Distance between two points on Earth doesn't increase with altitude?

[biologist]

TL;DR Summary

Any length over the surface of a sphere is an arc length. Arc length increases with radius. On Earth, this would translate to increase in distance (arc length) with increase in altitude (radius). This is not the case based on GPS results.

Consider the following example:
Point A has coordinates 45 lat, 0 long. Point B has coordinates 45 lat, 2 long. Both points are 5000 ft above sea level. The distance between them is X.
Point C has coordinates 45 lat, 100 long. Point D has coordinates 45 lat, 102 long. Both points are at sea level. The distance between them is Y.

X should be greater than Y because the difference in altitude on a sphere means a difference in radius, which means a difference in arc length. This does not seem to be the case for distances on earth, according to GPS.

Here is the equation for the difference between any distance X and distance Y on Earth assuming both distances are:
1. At the same latitude.
2. Span the same amount of degrees in longitude.

X-Y =.017 x (degrees of longitude between both distances) x (difference in altitude) x cos(latitude)

For the first example, the difference would be:
X-Y = .017(2)(5000)cos(45) = 120.208 ft

The equation has a straightforward geometric derivation that I'll provide at the end of this post. Assuming the equation is valid, however, here's an example where, using GPS, there appears to be no difference between distances X and Y:

Distance X spans across Issyk Kul, a lake in Kyrgyztan, between point A (42.446918,76.18375419) and point B (42.446918,77.86265619). The lake is 5272 ft above sea level.

Distance Y spans across the Caspian Sea between point C (42.446918,47.96593019) and point D (42.446918,49.64483219) at 92 ft below sea level.

Using GPS (Earth map online service (https://satellites.pro/)) both distances are equal to 85.60 miles. According to the above equation, there should be a difference of .02 miles, which should be within GPS accuracy. Both distances span the same amount of longitudinal degrees. The distances are both over bodies of water to rule out surface irregularities.
Looking for possible explanations for this.DERIVATION:
radius of a circle of longitude = r = (radius of earth) x cos(latitude)
radius of a circle of longitude above sea level = R = (radius of Earth + altitude) x cos(latitude)
R - r = (difference in altitude) x cos(latitude)

Arc length at sea level = l = 2pi x (r) x (change in longitude/360)
Arc length above sea level = L = 2pi x (R) x (change in longitude/360)
L - l = 2pi x (change in longitude/360) x (R-r) = .017 x (change in longitude) x (R-r) = .017 x (change in longitude) x (difference in altitude) x cos(latitude)

 

[jbriggs444]

Nothing here indicates the use of GPS. [GPS, satellite imagery and turn by turn navigation are three different things].

You have provided two pairs of two-dimensional coordinates (latitude and longitude). The third coordinate for each end point is presumably taken based on the geoid (sea level).

 

[fresh_42]

Let's calculate a bit.
\begin{align*}
5,000 \text{ ft.} \approx 1,500\text{ m} &\Longrightarrow \dfrac{6,360}{6,361.5} \approx 0,9997642
\end{align*}
Thus the radii change by $0.02358 \text{ %}$. How confident are you that the computation on the website takes such a tiny difference into consideration?

And the Earth isn't a sphere. How do you even know that your heights at different points reflect different radii? It could easily be that the arc length at sea level is by far larger than $2\pi\cdot 1,500\text{ m}\approx 6.5\text{ km}$ at some other point on high mountains.

 

[Baluncore]

X should be greater than Y because the difference in altitude on a sphere means a difference in radius, which means a difference in arc length. This does not seem to be the case for distances on earth, according to GPS.

Maps are not plotted on 3D stretchy paper, with lumps where there are mountains. All points are reduced to the reference ellipsoid before being plotted. For that reason there should be no difference in the arc distance as a function of height, because the arc is always reduced to and measured along the same reference ellipsoid.

 

[biologist]

All points are reduced to the reference ellipsoid before being plotted.

That would definitely explain the issue, but are there any references for this assumption? I wasn't aware that Google and Apple Maps did this for their distance calculations.

This seems at odds with the fact that these mapping services provide highly accurate latitude and longitude points (47.96593019 longitude, for example).

 

[Baluncore]

This seems at odds with the fact that these mapping services provide highly accurate latitude and longitude points (47.96593019 longitude, for example).

A (Lat,Long) pair specifies a line from the centre of the Earth to infinity. The distance between two such lines must be specified as an angle, or the length of a great circle on a reference surface.

How would you define and measure an arc length between two points at different heights? You can really only measure the straight line distance, or reduce everything vertically to a reference surface.

When you buy a block of land you buy an area of the reference ellipsoid, where the corner points are specified effectively by (Lat,long), without any height or slope being specified or considered. Everything is reduced to the national mapping grid, which is referenced to (Lat,long), and the GPS reference ellipsoid, WGS84.

 

[jbriggs444]

How would you define and measure an arc length between two points at different heights?

One could measure it along a sort of analog to a rhumb line -- a great circle path with a constant pitch angle above or below the horizontal.

Not that I am advocating such an approach. Just offering the possibility.

 

[Baluncore]

This seems at odds with the fact that these mapping services provide highly accurate latitude and longitude points (47.96593019 longitude, for example).

That is only 10 digits.
Napoleon originally decided, it would be 10,000 km from the North Pole to the Equator through Paris. For 1 m accuracy that would require 7 digits. For 1 mm accuracy it needs another 3 digits, making 10 digits.

1 mm is about the limit of differential GPS, but for VLBI radio astronomy, the vertical component of the solid Earth tide is over 100 times bigger than that.
https://en.wikipedia.org/wiki/Earth_tide

Not that I am advocating such an approach. Just offering the possibility.

One could also find the midpoint, and halve the height difference, recursively, then sum the many straight segments. But if there is nothing that follows such a path, then that distance metric between the two points is next to useless. You may as well use a straight line.

 

[Baluncore]

Distance X spans across Issyk Kul, a lake in Kyrgyztan, between point A (42.446918,76.18375419) and point B (42.446918,77.86265619). The lake is 5272 ft above sea level.

Distance Y spans across the Caspian Sea between point C (42.446918,47.96593019) and point D (42.446918,49.64483219) at 92 ft below sea level.

I take your GPS Lat, Lng, Hgt; and convert height to metres.
Then I accurately convert WGS84 LLH to Earth centred cartesian xyz.
Note: +x is Null Island, +y is Indian Ocean, and +z is North Pole.
Using Pythagoras I find the straight line distance between the two points.
Then I subtract them to see the difference due to different altitude.

Point A
Lat= 42.44691800° Lng= 76.18375419° Hgt= 1606.906 metre
x = 1125941.440 y = 4578407.935 z = 4283450.056 metre
Point B
Lat= 42.44691800° Lng= 77.86265619° Hgt= 1606.906 metre
x = 991319.097 y = 4609430.528 z = 4283450.056 metre
Dist A-B = 138150.5577 metre

Point C
Lat= 42.44691800° Lng= 47.96593019° Hgt= -28.042 metre
x = 3156108.012 y = 3501024.248 z = 4282346.619 metre
Point D
Lat= 42.44691800° Lng= 49.64483219° Hgt= -28.042 metre
x = 3052179.527 y = 3591989.519 z = 4282346.619 metre
Dist C-D = 138115.2077 metre

Difference = 35.350 metre due to altitude.

 

[pbuk]

Summary:: Any length over the surface of a sphere is an arc length. Arc length increases with radius. On Earth, this would translate to increase in distance (arc length) with increase in altitude (radius). This is not the case based on GPS results.

Geodesy is a much more complicated subject than you appreciate so don't believe that the equations you have worked out apply to an arbitrary degree of precision. It is a really interesting subject though, so if you want to investigate it start with that link and learn about rhumb lines, great circle distance and haversine formula, reference ellepsoids and Vincenty's formulae, and ultimately tidal and local topographical and gravitational effects.

 

[pbuk]

Using Pythagoras I find the straight line distance between the two points.

Which is the distance in Euclidean 3-space: @biologist this is of course irrelevant to people living on the irregular surface of a rotating approximately ellipsoidal lump of mostly molten rock of uneven density.

All these features conspire such that unless you make many corrections, distances of more than, say, 100km can only be measured to within about 0.5%. And for almost all purposes this doesn't matter because when traveling between two points the uncertainty of practical factors (mainly wind and ocean current if relevant) is much greater than this.

 

[Baluncore]

Which is the distance in Euclidean 3-space: @biologist this is of course irrelevant to people living on the irregular surface of a rotating approximately ellipsoidal lump of mostly molten rock of uneven density.

If a measurement is made in a straight line, and it is repeatable, then it is meaningful and demonstrates that the measurement is well specified and understood. It may be irrelevant to you, but it is not irrelevant to the mapping of the country, engineering, or astronomy.

All these features conspire such that unless you make many corrections, distances of more than, say, 100km can only be measured to within about 0.5%.

International radio astronomy VLBI measures the position of the continents to better than 10 mm as they race across the face of the Earth at speeds measured in inches per year.
A distance of 100 km within a continent can be repeatedly measured to an accuracy of a 10 mm with optical surveying equipment, and verified by GPS. That is about 0.00001% .

Just because you don't need something, does not mean you should destroy it, or gaslight it's practitioners.

@pbuk What have you got against straight lines and geodesy ?
Are the computations too complex for you to understand, perform, or apply ?

 

[pbuk]

Fair challenges @Baluncore.

If a measurement is made in a straight line, and it is repeatable, then it is meaningful and demonstrates that the measurement is well specified and understood.

Agreed, provided that it IS well understood: the misconceptions in the OP illustrate well the fact that many aspects of geodesy are not generally well understood.

It may be irrelevant to you, but it is not irrelevant to the mapping of the country, engineering, or astronomy.

To the mapping of a country (which is usually relative to one or more secant planes) agreed, but I think it is a pretty specialised kind of engineering that uses straight lines over these kind of distances (particle accelerators sure, do you have any other examples?) and where does astronomy come into measurements between points on the Earth? (Hmmm, parallax baselines I suppose at a pinch, and of course VLBI - OK I'll give you that).

International radio astronomy VLBI measures the position of the continents to better than 10 mm as they race across the face of the Earth at speeds measured in inches per year.
A distance of 100 km within a continent can be repeatedly measured to an accuracy of a 10 mm with optical surveying equipment, and verified by GPS. That is about 0.00001% .

Yes, I didn't say what I meant (although I did of course mean what I said ). I should have said:

All these features conspire such that unless you make many corrections, distances of more than, say, 100km measured using map coordinates only relate to distances on the ground to within about 0.5% (see for instance https://www.cambridge.org/core/jour...-comparisons/E757B940C8C622A064276AC33CDC15C0)​

Just because you don't need something, does not mean you should destroy it, or gaslight it's practitioners.

I did not intend to do either and so I am sorry that that is the way it came across.

@pbuk What have you got against straight lines?

My inability to travel in them due to the factors I mentioned

Are the computations too complex for you to understand, perform, or apply ?

No, however I believe they are more complex than the OP realized, which is what I wanted to highlight.

 

[Baluncore]

... , but I think it is a pretty specialised kind of engineering that uses straight lines over these kind of distances (particle accelerators sure, do you have any other examples?)

Straight lines are used for triangulation and surveying because light travels in straight lines, even if it does wander up and down predictably. Curves are set out using straight lines.
Tunneling and building high speed railways and bridges involves triangulation using many straight line segments. Indeed, big engineering is deliberately bent to eliminate expansion and contraction in straight lines. The final engineering does not need to be straight, but the surveying does.

All these features conspire such that unless you make many corrections, distances of more than, say, 100km measured using map coordinates only relate to distances on the ground to within about 0.5%

Geodesy has used an ellipsoid for the last 200 years, not a sphere, nor a flat map that represents a spherical surface. The rhumbline is a crude approximation used by sailors to navigate between two points on the Earth, by using a fixed compass heading, and is in no way applicable to geodesy.

The paper you linked demonstrates that treating the Earth like either a flat sheet, or a sphere, or navigating by rhumbline, will give 0.5% errors compared to the truth of a great circle on the ellipsoid. That is good enough for a sailor.

It is the spherical assumption and the rhumbline that give the errors, not the geodesy. Geodesy provides the ultimate reference.

 

[biologist]

How would you define and measure an arc length between two points at different heights? You can really only measure the straight line distance, or reduce everything vertically to a reference surface.

This is why I asked for some sort of reference that would explain exactly how Google and Apple Maps determine distance. We're limited by the numbers available to us and current models of the Earth, but these companies have access to fairly advanced satellite technology that should be able to determine arc lengths over a distance of less than 100 miles.

I might have made a mistake assuming a publicly available satellite map would be accurate to .01 miles, but those are the numbers they provided. For example, it is known that one side of the base of the Louvre pyramid is .02 miles, and Apple Maps accurately gives this length.

Google Maps also has a "terrain" feature that provides elevation models. I've heard it said that their travel distance calculators also take elevation into account.

If you're asking how I would personally measure arc length over an ellipsoid Earth, I think if I had access to satellites I probably wouldn't find it too hard to model the contour of irregular surfaces by comparing the altitude of a particular satellite of known coordinates to its distance from the Earth's surface repeatedly. But this is just an idea.

In my example, though, the oblate ellipsoid model was taken into account by having all points over the same latitude. This ensures a steady radius at sea level (which apparently deviates from the Geoid by plus or minus 2-3 meters). If the points had been over the same longitude, for example, and different latitudes, then the arc length would require a more complicated integral equation to determine because radius changes with latitude.

I appreciate the discussion so far btw, great points.

 

[Baluncore]

We don't know what you are using the data for, so can't guess how much accuracy you might need.

If you can use a GPS receiver to get a location, or zoom in and get GPS from a map, then you might be better using those GPS LLH numbers to do the calculations separately, independent from the source of data.

If you want code that turns WGS84 GPS LLH into Earth centred XYZ, or XYZ back into LLH, it is available.

 

[pbuk]

This is why I asked for some sort of reference that would explain exactly how Google and Apple Maps determine distance

Google uses the haversine formula as mentionted above, see https://cloud.google.com/blog/products/maps-platform/how-calculate-distances-map-maps-javascript-api. I would imagine Apple is the same.

This does not take altitude into account and measures the great circle distance on a spherical geoid this gives a potential c.0.5% inaccuracy vs. the WGS84 reference ellipsoid (see reference in previous post). If we actually want to travel between two points then we have to take into account many more factors than this difference (or the difference in altitude) and so this is not a problem. If you want a more accurate calculation then you need to use more advanced GIS software.

these companies have access to fairly advanced satellite technology that should be able to determine arc lengths over a distance of less than 100 miles.

GPS technology determines position, not arc length, it is up to you how you determine the distance between the positions according to your need and as you can see there are at least 5 different calculations depending on your need (rhumb line, great circle, great ellipse, 3D Euclidean and route length).

I might have made a mistake assuming a publicly available satellite map would be accurate to .01 miles, but those are the numbers they provided. For example, it is known that one side of the base of the Louvre pyramid is .02 miles, and Apple Maps accurately gives this length.

There is a difference between accuracy and precision.

 

[pbuk]

I feel an Insight coming on something like this:

• Rhumb line distance
  This is the distance measured on an ordinary map, which assumes the Earth is flat. Over short distances this is a good enough approximation for most purposes and was the default before the age of computers. It is calculated as the 2D Euclidean distance on the flat map using Pythagoras' theorem.
• Great circle distance
  This is the distance measured on the surface of a spherical model of the Earth. It is more accurate than the rhumb line distance and just as easy for a computer to calculate so this is what is used most of the time by computer systems. It is calculated using the haversine formula.
• Great ellipse distance
  This is the distance measured on the surface of an ellipsoidal model of the Earth (GPS positions use the WGS84 reference ellipsoid). It is more accurate than the great circle distance by up to 0.5% however it cannot be calculated analytically and requires an iterative method (Vincenty's formulae).
• 3D Euclidean distance
  This is the distance between two points in space which for most practical purposes can be considered the length of a laser beam between those two points. For some purposes such as measuring the baseline for Very Long Baseline Interferometry (VLBI, an astronomical observation) this is the distance we are interested in. As we can now determine position very accurately with GPS, this distance can be calculated to a high degree of accuracy using (the 3D version of) Pythagoras' theorem.
• Route length
  If we want to travel between two points on land then the route we take is subject to a number of constraints: at least the topography of the terrain, but often mainly the path of roads or rails as appropriate. Calculating minimim route length is a potentially hard calculation involving summing the length of many distances which have often been measured by a vehicle actually traveling along each section of the path. The relevant distances and travel times for different modes of transport are also provided by services such as Google Maps.
• Distance run
  Ships travel through water that is moving due to tidal and other currents. Similarly aircraft travel through air that is moving with the added complication of different rates and directions at different altitudes. For a ship the actual distance traveled through the water is known as distance run (I am not sure if this measure is used for an aircraft where flight time is the more important navigational metric). This can be estimated in advance from knowledge of a ship's speed through the water and mapping of ocean currents provided by agencies such as the UK Hydrographic Office (UKHO) and the NOAA in the US, and is also the distance measured by the ship's log.

 

[Baluncore]

An unspecified problem has many possible solutions. Too many to list.