversed means that we cannot even say that the pseudorange represents an expanding front of constant phase. [[User:Siafu|siafu]] ([[User talk:Siafu|talk]]) 09:15, 27 June 2015 (UTC)

:::::{{reply|Siafu}} Let's assume vacuum propagation, otherwise we won't get anywhere, shall we? [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 03:20, 28 June 2015 (UTC)

Well at least we are discussing the issues. That seems to indicate some progress. I agree with [[User:Woodstone|Woodstone]] that a pseudorange is just a distance and that the basic equations given above describe spheres (=spherical surfaces) for each fixed value of the (unknown) clock bias b. I agree with [[User:Fgnievinski|Fgnievinski]] that the discussion should be restricted to the geometrical [[three-dimensional space]] -- no 4D [[space-time]]. I think that we should recognize b as the clock bias. And that there is one and only one clock bias. Therefore we should talk about one and only one value of b. There is no reason to drag the reader through what may be envisioned as happening in an iterative procedure with b changing. The pseudorange, <math>p_i</math> is the measured quantity, equal to <math> \left ( \tilde{t} - s_i \right )c</math> where

<math> \left ( \tilde{t}\right)</math> denotes the indicated time of reception, <math> \left ( s_i \right )</math> denotes the time of transmission and <math>c</math> denotes the speed of light.. But the pseudorange typically contains large errors because of the large velocity of light along with the fact that there are inaccuracies in the receiver clock. The corrections to the pseudoranges are given by :<math>\bigl( - b c\bigr), \; i=1,2,\dots,n</math> where b is the clock bias at least to a highly accurate approximtion. The distances from receiver to satellites (at least to an accurate approximation are

<math>\bigl( \left( \tilde{t} - s_i \right ) c - bc\bigr), \; i=1,2,\dots,n</math> . The squares of these distances are equal to the sum of the squares of the three components of the distances from receiver to the satellites. Thus these equations appear to be describing the surfaces of spheres as pointed out by [[User:Woodstone|Woodstone]]. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 17:26, 26 June 2015 (UTC)

:As the article states clearly, the equations to be solved are:

::find (x, y, z, b) such that r_i (x, y, z) + bc = p_i for i=1,...,n

:The clock bias is unknown and needs to be solved for together with its geo-coordinates. There are four unknows to be found, in a 4-dimensional space. With 4 satellites there is a unique solution, with more than 4 there would be one in an ideal world, without any disturbances (such as retardation in the atmosphere). In practice there will be no solution, so for more than 4 the set of equations is transformed into a least squares form, still solving for 4 unknowns x, y, z, b. &minus;[[User:Woodstone|Woodstone]] ([[User talk:Woodstone|talk]]) 07:06, 26 June 2015 (UTC)

::That's one of the most concise, understandable explanations I've heard yet. I would say that first, before launching in to the math. [[User:Kendall-K1|Kendall-K1]] ([[User talk:Kendall-K1|talk]]) 10:34, 26 June 2015 (UTC)

:::{{reply|Kendall-K1}} Let's neglect errors sources, other than clock bias, such that there's always a unique solution (except for degenerate cases, such as two coinciding satellites), so that least squares is put out of scope, shall we? [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 19:07, 26 June 2015 (UTC)

:{{reply|RHB100}} (i) agreed on 4D outside the scope; (ii) agreed that ''b'', clock bias, is unique (per epoch), meaning we're neglecting satellite clock bias and considering only receiver clock bias; (iii) agreed that pseudorange is equivalent to time elapsed ([[time of flight]]), as measured by a biased receiver clock and an unbiased satellite clock; (iv) disagreed that we need to consider "large [variations in] velocity [sic, speed] of light" -- let's consider idealized scenario only; (v) agreed that the range resulting from a pseudorange corrected for the clock bias can be interpreted as a radius, thus specifying a spherical surface; (vi) disagreed on not considering ''b'' variations: although ''b'' evaluates to a single constant, that value is unknown beforehand; so the whole point of the exercise is how to adapt the spherical geometry -- strictly valid only for ranges -- to pseudoranges. I proposed the interpretation that spheres may be enlarged or shrank uniformly (i.e., by the same radial change amount) until their intersection yields a valid receiver position solution (and the unknown clock bias). I'd like to hear from you whether or not you accept that pseudorange, when uncorrected for clock bias, cannot strictly be interpreted geometrically as a [[spherical surface]], as there's one too many free variable. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 19:07, 26 June 2015 (UTC)

::Yes, this is a concise statement of the problem and its solution which I agree with. This is useful on the talk page where we usually have people who understand the concise mathematical notation. Probably in the article we should use the less concise notation since many readers may be unfamiliar with concise mathematical terminology. It seems that we may be coming to an agreement with regard to our understanding of the mathematics of GPS. I don't want to get involved in another edit war and I realize that there may be some hurt feelings and left over hostility from the last edit war. However, it seems that now is the time to ask the question, should we envision this problem geometrically as being to find the near intersection of the surfaces of four or more spheres? Even if there are more than four spheres the solution should be the intersection or near intersection of all of these spherical surfaces. This geometric interpretation I think would enhance understanding of GPS. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 19:08, 26 June 2015 (UTC)

::: The problem is: even if we all agree on a new interpretation, we cannot mention it in the article page, unless it can be externally sourced explicitly to a published work: [[WP:VERIFIABILITY]]. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 19:28, 26 June 2015 (UTC)

:::: Well we do have it sourced within Wikipedia as I have shown above. Furthermore we do not have to source things like 2+2=4. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 19:59, 26 June 2015 (UTC)

::::: {{reply|RHB100}} Would you please familiarize yourself with [[WP:CIRCULAR]]. Also, do you realize that above you proved yourself wrong when you agreed that pseudoranges cannot be interpreted as spherical surfaces (unless they are corrected for the clock bias, i.e., unless pseudoranges are actually synchronized ranges, or pure geometrical distances)? [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 20:36, 26 June 2015 (UTC)

:::: Fgnievinski|talk]]), another source is [http://web.archive.org/web/20050316051910/http://www.siam.org/siamnews/general/gps.htm The Mathematics of GPS]. This article clearly states that they are spheres around satelllites. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 20:19, 26 June 2015 (UTC)

::::: {{reply|RHB100}} That article (which is cited in the [[GPS]] article) states: "In principle, three '''distance''' measurements should be enough. They specify spheres around three satellites, and the receiver lies at the point of intersection." This is already covered in section [[Global Positioning System#Spheres]]. Later it says: "The problem is that the receiver clock is not perfectly in sync with the satellite clock. This would cause a major error — the uncorrected reading is called a '''pseudorange'''."; nowhere those authors ascribe spherical surfaces to clock-biased pseudoranges -- only to synchronized ranges or geometrical distances. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 20:33, 26 June 2015 (UTC)

:::: Here is another source [http://gis.stackexchange.com/questions/12866/why-does-gps-positioning-require-four-satellites| Geographic Information Systems]

::::: {{reply|RHB100}} Now please read [[WP:NOTRELIABLE]]. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 20:38, 26 June 2015 (UTC)

:::: And here is another very good source at [http://gauss.gge.unb.ca/gpsworld/EarlyInnovationColumns/Innov.1991.07-08.pdf The Mathematics of GPS by Richard Langley] <ref name=Langley>Richard Langley, The Mathematics of GPS, [http://gauss.gge.unb.ca/gpsworld/EarlyInnovationColumns/Innov.1991.07-08.pdf], 1991</ref>

::::: {{reply|RHB100}} Indeed another excellent source, that I recommend everyone to read; it further documents the distinction between range and pseudorange, and how the spherical surface interpretation strictly only applies to the former:

:::::* "With synchronized clocks, simultaneous range measurements to three GPS satellites produce a determination of a receiver's position. Each '''range''' measurement can be portrayed as the radius of a sphere centered on a particular satellite"

:::::* "When we started our analysis, we assumed that the clock in the GPS receiver was synchronized with the clocks in the satellites. This assumption, however, is fallacious. (...) The ranges measurements [that] the receiver makes are biased by the receiver and satellite clock errors and therefore are referred to as '''''pseudoranges'''''. (...) Because of this error, the three spheres with radii equal to the measured pseudoranges ... will not intersect at a common point. However, if the receiver clock error, dT, can be determined, then the pseudoranges can be corrected and the position of the receiver determined. The situation, compressed into two dimensions, is illustrated in Figure 2."

::::: May I say that his Fig. 2 seems to show exactly the annular regions that I proposed above. Yet the author doesn't draw that conclusion explicitely, and we cannot put words in his mouth ([[WP:SYNTHESIS]]). [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 00:13, 27 June 2015 (UTC)

[[User:Fgnievinski|Fgnievinski]] The statements you make in [[Global Positioning System#Spheres]] is not supported by the article. You have written something that you have made up which is nothing more than editorializing on your part. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 00:32, 27 June 2015 (UTC)

[[User:Fgnievinski|Fgnievinski]], your writing is confusing. You say, "Indeed another excellent source, that I recommend everyone to read; it further documents the distinction between range and pseudorange, and how the spherical surface interpretation strictly only applies to the former". What I have been saying all along is that the equations with the range to the target which is at least approximately, bc+pseudorange, on the right side are equations of sphees. But you seem to be arguing against your own words. Your writing is so confusing, it is extremely difficult to understand what you are talking about. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 00:54, 27 June 2015 (UTC)

[[User:Fgnievinski|Fgnievinski]], you say "When we started our analysis, we assumed that the clock in the GPS receiver was synchronized with the clocks in the satellites. This assumption, however, is fallacious." Now what are you talking about. I never made any such assumption. I assumed that the solution for the value of the clock bias, b, was a highly accurate approximation to the true clock bias. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 01:01, 27 June 2015 (UTC)

: {{reply|RHB100}} Those were direct quotations from the article you provided a link (that's what double quotes are used for) -- I'm surprised you forgot the content of the article that you just read. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 01:36, 27 June 2015 (UTC)

::[[User:Fgnievinski|Fgnievinski]], You're still making vague and ambiguous statements. You say, "Those were direct quotations from the article you provided a link." What do you mean by "Those" I don't have the vaguest idea what you are talking about. What article that I made reference, are you talking about. You're so vague, I can't tell what you are talking about. I don't recall making reference to any article which said, "When we started our analysis." [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 02:11, 27 June 2015 (UTC)

:::Maybe this will help. [//en.wikipedia.org/w/index.php?title=Talk:Global_Positioning_System&diff=prev&oldid=668817171]. [[User:Burninthruthesky|Burninthruthesky]] ([[User talk:Burninthruthesky|talk]]) 07:29, 27 June 2015 (UTC)

::[[User:Fgnievinski|Fgnievinski]], you say, "Also, do you realize that above you proved yourself wrong when you agreed that pseudoranges cannot be interpreted as spherical surfaces (unless they are corrected for the clock bias." Again I don't know what you are talking about. Where did I make such an agreement. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 02:11, 27 June 2015 (UTC)

:::Here [//en.wikipedia.org/w/index.php?title=Talk:Global_Positioning_System&diff=668800910&oldid=668800819]. [[User:Burninthruthesky|Burninthruthesky]] ([[User talk:Burninthruthesky|talk]]) 07:29, 27 June 2015 (UTC)

::::When I made that statement at 19:08, 26 June 2015 (UTC), I had not even seen the statement of [[User:Fgnievinski|Fgnievinski]] at 19:07, 26 June 2015 (UTC). When I said, "Yes, this is a concise statement of the problem and its solution which I agree with", I was agreeing with the statement of [[User:Woodstone|Woodstone]] at 07:06, 26 June 2015 (UTC). I was not expressing agreement with [[User:Fgnievinski|Fgnievinski]]. The 19:07, 26 June 2015 (UTC) of [[User:Fgnievinski|Fgnievinski]] got put on the talk page after I got started writing but before I finished. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 00:12, 28 June 2015 (UTC)

:::::{{reply|RHB100}} OK, so could you please take the time and respond to each issue raised above: (i), (ii), (iii), (iv), (v), (vi). Otherwise we don't know exactly what you are agreeing and disagreeing about. Thanks. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 03:20, 28 June 2015 (UTC)

{{outdent}}

Continues at [[#Simple calculation precedure]]. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 23:10, 28 June 2015 (UTC)

=== Simple calculation precedure ===

A simplified produre of calculation&mdash;ignoring for argument's sake any relativistic effects, inhomogeneous media, movement of the receiver and technical errors&mdash;is represented as follows:

{|table class="wikitable"

| ''s_i'' || the timestamp on message from satellite ''i''

|-

| ''t_i'' || the time of onboard receiver clock at reception of message from satellite ''i''

|-

| ''p_i'' = c &middot; (''t_i'' - ''s_i'') || the pseudorange for satellite ''i''

|-

| (x_i, y_i, z_i) || the computed position of satellite ''i'' at sending the message

|-

| ''r_i'' (''x, y, z'') = sqrt((''x'' - ''x_i'')² + (y - y_i)² + (z - z_i)²) || distance at point (''x, y, z'') from satellite ''i''

|-

| ''p_i'' - ''w'' || the distance between receiver and satellite ''i'' for synchronization error ''w''

|-

| ''r_i'' (''x, y, z'') + ''w'' = ''p_i'' || the equation for satellite ''i''

|-

| (''x_r, y_r, z_r, w_r'') || the solution of a set of these equations with ''i'' = 1, ..., ''n'', n >= 4

|-

| (''x_r, y_r, z_r'') || position of receiver

|-

| ''w_r'' / c || the clock synchronization error

|}

Here all elements are present and can be discussed.

&minus;[[User:Woodstone|Woodstone]] ([[User talk:Woodstone|talk]]) 11:59, 28 June 2015 (UTC)

: Thanks to {{u|Woodstone}} for the initiative. The only minor wording issue that I'd raise is with regard to ''s_i'', which I'd call simply the transmission time, for the following reasons. First, to avoid confusion with the satellite broadcast navigation message contents. Secondly, to avoid suggesting that it's the satellite who tells what was the transmission time, which is actually derived by the receiver (based on a comparison between, on the one hand, the pseudorange code sequence impressed on the carrier wave by the satellite, and on the other hand, a code replica synthesized internally by the receiver) -- the satellite couldn't possibly know at what time each different receiver is going to receive its continuously broadcast signal. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 00:03, 29 June 2015 (UTC)

I don't see the need for this new symbol, w. Why don't we just continue to call it bc for consistency with what we have used in the past? [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 22:00, 28 June 2015 (UTC)

: I agree with {{u|RHB100}} on preferring ''bc'' for clock bias (in meters). [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 00:03, 29 June 2015 (UTC)

:: (Woodstone:) The letter ''b'' has a stong flavor of a constant and has lead to confusion; using ''w'' resolves this issue and makes a nice set (''w, x, y, z'') of unknowns to be solved for.

::: I won't fuss over ''w=bc''. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 01:52, 30 June 2015 (UTC)

(i) RHB100 agreed on 4D outside the scope;{{anchor|4D}}

: OK. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 00:03, 29 June 2015 (UTC)

:: (Woodstone:) The equations solved are fundamentally 4D, having 4 unknowns.

::: Yes, true, and that perspective is already well described in [[GPS#Spherical cones]]; let's narrow the dispute to [[GPS#Spheres]]. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 01:52, 30 June 2015 (UTC)

(ii) RHB100 agreed that ''b'', clock bias, is unique (per epoch), meaning we're neglecting inconsistencies between satellite clocks.

: OK. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 00:03, 29 June 2015 (UTC)

:: (Woodstone:) ok, we should collect all ignored effects somewhere.

::: Let's leave that for a future discussion of [[GNSS positioning calculation]]. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 01:52, 30 June 2015 (UTC)

(iii) RHB100 agreed that pseudorange is equivalent to measured time elapsed (time of flight)*(speed of flight.

: OK. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 00:03, 29 June 2015 (UTC)

:: (Woodstone:) with caveat that is not the real time of flight, but the time as measured initially.

::: I think {{u|Woodstone}}'s caveat means that pseudorange is not just the time of flight as it would be measured with synchronized clocks, i.e., it's actually the time of flight as measured with unsynchronized clocks. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 01:52, 30 June 2015 (UTC)

(iv) RHB100 takes no position on this since its not too clear what is being talked about - Fgnievinski statement that we need to consider "large [variations in] velocity [sic, speed] of light" -- let's consider idealized scenario only;

: You said "But the pseudorange typically contains large errors because of the large velocity of light", to which I replied: "disagreed that we need to consider "large [variations in] velocity [sic, speed] of light" -- let's consider idealized scenario only". Do you agree on considering only vacuum propagation? [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 00:03, 29 June 2015 (UTC)

:: (Woodstone:) agree to ignore a stated list of effects, including variation in speed of light in atmosphere.

(v) RHB100 says any distance can be interpreted as a radius and that a radius and a positon for the center uniquely determine a sphere - Fgnievinski statement that the range resulting from a pseudorange corrected for the clock bias can be interpreted as a radius, thus specifying a spherical surface;

: OK. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 00:03, 29 June 2015 (UTC)

:: (Woodstone:) True for a fixed ''b'', not true for the process of solving the set of equations.

(vi) RHB100 takes no position on this since the meaning of b variations is unclear. b does not vary in analytical method such as the Bancroft method. b can be envisioned as varying in an iterative method but we shouldn't confuse the Problem statement section by talking about the details of the computational procedure - Fgnievinski statement not considering b variations: although b evaluates to a single constant

[[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 22:36, 28 June 2015 (UTC)

: Let's further discuss this point (vi) at [[#rescue]] below. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 00:03, 29 June 2015 (UTC)

:: (Woodstone:) ''b'' is a variable in the equations, regardless how they are solved.

OK, so given the table above, it seems that we all agree on the following. In the case of ''b=0'' (i.e., perfect or synchronized clocks), the spherical surface interpretation is valid. Furthermore, in this situation there also applies the solution principle stating that the receiver position can be found at the intersection of three such spheres. This idealized scenario is an explanation worth of mention in the article. Agreed? [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 00:03, 29 June 2015 (UTC)

:(Woodstone:) Agreed; but as has been noted before "spherical surface" is a pleonasm. A "sphere" in mathematics is always a surface. The filled object is mathematically called a "ball".

:: That's mathematicians' jargon, likely confusing for the the general audience -- the primary meaning of [[ball]] is not [[ball (mathematics)]]. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 01:52, 30 June 2015 (UTC)

:::(Woodstone:) Nevertheless, we should perhaps mention only once that by "sphere" we mean "spherical surface" throughout.

::::Agreed. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 05:19, 2 July 2015 (UTC)

Now for the parts where we likely disagree: the case of pseudoranges, that are affected by receiver clock synchronization error or bias -- which is held fixed to a constant though unknown value. The spherical surface interpretation remains applicable -- one can certainly envision spheres centered at satellites, with radii equal to the respective pseudoranges. BUT, the spherical intersection position solution principle is no longer valid for pseudoranges. Little is known and published about the number of resulting intersections (e.g., if it's zero, one, or more), and whether or not the true receiver position lies in the vicinity of those spurious pseudorange sphere intersections. (In fact, Fig.2 in [http://gauss.gge.unb.ca/gpsworld/EarlyInnovationColumns/Innov.1991.07-08.pdf Langley (1991)] illustrates a counter-example: the true receiver position is located outside the [[convex hull]] of pseudorange spherical intersections -- in the figure, the point is outside the triangle.) Therefore, I think it's problematic to state that one can "envision this problem geometrically as being to find the near intersection of the surfaces of four spheres", as defended extensively by RHB100. I also think this wording would confuse the reader into misinterpreting the approximate intersection as some sort of least squares solution, which it is not: with four satellites, the solution is exact. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 00:03, 29 June 2015 (UTC)

:(Woodstone:) This section may be key to mutual understanding. In the table above, the pseudorange is strictly a constant, derived from measurement. It does not depend on ''b'' as in the equations. The way it is described just above here, it looks like you are talking about a pseudorange as a function of ''b'':

::''p_i''(''b'') = ''t_i'' - ''bc'' - ''s_i'', and solve for

:: ''r_i'' (''x, y, z'') = ''p_i(b)''

:which is not invalid, but unusual and still does not support viewing spheres, because the seeming radius is not a constant.

::Agreed the equations above are wrong. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 01:52, 30 June 2015 (UTC)

:(Woodstone:) The second part of the section I do not understand. Four spheres rarely have an intersection at all. Three spheres have at most two intersection points. It would be mere coincidence if a fourth passes through either. That's what the variable ''b'' is for. Only for one value of ''b'' is there a common intersection. &minus;[[User:Woodstone|Woodstone]] ([[User talk:Woodstone|talk]]) 06:39, 29 June 2015 (UTC)

:: Agreed that speaking about near intersection of pseudorange-based spheres is problematic and to be avoided; by the way, what did you think of Fig.2 of Langley (1991)? [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 01:52, 30 June 2015 (UTC)

{{anchor|rescue}} Finally, can we salvage a valid spherical geometry interpretation for pseudoranges (not ranges)? Is the proposal based on enlarging/shrinking pseudorange spheres into geometrical/synchronized ranges useful? [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 00:03, 29 June 2015 (UTC)

I don't think we should talk about the intersection of spheres which have pseudoranges as their radii. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 18:47, 29 June 2015 (UTC)

: Agreed. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 01:52, 30 June 2015 (UTC)

::(Woodstone:) Indeed.

We should only talk about the intersection of spheres which have ''t_ic'' - ''bc'' - ''s_ic'' as their radii. Fig.2 in [http://gauss.gge.unb.ca/gpsworld/EarlyInnovationColumns/Innov.1991.07-08.pdf Langley (1991)] shows an example in which the spheres with radii equal to the pseudoranges do not intersect at the solution as expected, but the spheres in which the radii are corrected for clock bias do intersect at the solution. The equations in the problem description section are completely consistent with that. These equations,

<math>(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2 = \bigl([ \tilde{t} - b - s_i]c\bigr)^2, \; i=1,2,\dots,n</math>

or in terms of ''pseudoranges'', <math> p_i = \left ( \tilde{t} - s_i \right )c</math>, as

:<math>\sqrt{(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2} + bc = p_i, \;i=1,2,...,n</math> These equations have the correction terms for clock bias, <math> - bc, </math> applied to the measured pseudoranges,

<math> p_i = \left ( \tilde{t} - s_i \right )c</math>

([[User talk:RHB100|talk]]) 18:47, 29 June 2015 (UTC)

: The difficulty is that ''b'' is unknown ''a priori'', so this interpretation doesn't help in finding a position solution given pseudorange measurements -- you can't, e.g., solve first for the time component ''b'' alone then use it to correct ''p-bc'' to finally apply the spherical intersection algorithm to obtain the remanining geometrical component; it's a simultaneous space-time solution. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 01:52, 30 June 2015 (UTC)

: The only statement that I'd find acceptable for inclusion in the article is something like: "''a posteriori'', the position solution previously obtained can be envisioned as lying at the intersection of satellite-centered spheres having radii equal to clock-bias-corrected pseudoranges (as in the idealized case of synchronized ranges)." {{u|RHB100}}, would such a statement settle your contention? [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 01:52, 30 June 2015 (UTC)

::(Woodstone:) This defeats the idea of clock bias in the equations. If the actual value of ''b'' is known, we calculate with "(true) ranges", not "pseudoranges".

:::Discussion continues [[#true&trivial|below]]. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 05:19, 2 July 2015 (UTC)

Again we should not talk about the computational details in the problem description section. We should just talk about b as the unique clock bias. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 18:47, 29 June 2015 (UTC)

: Agreed. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 01:52, 30 June 2015 (UTC)

::(Woodstone:) This does not make sense. In the equations ''b'' is not a constant, but a variable.

:::{{anchor|terminology}}There's a terminology problem there: let's call ''b'' an "unknown"; some call it a "constant" (which confuses with [[literal value]]), others call it "variable", which gets confused with different values per satellite. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 05:19, 2 July 2015 (UTC)

Here are some statements on which we may be able to agree.

(1)The material in [http://gauss.gge.unb.ca/gpsworld/EarlyInnovationColumns/Innov.1991.07-08.pdf| The Mathematics of GPS BY Richard Langley] down through the section titled, Linearization of the pseudorange equations, provides a correct description of the GPS mathematical problem, its solution, and its geometry. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 18:47, 29 June 2015 (UTC)

: Agreed. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 01:52, 30 June 2015 (UTC)

::(Woodstone:) We are now discussiong the presentations here, not in some book.

::: I'd think it's more likely we reach agreement based on an external quotation than using our own words. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 05:19, 2 July 2015 (UTC)

(2) The GPS mathematical problem is most clearly envisioned as a problem involving four unknowns in three dimensional space. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 18:47, 29 June 2015 (UTC)

: Please keep the discussion about this point restricted to [[#4D]] above. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 01:52, 30 June 2015 (UTC)

::(Woodstone:) Indeed.

(3a) The problem description section should include the equations to be solved and the properties of the solution. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 18:47, 29 June 2015 (UTC)

: The article is already too big; I think the discussion here is starting to overflow into [[GNSS positioning calculation]]. I'd like to finish the discussion about section [[GPS#Geometric interpretation]] first. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 01:52, 30 June 2015 (UTC)

::(Woodstone:) Properties of the solution don't really belong here.

:::We should point out here as in the paper by Richard Langley that the solution is near the intersection of 4 spheres. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 18:16, 30 June 2015 (UTC)

:::: Discussion of this point continues [[#true&trivial|below]]. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 05:19, 2 July 2015 (UTC)

(3b) Details of the solution method should be hidden from the reader in this section with only links or names of the solution method. We should not drag the reader through what may be envisioned as occurring in the solution process such as b, the clock bias, taking on different values in an iterative solution method. This allows the reader to concentrate on the meaning of the equations and the properties of the solution without the distraction of computer algorithms. The mathematics of GPS can be better understood when you think just about that. Then in a section devoted to solution methods if included, the reader can think exclusively about the solution method and better understand that. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 21:23, 29 June 2015 (UTC)

: Agreed. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 01:52, 30 June 2015 (UTC)

::(Woodstone:) The nature of ''b'' has nothing to do with the solution method. It is always a variable in the equations. Whether they are solved analytically or iteratively has no bearing on the concept. &minus;[[User:Woodstone|Woodstone]] ([[User talk:Woodstone|talk]]) 07:28, 30 June 2015 (UTC)

:::Well, Woodstone, what we don't need are these confusing comments that b varies to form light cones or something like that. We should point out that b is the unique clock bias and leave out the comments that b is a variable. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 18:16, 30 June 2015 (UTC)

::::With exactly the same reasoning (and non-validity) you could claim that (''x,y,z'') is the unique position of the receiver. Why do you single out ''b'', but consider ''x'', ''y'', and ''z'' to form a sphere? They are all four fully equivalently unknown in the 4 (or more) equations. &minus;[[User:Woodstone|Woodstone]] ([[User talk:Woodstone|talk]]) 17:07, 1 July 2015 (UTC)

{{anchor|true&trivial}}

Above Fgnievinski said 'The only statement that I'd find acceptable for inclusion in the article is something like: "''a posteriori'', the position solution previously obtained can be envisioned as lying at the intersection of satellite-centered spheres having radii equal to clock-bias-corrected pseudoranges (as in the idealized case of synchronized ranges)." {{u|RHB100}}, would such a statement settle your contention?"

Now that statement appears to be generally correct but it is excessively wordy. The straightforward way of making the statement follows:

The solution of these equations is at the intersection of the surfaces of spheres centered at the satellites with radii equal to the clock-bias-corrected pseudoranges. When we talk about solutions, we don't need to redundant by saying ''a posteriori''. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 18:37, 30 June 2015 (UTC)

:It does not make much sense either way. The pseudorange corrected for the clock bias is the true range. Talking about spheres does not clarify anything. It essentially would say that the receiver is at the "true range" distance from the satellite, which is true, but hardly informative. &minus;[[User:Woodstone|Woodstone]] ([[User talk:Woodstone|talk]]) 17:04, 1 July 2015 (UTC)

::Agreed with {{u|Woodstone}} it's true and trivial; but I'd still like to have it added to the article, at least to avoid recurring and tiring edit wars with {{u|RHB100}}. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 04:55, 2 July 2015 (UTC)

=== Mathematics of GPS ===

[http://gauss.gge.unb.ca/gpsworld/EarlyInnovationColumns/Innov.1991.07-08.pdf Langley (1991)| The Mathematics of GPS by Richard Langley] is a very good paper as we have agreed. Here is a quote from this document:

"The expression under the square root sign is the true range to the satellite. It is actually a representation of the sphere centered on coordinates x,y,z, the position of the satellite." It seems that we should be able to take this quote with notational changes and use it in our Problem description section. We would of course give Langley credit with the appropriate reference. Can anyone with good intentions disagree with that? [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 19:59, 1 July 2015 (UTC)

: Agreed, let's do it and move on with our lives. Would you please craft a sentence for inclusion at the end of section [[GPS#Spheres]]. Thanks. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 05:02, 2 July 2015 (UTC)

Alright, I'll do that. I just noticed this. I'll do it tomorrow. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 05:26, 2 July 2015 (UTC)

:No let's not, because it is incorrect. How can a single expression represent a sphere? In order to describe a sphere an equation is needed. ''r''(''x_r, y_r, z_r'') is the true range. A sphere would be described by {{nobreak|''r''(''x, y, z'') {{=}} ''r''(''x_r, y_r, z_r'')}}. &minus;[[User:Woodstone|Woodstone]] ([[User talk:Woodstone|talk]]) 08:38, 2 July 2015 (UTC)

::''r=p-bc''. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 18:01, 2 July 2015 (UTC)

Well [[User:Woodstone|Woodstone]], I think Richard Langley is better qualified than you. Read his paper. If you still don't understand it then tell us what it is that you can't understand. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 18:45, 2 July 2015 (UTC)

:{{ping|RHB100}} Your response is bordering on ''[[ad hominem]]'' attack; let's not derail this carefully crafted negotiation. [[WP:NPA]], please. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 18:51, 2 July 2015 (UTC)

Below is what I propose adding to the article. We are taking the statements of Langley and making the appropriate changes. The concern of [[User:Woodstone|Woodstone]] was taken care of by changing from referring to an expression to referring to an equation. The concern of [[User:Woodstone|Woodstone]] seemed to be not nearly the problem that [[User:Woodstone|Woodstone]] seemed to think as it was very easy to fix. The appropriate notational changes were made. I think the appropriate place for adding the statement below is in the Spheres section under Geometric Interpretation [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 20:04, 2 July 2015 (UTC)

The equations,

<math>(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2 = \bigl([ \tilde{t} - b - s_i]c\bigr)^2, \; i=1,2,\dots,n</math>

represent [[sphere#Equations in three-dimensional space| spheres]] centered on the satellites with coordinates <math>x_i, y_i, z_i, \; i=1,2,\dots,n</math> and with radii given by

<math>\sqrt{(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2} = [ \tilde{t} - b - s_i]c, \; i=1,2,...,n</math>.

<ref name="Langley">{{cite article |title=The Mathematics of GPS |author=Richard Langley |work=GPS World|date=1991 |url=http://gauss.gge.unb.ca/gpsworld/EarlyInnovationColumns/Innov.1991.07-08.pdf }}</ref>

[[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 20:04, 2 July 2015 (UTC)

:That seems pretty verbose to me, not with words, but there's a lot of symbols. Can't we just say: "Each pseudorange corrected for the same unknown clock bias equals the actual range, and these similarly represent spheres centered at each satellite position."<ref name="Langley"/> [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 22:04, 2 July 2015 (UTC)

Its the equations that represent spheres, [[User:Fgnievinski|Fgnievinski], not the corrected pseudoranges. The symbols are simple and straightforward. The equations are simple and straightforward. The statement you make is unclear since you appear to be calling corrected pseudoranges, spheres. This is a clear and concise statement, [[User:Fgnievinski|Fgnievinski]. The statement takes about two lines. You have not pointed out anything that is not clear and precise in this statement. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 23:30, 2 July 2015 (UTC)

:{{reply|RHB100}} I'm just suggesting that we make use of the notation whose definition we had agreed earlier: range is <math>r=\sqrt{(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2}</math> and pseudorange is <math>p = [ \tilde{t} - s_i]c</math>. So where you write "with radii given by <math>\sqrt{(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2} = [ \tilde{t} - b - s_i]c</math>", I'd write, more succinctly, "with radii given by <math>r_i = p_i - bc</math>." [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 00:02, 3 July 2015 (UTC)

[[User:Fgnievinski|Fgnievinski]], I am unable to find any statement in the Langley paper that resembles the statement you made. Are you claiming that your statement comes from the Langley paper? If so where is it in the Langley paper? [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 23:39, 2 July 2015 (UTC)

:{{reply|RHB100}} Sure, here it is: "The pseudorange measurement [''p''] made by the receiver, in units of distance, is on the left-hand side of each of the equations. The expression under the square root sign is the true range [''r''] to the satellite. It is actually a representation of the sphere centered on coordinates ''x,y,z'', the position of the satellite. (...) The term ''c dT'' [on the right-hand side] is the contribution to the pseudorange [''p''] from the receiver clock offset, ''dT''." His pseudorange equation reads ''p=r-c dT'', which is equivalent to ours ''p=r+bc'' except for an innocuous algebraic sign in ''b=-dT''. And of course, ''p=r+bc'' is equivalent to ''r=p-bc'' via [[equivalence relation]]s. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 00:02, 3 July 2015 (UTC)

I think the language you used differed substantially from Langley, [[User:Fgnievinski|Fgnievinski]]. This notation you say we have agreed upon has been done only informally to some extent on the talk page. Its only to save time when writing on the talk page. The equations we are talking about are now in only the Problem description section of the article. Since the concise addition I have written should go in the Spheres section under Geometric interpretation, we should just use a line or two to repeat the equations in order to avoid the risk of misinterpretation. There is no reason we should try to obtain the dubious benefit of saving half a line by using abbreviations which we have not yet defined. Also, keep in mind that there is a slight error that Langley is making that Woodstone mentioned and that is he says that an expression rather than an equation represents a sphere. The concise statement that I made is straightforward, clear, and to the point. Let's not screw it up by changing it. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 01:15, 3 July 2015 (UTC)

:{{reply|RHB100}} Section [[GPS#Problem description]] already defines ''p'', so that section is the best place to introduce ''r'', whose definition I'm sure is undisputed. Section [[GPS#Spheres]] follows immediately, so there's no risk of misinterpretation. (As an aside, I think it's hair splitting to distinguish between equation and expression; the nuance will most certainly be lost in the reader, and to say that Langley got this wrong borders on pedantic.) Pragmatically, I'm only suggesting that your draft:

:: The equations, <math>(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2 = \bigl([ \tilde{t} - b - s_i]c\bigr)^2, \; i=1,2,\dots,n</math> represent [[sphere#Equations in three-dimensional space| spheres]] centered on the satellites with coordinates <math>x_i, y_i, z_i, \; i=1,2,\dots,n</math> and with radii given by <math>\sqrt{(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2} = [ \tilde{t} - b - s_i]c, \; i=1,2,...,n</math>.<ref name="Langley"/>

:be made more concise:

:: The pseudorange equations [[#Problem descrption|above]] represent [[Sphere#Equations in three-dimensional space|spheres]] centered at the satellite positions (<math>x_i, y_i, z_i</math>) with radii given by <math>r_i=p_i-bc</math>.<ref name="Langley"/>

: We got to give-and-take, otherwise we'll get stuck here forever. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 01:34, 3 July 2015 (UTC)

Well alright [[User:Fgnievinski|Fgnievinski]], here is somehting we can do. We have used the term navigation equations above and if we give them a name this is what they should be called.

:The navigation equations [[#Problem descrption|above]] represent [[Sphere#Equations in three-dimensional space|spheres]] centered at the satellite positions (<math>x_i, y_i, z_i</math>) with radii given by <math>r_i=p_i-bc</math> and also by the square root term in these equations.<ref name="Langley"/>

This is certainly a very concise statement and it provides the essential information including that the radii are also equal to the square root term as Langley did. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 03:51, 3 July 2015 (UTC)

:{{reply|RHB100}} How about this:

::Each of the navigation equations [[#Problem descrption|above]] represents a [[Sphere#Equations in three-dimensional space|sphere]] centered at the satellite positions (<math>x_i, y_i, z_i</math>) with radii given by <math>r_i=p_i-bc=\sqrt{(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2}</math>.<ref name="Langley"/>

:Deal? [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 04:21, 3 July 2015 (UTC)

::Certainly not. The proposals above again (still) have the fallacy of confusing equations with solutions. May I make a counter-proposal, that may work for us all. At the beginning of the equations section, we introduce the functions {{nobreak|''r_i''(''x, y, z'')}} as described here. We can then say: for every constant ''d'', the equation {{nobreak|''r_i''(''x, y, z'') {{=}} ''d''}} represents a sphere around satellite ''i'' with radius d. So if the clock bias ''b'' would be known, the receiver would be somewhere on the sphere with equation {{nobreak|''r_i''(''x, y, z'') {{=}} ''p_i'' &minus; ''b''c}}. It is very similar to your sproposal, but using the word constant first and the conditional later make a huge difference to me. &minus;[[User:Woodstone|Woodstone]] ([[User talk:Woodstone|talk]]) 13:07, 3 July 2015 (UTC)

::Finally we could then turn around and declare b as 4th unknown and symbolically move it to the left-hand side to get the actual equations to be solved from at least 4 staellites: {{nobreak|''r_i''(''x, y, z'') &minus; ''b''c{{=}} ''p_i''}}.

:::{{reply|Woodstone}} I think you're alluding to a previous [[#terminology]] difficulty; how about this:

::::Each of the navigation equations [[#Problem descrption|above]] represents a [[Sphere#Equations in three-dimensional space|sphere]] centered at the satellite positions (<math>x_i, y_i, z_i</math>) with fixed radii given by <math>r_i=p_i-bc=\sqrt{(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2}</math>, based on measured ''p'' and if ''b'' is known.<ref name="Langley"/>

:::If that doesn't please you, would you be so kind as to provide the literal content of what would be inserted in a forthcoming {{Tl|edit protected}} request. Thanks. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 18:18, 3 July 2015 (UTC)

::::Continues at [[#Proposed "equations" section]]. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 21:05, 4 July 2015 (UTC)

Alright [[User:Fgnievinski|Fgnievinski]], I like that equation,

:<math>r_i=p_i-bc=\sqrt{(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2}</math>.<ref name="Langley"/>

Let's go ahead and do it. This is concise and to the point. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 17:26, 3 July 2015 (UTC)

{{reflist-talk}}

== Proposed "equations" section ==

<i>{{User|Woodstone}} proposes the following rewording of the equations section (with which the intro of the geometric part is made redundant):</i>

The receiver uses messages received from satellites to determine the satellite positions and time sent. The ''x, y,'' and ''z'' components of satellite position and the time sent are designated as [''x_i, y_i, z_i, s_i''] where the subscript ''i'' denotes the satellite and has the value 1, 2, ..., ''n'', where ''n''&nbsp;&ge;&nbsp;4. When the time of message reception indicated by the on-board receiver clock is ''t&#771;_i'', the true reception time is {{nobreak|1=''t_i'' = ''t&#771;_i'' - ''b''}}, where ''b'' is the receiver's clock bias from the much more accurate GPS system clocks employed by the satellites. The receiver clock bias is the same for all received satellite signals (assuming the satellite clocks are all perfectly synchronized). The message's transit time is {{nobreak|1=''t&#771;_i'' - ''b'' - ''s_i''}}, where ''s_i'' is the satellite time. Assuming the message traveled at [[Speed of light|the speed of light]], ''c'', the distance traveled is {{nobreak|1=(''t&#771;_i'' + ''b'' - ''s_i'') ''c''}}. <!--(''t~_i + b − t_i'')''c''.-->

For each satellite, define:

:<math>r_i(x,y,z) = \sqrt {(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2}</math>

Then the receiver would have to be located somewhere on each of the spheres given by:

:<math>r_i(x,y,z) = (t_i - s_i) c </math>

or in terms of ''pseudoranges'', <math> p_i = ( \tilde{t}_i - s_i )c</math>, satisfy the equations:

:<math>r_i(x,y,z) + b c = p_i, \;i=1,2,...,n</math>

Since the equations have four unknowns (''x, y, z, b'')&mdash;the three components of GPS receiver position and the clock bias&mdash;signals from at least four satellites are necessary to attempt solving these equations. They can be solved by algebraic or numerical methods. Existence and uniqueness of GPS solutions are discussed by Abell and Chaffee. When ''n'' is greater than 4 this system is overdetermined and a fitting method must be used.

&minus;[[User:Woodstone|Woodstone]] ([[User talk:Woodstone|talk]]) 09:34, 4 July 2015 (UTC)

:[[User:Woodstone|Woodstone]], all you appear to be doing is copying what we already have in the Problem description section under Navigation equations and changing the name of the section. You don't say anything about what you think you are accomplishing and I do not see anything you have improved. You are introducing inconsistent notation. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 20:45, 4 July 2015 (UTC)

:{{reply|Woodstone}} I'm against speaking of spheres outside of section [[GPS#Geometrical interpretation]] and sub-section [[GPS#Spheres]].

:At first glance it may have seemed as if you were proposing a full rewrite of section [[GPS#Problem description]], but at a closer look, the only modification is word "offset" for "bias", and where it was:

<blockquote>

For n satellites, the equations to satisfy are:

:<math>(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2 = \bigl([ \tilde{t} - b - s_i]c\bigr)^2, \; i=1,2,\dots,n</math>

or in terms of ''pseudoranges'', <math> p_i = \left ( \tilde{t} - s_i \right )c</math>, as

:<math>\sqrt{(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2} + bc = p_i, \;i=1,2,...,n</math>

</blockquote>

:it became:

<blockquote>

For each satellite, define:

:<math>r_i(x,y,z) = \sqrt {(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2}</math>

Then the receiver would have to be located somewhere on each of the spheres given by:

:<math>r_i(x,y,z) = (t_i - s_i) c </math>

or in terms of ''pseudoranges'', <math> p_i = ( \tilde{t}_i - s_i )c</math>, satisfy the equations:

:<math>r_i(x,y,z) + b c = p_i, \;i=1,2,...,n</math>

</blockquote>

:Am I missing any other modification? [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 21:21, 4 July 2015 (UTC)

===Modification of Spheres section===

We agreed on a statement for the Spheres section under Geometric interpretation above. In addition to this statement, we should add a statement about the solution which comes from the Langley paper. The Spheres section should be changed to read:

:Each of the navigation equations [[#Problem descrption|above]] represents a [[Sphere#Equations in three-dimensional space|sphere]] centered at the satellite positions (<math>x_i, y_i, z_i</math>) with radii given by <math>r_i=p_i-bc=\sqrt{(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2}</math>.<ref name="Langley"/> The solution of these equations is at the intersection or near intersection of the spheres they represent.

Talking about the solution being at the intersection is a geometric interpretation. This type of language is in keeping with the meaning of the Langley paper and aids in understanding. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 21:14, 4 July 2015 (UTC)

:{{reply|RHB100}} Thanks for discussing section-specific changes; let's put the proposal in context:

<blockquote>

The measured ranges, called pseudoranges, contain clock errors. In a simplified idealization in which the ranges are synchronized, these true ranges represent the radii of spheres, each centered on one of the transmitting satellites. The solution for the position of the receiver is then at the intersection of the surfaces of three of these spheres.<ref>http://web.archive.org/web/20050316051910/http://www.siam.org/siamnews/general/gps.htm</ref> If more than the minimum number of ranges is available, a near intersection of more than three sphere surfaces could be found via, e.g. least squares.

<i>

Each of the navigation equations [[#Problem descrption|above]] represents a [[Sphere#Equations in three-dimensional space|sphere]] centered at the satellite positions (<math>x_i, y_i, z_i</math>) with radii given by <math>r_i=p_i-bc=\sqrt{(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2}</math>.<ref name="Langley"/> The solution of these equations is at the intersection or near intersection of the spheres they represent.

</i>

</blockquote>

:where the new part is in ''italics''. I'd agree with the last sentence in the new second paragraph, but don't you think it'd duplicate the last sentence in the existing first paragraph? [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 21:36, 4 July 2015 (UTC)

::The last two sentences of the first paragraph should be eliminated since you are talking in terms of only three spheres and the Langley paper makes clear that three spheres are inadequate. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 23:55, 4 July 2015 (UTC)

:::Not at all! Langley states explicitly:

::::''"With synchronized clocks, simultaneous range measurements to three satellites produce a determination of a receiver's position. Each range measurement can be portrayed as the radius of a sphere centered on a particular satellite"'' and also ''"Let's assume that the clock in the receiver is synchronized with the clock in the satellite... With a single such measurement of the distance or range to the satellite, we can determine something about the position of the receiver: it must lie somewhere on a sphere centered on the satellite with a radius equal to the measured range..."'' It is only much later that he introduces pseudoranges: ''"When we started our analysis, we assumed that the clock in the GPS receiver was synchronized with the clocks in the satellites. This assumption, however, is fallacious. (...) The ranges measurements [that] the receiver makes are biased by the receiver and satellite clock errors and therefore are referred to as pseudoranges. (...) Because of this error, the three spheres with radii equal to the measured pseudoranges ... will not intersect at a common point. However, if the receiver clock error, dT, can be determined, then the pseudoranges can be corrected and the position of the receiver determined."''

:::To keep in the spirit of the original source, we must speak of spheres first in terms of idealized ranges, and only later demonstrate how pseudoranges can be corrected so as to correspond to that geometrical interpretation. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 01:30, 5 July 2015 (UTC)

::::[[User:Fgnievinski|Fgnievinski]], above I state

:::::The last two sentences of the first paragraph should be eliminated since you are talking in terms of only three spheres and the Langley paper makes clear that three spheres are inadequate. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 23:55, 4 July 2015 (UTC)

::::Then you state

:::::Not at all! Langley states explicitly:"With synchronized clocks, simultaneous range measurements to three satellites produce a determination of a receiver's position."

::::But you are clearly quoting Langley out of context. He is only stating this before leading up to his conclusion at least four satellites are required. Therefore when you say "Not at all", contradicting my statement that the Langley paper makes clear that three spheres are inadequate, you are incorrect. In fact your own post where you quote Langley as saying, "Because of this error, the three spheres with radii equal to the measured pseudoranges ... will not intersect at a common point", verifies my statement that the Langley paper makes clear that three spheres are inadequate and that your contradiction "Not at all!" is false. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 23:56, 6 July 2015 (UTC)

:::::I disagree with your interpretation of Langley's paper. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 16:25, 7 July 2015 (UTC)

:: Also you don't need to talk about solution methods here since it's discussed outside of Geometric interpretation. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 23:55, 4 July 2015 (UTC)

:::Agreed with removing "least squares". [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 01:30, 5 July 2015 (UTC)

:: The first sentence of the first paragraph should also be removed since we have a much clearer description of what we're talking about in the new material. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 23:55, 4 July 2015 (UTC)

::Agreed. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 01:30, 5 July 2015 (UTC)

:::{{reply|RHB100}} OK, so I guess we're down to:

<blockquote>

In a simplified idealization in which the clocks are synchronized, ranges represent the radii of spheres, each centered on one of the transmitting satellites. The solution for the position of the receiver is then at the intersection three of these spheres. If more than the minimum number of ranges is available, a near intersection could be found. <ref>http://web.archive.org/web/20050316051910/http://www.siam.org/siamnews/general/gps.htm</ref>

In the case of clock-corrupted pseudoranges, each of the navigation equations [[#Problem descrption|above]] represents a [[Sphere#Equations in three-dimensional space|sphere]] centered at the satellite positions (<math>x_i, y_i, z_i</math>) with radii given by <math>r_i=p_i-bc=\sqrt{(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2}</math>.<ref name="Langley"/> The solution of these equations is at the intersection or near intersection of the spheres they represent.

</blockquote>

:::I still think the last sentences of each first and second paragraphs are near duplicates. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 01:30, 5 July 2015 (UTC)

{{reflist-talk}}

===Modification of Problem description section===

{{reply|Woodstone}} How about this, where it was:

<blockquote>

Assuming the message traveled at [[Speed of light|the speed of light]], ''c'', the distance traveled is {{nobreak|1=(''t&#771;'' - ''b'' - ''s_i'') ''c''}}.

For n satellites, the equations to satisfy are:

:<math>(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2 = \bigl([ \tilde{t} - b - s_i]c\bigr)^2, \; i=1,2,\dots,n</math>

or in terms of ''pseudoranges'', <math> p_i = \left ( \tilde{t} - s_i \right )c</math>, as

:<math>\sqrt{(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2} + bc = p_i, \;i=1,2,...,n</math>

</blockquote>

it'd become:

<blockquote>

Assume the message traveled at [[Speed of light|the speed of light]], ''c''.

For each satellite (indexed by <math>i=1,2,...,n</math>), define the ''range'' as:

:<math>r_i(x,y,z) = \sqrt {(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2}</math>

and the ''pseudorange'' as:

:<math> p_i = \left ( \tilde{t} - s_i \right )c = r_i + bc</math>

</blockquote>

If you insist on the distinction between expression and equation, that would have to be done in section [[GPS#Spheres]]. I made a proposal before, which involved inserting words "fixed", "measured", and "known" in "fixed radii given by <math>r_i=p_i-bc=\sqrt{(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2}</math>, based on measured ''p'' and if ''b'' is known" -- why is that not OK? Do you contend that the value <math>r_i</math> will not also be understood as the expression <math>r_i(x,y,z)</math>? [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 21:55, 4 July 2015 (UTC)

:This change as shown above completely eliminates the equations to be satisfied which is the real problem description. I don't care if you add a definition of <math>r_i(x,y,z)</math> which is all you seem to be adding but don't remove the equations to be satisfied. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 00:03, 5 July 2015 (UTC)

::{{reply|RHB100}} The definition of range is certainly useful. The only thing eliminated were the words "equations to satisfy", but the equations themselves haven't changed at all: it was only a square root that was taken, and a few terms shuffled from left- to right-hand side of the equality sign. I don't know what exactly you are missing. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 01:34, 5 July 2015 (UTC)

:::[[User:Fgnievinski|Fgnievinski]], just in eliminating the words, "equations to satisfy" you've eliminated a statement of the Problem description. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 03:41, 5 July 2015 (UTC)

::::{{reply|RHB100}} Alright:

<blockquote>

Assume the message traveled at [[Speed of light|the speed of light]], ''c''.

The equations to satisfy are as follows.

For each satellite (indexed by <math>i=1,2,...,n</math>), define the ''range'' as:

:<math>r_i(x,y,z) = \sqrt {(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2}</math>

and the ''pseudorange'' as:

:<math> p_i = \left ( \tilde{t} - s_i \right )c = r_i + bc</math>

</blockquote>

:::: How about as quoted above? [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 03:48, 5 July 2015 (UTC)

{{outdent}}

Licensed Professional Engineer, [[User:RHB100|RHB100]], says, here are the equations,

For n satellites, the equations to satisfy are:

:<math>(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2 = \bigl([ \tilde{t} - b - s_i]c\bigr)^2, \; i=1,2,\dots,n</math>

or in terms of ''pseudoranges'', <math> p_i = \left ( \tilde{t} - s_i \right )c</math>, as

:<math>\sqrt{(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2} + bc = p_i, \;i=1,2,...,n</math>

These equations should not be modified. These equations are in a form in which they can be more easily understood by readers. They are also in a form which allows easier comparison with other documents such as the Langley paper. Also if you look at the Langley paper you will see that abbreviations of the equations are not used and it is a real good paper. On the other hand if you look at the GNSS article you will see that very terse notation is used and it is a terrible article, virtually unreadable. We should devote our efforts to maintaining the superiority of the GPS article over the inferior GNSS article. GPS was developed by Americans using the money of American taxpayers. GPS shows American technical superiority in navigation and position finding. This should give us the incentive to maintain that same technical superiority of our GPS article over the GNSS article. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 18:05, 5 July 2015 (UTC)

:Your statement above convinced me not to further engage is discussions about GPS or GNSS with you. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 16:34, 7 July 2015 (UTC)

Woodstone made the statement,

:Then the receiver would have to be located somewhere on each of the spheres given by:

:<math>r_i(x,y,z) = (t_i - s_i) c </math> above.

This is a terrible statement which could only be true if the clock bias, b, were zero and that were no other errors. Therefore we should avoid these types of changes. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 18:05, 5 July 2015 (UTC)

: It'd help reaching a compromise if you could show any sign of goodwill towards suggestions from other people. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 16:34, 7 July 2015 (UTC)

Well what do we want to discuss. We were discussing the spheres section and then we got diverted from that by a proposal to change the Problem description section made on the 4th of July. Do you want to go back and try to resolve our differences on the Spheres section under Geometric interpretation? [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 17:02, 7 July 2015 (UTC)

:The "terrible" statement was made in a context where t_i is unambiguously the true arrival time of the message, not the measured arrival time, which has a tilde. The true range is precisely the only case where the equations represent spheres. That is why I think that fits perfectly in the buildup towards the practical eqautions involving also a clock bias. In my view defining the functions {{nobreak|''r_i''(''x, y, z'')}} as preparation can help in clarifying the structure of the equations more than big expressions with square roots. &minus;[[User:Woodstone|Woodstone]] ([[User talk:Woodstone|talk]]) 17:27, 7 July 2015 (UTC)

Well alright [[User:Woodstone|Woodstone]], if you are talking about the true arrival time that does provide an explanation of what you are talking about. But I don't see that definition of t_i in the article as it is now. I don't agree with the statement, "The true range is precisely the only case where the equations represent spheres". If you look at the paper by Langley, you will see where he shows a figure in which there are spheres with radii equal to the pseudoranges, but they do not intersect at the solution. In the same figure, he shows spheres with corrected radii intersecting at the solution. Also The equations,

:<math>(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2 = \bigl([ \tilde{t} - b - s_i]c\bigr)^2, \; i=1,2,\dots,n</math>

which are in the article, show that they define spheres for all cases, except for the trivial case in which the right side is zero. I think the reader is less likely to be confused if he can see the equations directly without renaming for extra conciseness. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 18:29, 7 July 2015 (UTC)

:Those equations only represent spheres if the RHS is a constant. When looking to find ''b'', this is not the case. Compare to the simplified case:

::<math>(x-x_i)^2 + (y-y_i)^2 + z = 1</math>

:Do you claim that equation in (''x, y, z'') describes a sphere, or circle? Probably not, because it describes a ''cone''. &minus;[[User:Woodstone|Woodstone]] ([[User talk:Woodstone|talk]]) 09:12, 8 July 2015 (UTC)

I understand that in 3 dimensional space, your equation represents a cone when z < 1. For the purposes of GPS understading and description we should interpret the equations,

:<math>(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2 = \bigl([ \tilde{t} - b - s_i]c\bigr)^2, \; i=1,2,\dots,n</math>

as 3 dimensional curves (i.e. spheres) in which the terms on the right side take on their defined meanings, <math>\tilde{t}</math> as time of reception, b as clock bias, and <math>s_i</math> as time of transmission. The interpretation in which there are expanding or contracting spheres because of the right side of the equation changing should not be used because it does not help in understanding GPS. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 17:28, 8 July 2015 (UTC)

:On the contrary, in order to understand why at least 4 satellites are needed, it is essential to understand that all 4 of (''x, y, z, b'') are to solved simultaneously from the equations. If the value of ''b'' is supposed to be known, there would be no need for more than 3 equations. &minus;[[User:Woodstone|Woodstone]] ([[User talk:Woodstone|talk]]) 14:03, 9 July 2015 (UTC)

In speaking of these equations as defining spheres in 3 dimensional space, we are using the same type of explanation used in the Langley paper and other publications. To speak of these equations as representing spheres, we do not need to know the value of b, all we need to know is that there exists a solution. Existence follows from the fact that the receiver is some distance from each of the satellites and therefore the receiver is on the surfaces of spheres centered at the satellite. After pointing out that these equations represent spheres, we can then point out that the solution is at the intersection of these spheres. Sometimes in engineering, it is necessary to think abstractly. Many mathematicians understand the concept of abstract thinking. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 18:55, 9 July 2015 (UTC)

:You have not addressed my point above. Why are 4 equations needed if only 3 unknows are to be found? What is so special about ''b'' that makes it different from ''x, y'' and ''z''? The receiver does not by nature lie on any sphere. It has a fixed ___location in space (as well as a fixed clock bias). The essential point is that the equations solved are such that all four are solved from them simultaneously. The abstraction lies the fact that four unknowns (''x, y, x, b'') are postulated for which four equations, with coefficients expressed in measured data (''c, t_i, s_i, x_i, y_i, z_i'') need to be satisfied. &minus;[[User:Woodstone|Woodstone]] ([[User talk:Woodstone|talk]]) 08:54, 10 July 2015 (UTC)

[[User:Woodstone|Woodstone]], you are not telling me anything I don't already know. I know that at least 4 equations must be solved simultaneously for the 4 unknowns, (''x, y, x, b'') or else a least squares solution must be found using more than 4 equations. I don't think you fully comprehend the difference between knowing that a solution exists and knowing what the solution is. This may be somewhat of a subtle point. Do you understand the difference between knowing that a solution exists and knowing what the solution is? [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 17:32, 10 July 2015 (UTC)

:Finally we are making some progress, you now agree that the equations play in a 4 dimensional space (''x, y, z, b''). So in order for them to represent spheres, the terms in all 4 unknowns would need to be quadratic, added with positive coefficients and a non-zero right hand side. The equations discussed can be normalised to:

::<math>(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2 - c^2(b - p_i/c)^2 = 0</math>

:Clearly the quadratic term in ''b'' has the wrong sign, and the RHS is zero: these are not 4D spheres. Only holding ''b'' constant makes the equations 3D spheres, but why then not holding ''x'' (or ''y'' etc) constant and look at what the equations describes? All 4 are equally variable in the equations. &minus;[[User:Woodstone|Woodstone]] ([[User talk:Woodstone|talk]])

[[User:Woodstone|Woodstone]], nothing you are saying is of any value for the purposes of GPS, as far as I can tell. And it's certainly not interesting. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 05:41, 12 July 2015 (UTC)

:[[user:RHB100|RHB100]], please refrain from insulting people and check the mathematics of conical sections to confirm that the above equations unambiguously represent spherical cones, not spheres. &minus;[[User:Woodstone|Woodstone]] ([[User talk:Woodstone|talk]]) 06:54, 12 July 2015 (UTC)

[[User:Woodstone|Woodstone]], I say again what you are saying has nothing to do with GPS. These spherical cones have nothing to do with GPS. All references to spherical cones should be removed from the GPS article. What you are saying is certainly of no interest for the purposes of GPS. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 17:27, 14 July 2015 (UTC)

===Yet Another===

<del>

The only thing we are doing is defining the distance from satellite i to the receiver. Therefore there is no need for radical changes to the Problem description section which obscure the equations to be solved. Therefore I propose the following for the Problem description section which adds the definition without further changes:

--------------------------------------------------------------------------------------

The receiver uses messages received from satellites to determine the satellite positions and time sent. The ''x, y,'' and ''z'' components of satellite position and the time sent are designated as [''x_i, y_i, z_i, s_i''] where the subscript ''i'' denotes the satellite and has the value 1, 2, ..., ''n'', where ''n''&nbsp;&ge;&nbsp;4. When the time of message reception indicated by the on-board receiver clock is ''t&#771;'', the true reception time is {{nobreak|1=''t'' = ''t&#771;'' - ''b''}}, where ''b'' is the receiver's clock offset from the much more accurate GPS system clocks employed by the satellites. The receiver clock offset is the same for all received satellite signals (assuming the satellite clocks are all perfectly synchronized). The message's transit time is {{nobreak|1=''t&#771;'' - ''b'' - ''s_i''}}<!--, where ''s_i'' is the satellite time-->. Assuming the message traveled at [[Speed of light|the speed of light]], ''c'', the distance traveled is {{nobreak|1=(''t&#771;'' - ''b'' - ''s_i'') ''c''}}. <!--(''t~_i - b − t_i'')''c''.-->

For n satellites, the equations to satisfy are:

:<math>(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2 = \bigl([ \tilde{t} - b - s_i]c\bigr)^2, \; i=1,2,\dots,n</math>

or in terms of ''pseudoranges'', <math> p_i = \left ( \tilde{t} - s_i \right )c</math>, as

:<math>\sqrt{(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2} + bc = p_i, \;i=1,2,...,n</math> .<ref name=GPS_BASICS_Blewitt>section 4 beginning on page 15 [http://www.nbmg.unr.edu/staff/pdfs/Blewitt%20Basics%20of%20gps.pdf GEOFFREY BLEWITT: BASICS OF THE GPS TECHNIQUE]</ref><ref name=Bancroft>{{cite web|url=http://www.macalester.edu/~halverson/math36/GPS.pdf|archiveurl=http://web.archive.org/web/20110719232148/http://www.macalester.edu/~halverson/math36/GPS.pdf|archivedate=July 19, 2011|title=Global Positioning Systems|format=PDF|accessdate=October 15, 2010}}</ref>

We define :<math> r_i(x,y,z) = \sqrt{(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2}, \;i=1,2,...,n</math>, the distance from the satellite i to the receiver for later use.

Since the equations have four unknowns [''x, y, z, b'']&mdash;the three components of GPS receiver position and the clock bias&mdash;signals from at least four satellites are necessary to attempt solving these equations. They can be solved by algebraic or numerical methods. Existence and uniqueness of GPS solutions are discussed by Abell and Chaffee.<ref name="Abel1"/> When ''n'' is greater than 4 this system is overdetermined and a fitting method must be used.

With each combination of satellites, GDOP quantities can be calculated based on the relative sky directions of the satellites used.<ref>{{cite web|url=http://www.colorado.edu/geography/gcraft/notes/gps/gps.html#Gdop|title=Geometric Dilution of Precision (GDOP) and Visibility|first=Peter H.|last=Dana|publisher=University of Colorado at Boulder|accessdate=July 7, 2008}}</ref> The receiver ___location is expressed in a specific coordinate system, such as latitude and longitude using the [[WGS 84]] [[datum (geodesy)|geodetic datum]] or a country-specific system.<ref>{{cite web|url=http://www.colorado.edu/geography/gcraft/notes/gps/gps.html#PosVelTime|title=Receiver Position, Velocity, and Time|author=Peter H. Dana|publisher=University of Colorado at Boulder|accessdate=July 7, 2008}}</ref>

____________________________________________________________________________________

[[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 04:03, 5 July 2015 (UTC)

Woodstone made the statement,

:Then the receiver would have to be located somewhere on each of the spheres given by:

:<math>r_i(x,y,z) = (t_i - s_i) c </math> above. This is a terrible statement which could only be true if the clock bias, b, were zero and that were no other errors. Therefore we should avoid these types of changes. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 04:12, 5 July 2015 (UTC)

</del>

====Comment====

{{reply|RHB100}} I'll ignore the long new section above; I'll only consider specific changes, not complete rewrites. I'll not make an undue effort to understand your proposal. Only if you decide to abide by the [[Wikipedia:Talk page guidelines]], including threading and sectioning, I'll engage in discussion. Otherwise, the article will remain as it is, given the edit lock in effect. This discussion is restarting all too often, and I'm losing hope of convergence. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 05:10, 5 July 2015 (UTC)

:[[User:Fgnievinski|Fgnievinski]], when you make an abbreviated statement of what you are changing, it is sometimes unclear. To avoid confusion I made it clear by showing the entire section. If the Problem description section does not change, that is fine. It was Woodstone not me who made the time wasting proposal to change the Problem description section. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 05:46, 5 July 2015 (UTC)

::[[WP:THREAD]].[[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 05:53, 5 July 2015 (UTC)

Well I'll remove the above section if you want to ignore it. [[User:RHB100|RHB100]] ([[User talk:RHB100|talk]]) 17:38, 5 July 2015 (UTC)

*''Before'' anyone adds anything to the article be forewarned that if it is not agreed upon here (by which I mean if there is no consensus to move forward with the idea) and/or the change results in another round of reverting the article will be placed back on lockdown and the time for which you will all be unable to edit it will be extended. Therefore, it would be in everyone's best interest to discuss this to death to make certain that whatever you are going to do to the article is done with the majority consensus and that the disagreeing minority does not start or cause to be continued an edit or revert war. [[User:TomStar81|TomStar81]] ([[User talk:TomStar81|Talk]]) 17:57, 6 July 2015 (UTC)

{{reflist-talk}}