Plotting locations by GPS

Question

If I know the distance each of five points is to another

and for only one of those points I know the Lat & Lon (GPS)

based on that single point, how can I determine the Lat & Lon for for all points?

 

For example, I have 5 people I need to know where they are.

For only one of those people I know the exact location by GPS and therefore I know the Lat & Lon.

Since the other four are in a building, I have no GPS but I can determine the distance between each of them, including the distance to the one where I do have Lat & Lon data.

 

Can I calculate the relative position of each person based on their distances, one to another?

Answer 1

Draw 4 circles around each of the 5 points showing how far each of the other 4 points are.

Use the intersections of the circles to place the points relative to each other.

Then from the known point, determine the coordinates of each of the other 4 points.

 

I suspect you would not want to do this by drawing circles on a map.  You probably will want to do this by writing equations for the circles and therefor want to put everything into a matrix and use linear algebra to make things simpler to solve.

 

The unknowns will be the x and y coordinates for each of the 4 unknown points - 8 unknowns.  If you indeed have all the distances for Each one to Each of the others you will have 4+3+2+1 = 10 known distances plus the two known x and y coordinates of the known point.  12 knowns and 8 unknowns - plenty to solve the system of equations.

 

I am assuming the distances you know to each are horizontal distances since we are dealing only with 2 dimensions, Lat and Lon.  Knowing a 3 dimensional distance (if they were on different floors or in buildings of different heights) would mess things up. - You would have to draw spheres around each person instead of circles and then work all your equations in 3-space.

 

In 3-space case you will have 12 unknown x,y,z coordinates (Lat, Lon, Alt) and your system should still be solvable - but harder - since you will still have 10 known distances and now 3 known coordinates (Lat, Lon, Alt) - i.e. 12 unknowns and 13 knowns.

 

I hope this is enough to get you rolling.  I really don't want to write those equations - good luck :)

 

btw - you could google gps and you will find more ;and might even find the equations in at least a general form.