CODEWITHSUNDEEP
mathgeolocationgpsgeospatial
IElite
IElite

Reputation: 1848

Determining Waypoint from series of waypoints and distances

Giving a distance an a waypoint, how to i determine the new waypoint?

106642.947 feet from N 43° 11.543 W 073° 39.451 38425.157 feet from N 42° 50.883 W 073° 53.350 22804.598 feet from N 42° 54.613 W 073° 41.477 89405.494 feet from N 43° 08.800 W 073° 52.700 52595.477 feet from N 42° 47.361 W 073° 40.521 63324.857 feet from N 43° 03.150 W 073° 55.050 252303.651 feet fromN 43° 32.983 W 073° 24.283

No directions/headings or (azimuths)

Answer is great! Hints are good, your choice how you want to answer it

this is not homework, this is a g e o c a c h e p u z z l e

Upvotes: 0

Views: 658

Answers (1)

Eric O. Lebigot
Eric O. Lebigot

Reputation: 94485

Here is a first idea: since the distances are small compared to the radius of the earth (about 1/300 of the radius), you can simply use a map of the region you are interested in (with the same length scale in x and y) and draw circles with the given distances around the central locations you list. The point where they intersect gives you the location you want. You might want to draw this on a computer, on a grid with the coordinates.

If you want more precision, another idea is to calculate with a computer the intersection of only two of the four circles involved: the point you look for is one of them. By drawing one or two additional circles, you can see which of these two intersection points is the correct one.

Approximating the earth as being flat should give you errors of the order of 50,000 feet / 300^2 = 0.5 feet. This is because the mapping between longitude/latitude and a flat map is an odd function which has a correct first order term at the origin (by definition). Thus, corrective terms are two orders larger than the main term (the main term is first order, the next one is third order). So, since the correction to distances is expressed in powers of (circle radius)/(Earth radius), your error should be of length about (circle radius) / 300^2.

PS: here is how you can concretely do the drawings "manually" with the help of a computer and only some basic software:

  • First convert all your latitudes/longitudes in radians, as this makes mathematical operations simpler.
  • You determine a point which is "in the middle" of all your points, so that you know where to center your map; you can for instance take the barycenter of your longitude/latitudes. The latitude/longitude of this point are denoted by (l0, m0).
  • You can then calculate the approximate projected locations of all your 4 centers of latitude/longitude (l, m) in meters by neglecting the curvature of the earth and using the average Earth radius (this radius R is between 6,353 km and 6,384 km), like so: x(l, m) = R*cos(l0)*(m-m0) and y(l, m) = R*(l-l0). You can draw these locations on a map scaled in meters, on a computer (Inkscape, Mathematica, Illustrator, Matplotlib, etc. all can do that; or you can simply use a sheet of paper).
  • You can draw circles around these points, of the appropriate radii.
  • You then simply need to convert the location (in meters) of their intersection into latitude/longitude, by solving the x/y(l,m) functions above for (l, m). This gives you the latitude/longitude of the point you need.

Upvotes: 1

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