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Dear all,
I have been sailing on a cruiser which has a basic GPS: one can enter the WGS 84 coordinates of a waypoint, and the GPS displays in real time the distance and bearing of the waypoint, something like the one in the image attached.

As a theoretical physicist, there is a question that I have been asking myself for a while: I would like to know how bearing and distance are precisely calculated by the GPS. In particular, given a curve on the WGS84 ellipsoid that joins the current location of the boat (A) to the waypoint (B), one may choose this curve in different ways. For instance, a loxodrome (i) or a geodesic curve on the ellipsoid (ii), i.e., the shortest path between A and B, which would coincide with a great circle in the case where the ellipsoid reduces to a sphere.

In either case, the angle between the curve and the local meridian at A yields a bearing, and the length of the curve a distance. Does the bearing displayed on the GPS screen correspond to choice (i) or (ii), or something else? Or does it depend on the GPS settings, or on the GPS model?

Thank you very much for your help! :2 boat::2 boat::grin:grin:grin

PS I am aware that there is little discrepancy between the twos if the distance between the two points is small. Also, I found a related question on a forum years ago, but there seems to be no definite answer there...
 

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For practical purposes I believe GPS plotters assumes that the earth is flat. I believe when distance exceed some threshold the computations are great circle... and so the heading will change as you proceed on the path to the destination.
 

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Ive flown transport category aircraft for decades using both INS and GPS nav systems. I feel like I should know the answer to the OPs question. Im sure Ive glanced over it in reading the minutia of a particular nav system. Im fairly certain (most?) GPS uses the WGS84 model and will always plot great circle routes for course and heading.
 

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Excellent question. I thought gps determines your position by simple triangulation. More satellites above the horizon more opportunities for triangulation so position circle of uncertainty improves. That circle is really a sphere.
I thought the base map was projected as 2 dimensional but includes elevations. Therefore it’s really 3 dimensional. So each pixel has 3 variables. For navigational use on water (boat) additional variables such as depth, POIs etc. are added and on land (car) other variables are added. Using old school terms all courses on a boat, even short one, are great circle. However, this isn’t calculated as a great circle but rather as the shortest distance on a surface of varying height. Base map determines what’s permissible. Roads on land, water surface for us. Gps doesn’t care if you’re on water, land or flying.
 

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Until accurate clocks that would function at sea were invented folks sailed fixed latitudes. No need for changing course while following a fixed latitude. Given they were using magnetic compasses not a true shortest great circle but due to magnetic variation not a rhumb line great circle issue. Earth isn’t a perfect sphere mildly flattened not apple so you’re right rhumb and great circle will vary some depending upon what latitude. True even with a fluxgate compass. Think we need someone who knows more about the engineering of gps use to answer this question. It remains a good question. How does the software work? Understand how you get a position from it but don’t understand how the software uses that. Question remains how to get to the shortest great circle.
 

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Earth isn’t a sphere
Definition of great circle
: a circle formed on the surface of a sphere by the intersection of a plane that passes through the center of the sphere
specifically : such a circle on the surface of the earth an arc of which connecting two terrestrial points constitutes the shortest distance on the earth's surface between them
Strange but True: Earth Is Not Round
Credit: Gary S. Chapman Getty Images
As countless photos from space can attest, Earth is round—the "Blue Marble," as astronauts have affectionately dubbed it. Appearances, however, can be deceiving. Planet Earth is not, in fact, perfectly round.
This is not to say Earth is flat. Well before Columbus sailed the ocean blue, Aristotle and other ancient Greek scholars proposed that Earth was round. This was based on a number of observations, such as the fact that departing ships not only appeared smaller as they sailed away but also seemed to sink into the horizon, as one might expect if sailing across a ball says geographer Bill Carstensen of Virginia Tech in Blacksburg.
Isaac Newton first proposed that Earth was not perfectly round. Instead, he suggested it was an oblate spheroid—a sphere that is squashed at its poles and swollen at the equator. He was correct and, because of this bulge, the distance from Earth's center to sea level is roughly 21 kilometers (13 miles) greater at the equator than at the poles.
 

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Until accurate clocks that would function at sea were invented folks sailed fixed latitudes. No need for changing course while following a fixed latitude. Given they were using magnetic compasses not a true shortest great circle but due to magnetic variation not a rhumb line great circle issue. Earth isn't a perfect sphere mildly pear not apple so you're right rhumb and great circle will vary some depending upon what latitude. True even with a fluxgate compass. Think we need someone who knows more about the engineering of gps use to answer this question. It remains a good question. How does the software work? Understand how you get a position from it but don't understand how the software uses that. Question remains how to get to the shortest great circle.
I suspect that the computation considers the earth a sphere... and updates the computation based on current fix if the vessel is moving.
 

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I dont understand the problem. If the question is "How does GPS compute course and heading" thats pretty simple. Spherical geometry is how one computes angles on a sphere. Smart guys worked that out a long time ago. Geographic models such as the WGS84 are used to translate the earth surface into a 2d image that we use on our charts. I thought the OP was asking what model is used to model the earth surface, but maybe he needs to clarify.

Google Spherical Geometry and WGS 84.



.
 

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I dont understand the problem. If the question is "How does GPS compute course and heading" thats pretty simple. Spherical geometry is how one computes angles on a sphere. Smart guys worked that out a long time ago. Geographic models such as the WGS84 are used to translate a spherical surface into a 2d image that we use on our charts. I thought the OP was asking what model is used to model the earth surface, but maybe he needs to clarify.

Google Spherical Geometry and WGS 84.
Bot we all realize that for a waypoit/destination thousands of miles away... the ship's heading for the shortest path... will be changing as it proceeds. That is if you entered a waypoint 2,000 miles away that is not on the same lattitude... and your plotter told you to steer X° at the get go and you held THAT X° heading you would miss the mark!
 

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Thanks but thought it was a more basic question. Sure spherical geometry is used and perhaps modified to account for the earth being a spheroid rather than a sphere but what’s the mechanism? How is it done? Are all calculations off of a base map that’s spherical? Or off various Mercator projections? In some places due to tides and currents there’s significant perturbations with mounding of the sea so the surface isn’t spherical in that local area. Is that accounted for? Most chart plotters give you tides/currents/ depth at low tide but also current expected depth is in there. Perhaps of no concern at smaller latitudes near the equator but in NB, PEI or even Maine variation would impact on shortest course. The fastest course is a separate calculation and more important to us sailors but the question remains and is a good one.
 

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Thanks but thought it was a more basic question. Sure spherical geometry is used and perhaps modified to account for the earth being a spheroid rather than a sphere but what's the mechanism? How is it done? Are all calculations off of a base map that's spherical? Or off various Mercator projections? In some places due to tides and currents there's significant perturbations with mounding of the sea so the surface isn't spherical in that local area. Is that accounted for? Most chart plotters give you tides/currents/ depth at low tide but also current expected depth is in there. Perhaps of no concern at smaller latitudes near the equator but in NB, PEI or even Maine variation would impact on shortest course. The fastest course is a separate calculation and more important to us sailors but the question remains and is a good one.
I don't think any of your concerns are part of short distance calculations. I don't know what would define short distance. They simply don't matter. The precisions would have to be to many more decimal places! and no helmsmen or AP will work with that level of precision. No GPS plotter reports accuracies of more than 0.X places if that. Does your plotter tell you a course is 38.1° or it is 38°. What is the precision of the speed probably the same.

GPS can be very precise for fixes I believe... but that level of precision is not required for commercial and recreational navigation.
 

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Hello,

I really depends on your particular GPS.

This is from the user guide of my B&G Vulcan:

Navigation method:
Different methods are available for calculating the distance and bearing between any two points on a chart.
The Great circle route is the shortest path between two points. However, if you are to travel along such a route, it would be difficult to steer manually as the heading would constantly be changing (except in the case of due north, south, or along the equator).
Rhumb lines are tracks of constant bearing. It is possible to travel between two locations using Rhumb line computation, but the distance would usually be greater than if Great circle is used.

I get to select Great Circle or Rhumb Line.

I don't know if or how Garmin allows you to select any options.

Barry
 

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Hello,

The Great circle route is the shortest path between two points. However, if you are to travel along such a route, it would be difficult to steer manually as the heading would constantly be changing (except in the case of due north, south, or along the equator).
.

Barry
Good post Barry.

I think the heading on any lattitude... due east of west would be accurate as well as the equator. Once the heading crosses a lattitude the computations need to be great circle.

The distance does need spherical geometry because all paths are over a curved surface... assumed to be a sphere.
 

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As far as I know, the only line of latitude that is a great circle is the equator. All other lines of latitude are small circles. All meridians of longitude are great circles. The plane of the circle has to pass through the centre of the sphere for it to constitite a great circle.
 

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As I understand the question, the OP is asking whether calculated bearing and distance consider that the earth is not perfectly round. Clearly, the GPS knows where the destination is located. The question is one of accuracy and efficiency of the route to get there, but only on a theoretical basis. Of the roughly 25,000 mile circumference of the globe I think the equator and the longitudinal distances are only off by about 40 or 50 miles. Theoretical it is. You are going to arrive at the proper destination, you simply may be a short distance to the side of the most efficient course.

I believe the WGS84 datum considers the distance from the center of the earth, as well as it's two dimensional location on the surface. Therefore, it would stand to reason, it calculates distance and bearing, using this out of round shape. I can't say for sure, it does.
 
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So that’s the answer. All are great circle. Probably no compensation for 13 nm spheroid compression at the poles nor local anomalies due to local phenomena.
 
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