Calculating distance between two points using latitude longitude and altitude (elevation)

Question

I'm trying to calculate distance between two points, using latitude longitude and altitude (elevation).

I was using euklides formula in order to get my distance:

D=√((Long1-Long2)²+(Lat1-Lat2)²+(Alt1-Alt2)²)

My points are geographical coordinates and ofcourse altitude is my height above the sea. I only have lat and lng, I'm using GOOGLE API Elevation to get my altitude.

I'm developing an application which calculates my traveled distance (on my skis). Every application which I have used, gets distance traveled with included altitude. Like #Endomondo or #Garmin I cannot get my distance in 2D space because true distances are going to vary from the ones I've returned.

Which formula would be the best to calculate my distance ? Ofcourse with included altitude.

I'm writing my app in Python, with PostGis.

Ashmead

Why don't you store the coordinates in Postgis and use the ST_distance function? – Jacques

There are several articles that dispute the claim that Garmin and others use altitude in their calculations. trailhunger.com/info/articles/garmin-distance-calculation . In the equation you have there you are combining degrees and meters. – Microphysics

Hey, @Ashmead did you find any of the answers helpful? – Tevis

Answer 1

You can calculate distance between flat coordinates in, say, meters by using geopy package or Vincenty's formula, pasting coordinates directly. Suppose the result is d meters. Then the total distance travelled is sqrt(d**2 + h**2) where h is the change in elevation in meters.

Salimeter

Answer 2

I used the solution provided by John Moutafis but I didn't get a right answer.The formula needs some corrections. You will get the conversion of coordinates from Polar to Cartesian (x, y, z) at http://electron9.phys.utk.edu/vectors/3dcoordinates.htm. Use the above formula to convert spherical coordinates(Polar) to Cartesian and calculate Euclidean distance.

I used the following c# in a console app. Considering following dummy lat long

       double lat_1 = 18.457793 * (Math.PI / 180);
       double lon_1 = 73.3951930277778 *(Math.PI/180);
       double alt_1 = 270.146;

       double lat_2 = 18.4581253333333 * (Math.PI / 180);
       double lon_2 = 73.3963755277778 * (Math.PI / 180);
       double alt_2 = 317.473;

       const Double r = 6376.5 *1000; // Radius of Earth in metres

       double x_1 = r * Math.Sin(lon_1) * Math.Cos(lat_1);
       double y_1 = r * Math.Sin(lon_1) * Math.Sin(lat_1);
       double z_1 = r * Math.Cos(lon_1);

       double x_2 = r * Math.Sin(lon_2) * Math.Cos(lat_2);
       double y_2 = r * Math.Sin(lon_2) * Math.Sin(lat_2);
       double z_2 = r * Math.Cos(lon_2);

       double dist = Math.Sqrt((x_2 - x_1) * (x_2 - x_1) + (y_2 - y_1) *    
                               (y_2 - y_1) + (z_2 - z_1) * (z_2 - z_1));

Chuckwalla

This seems to only account for the radius of Earth in r, but should also include the elevation above Earth's surface for each point, e.g. r should be different for points 1 and 2 such that r1 = r + alt1 and r2 = r + alt2. With that modification, this seems more intuitive than the @John Moutafis's answer (being most directly comparable to the Haversine calculation for XY due to the assumption of a spherical Earth in this answer), although they may both end up being mathematically correct. – Vaivode

Answer 3

EDIT 2019: Since this answer, I composed a Q&A style example to answer similar questions (including this one as an example): How to calculate 3D distance (including altitude) between two points in GeoDjango.

In sort:

We need to calculate the 2D great-circle distance between 2 points using either the Haversine formula or the Vicenty formula and then we can combine it with the difference (delta) in altitude between the 2 points to calculate the Euclidean distance between them as follows:

dist = sqrt(great_circle((lat_1, lon_1), (lat_2, lon_2)).m**2, (alt_1 - alt_2)**2)

The solution assumes that the altitude is in meters and thus converts the great_circle's result into meters as well.

---

You can get the correct calculation by translating your coordinates from Polar (long, lat, alt) to Cartesian (x, y, z):

• Let:
  polar_point_1 = (long_1, lat_1, alt_1)
  and
  polar_point_2 = (long_2, lat_2, alt_2)

• Translate each point to it's Cartesian equivalent by utilizing this formula:

  x = alt * cos(lat) * sin(long)
  y = alt * sin(lat)
  z = alt * cos(lat) * cos(long)
  

  and you will have p_1 = (x_1, y_1, z_1) and p_2 = (x_2, y_2, z_2) points respectively.

• Finally use the Euclidean formula:

  dist = sqrt((x_2-x_1)**2 + (y_2-y_1)**2 + (z_2-z_1)**2)
  

Tevis

what unit of measure would this distance be in and how would you go about converting it back into say meters?? – Ellery

@Ellery Since we are using Google Elevation API, alt is in meters, therefore the dist will be in meters. – Tevis

ok cool so as long as my altitude is in meters i can just take the value of dist and it will be in meters? – Ellery

@Ellery Yes that is the case (You will have the same units as your alt variable)! – Tevis

Shouldn't the altitude be modified to account for earth radius when converting to Cartesian? I.e. use alt+earth_radius in the conversion equations? – Selene

@Selene Since the initial answer I have strudied better and came up with a better solution. Have a look at the edited answer and the linked Q&A as well. – Tevis