So, if we make the planar assumption, here’s the method

So, if we make the planar assumption, here’s the method for converting points with coordinates in latitude and longitude to points with coordinate in more familiar x and y.

Choose a reference point and find its lat/lon coordinates. Any point that is close to your operational area will work nicely, but it’s particularly convenient in the Northern Hemisphere to choose a point that is southeast of your operational area. That sets things up so that all of your x,y coordinates will be positive.

Find the relative coordinates of any point in your operational area by subtracting the coordinates of your reference point.

Calculate the conversion factor for changes in latitude and longitude to changes in distance at the reference point.  We’ll use the arc length formula for this, so s=rθs=rθ , where s is distance, r is radius, and θθis the angle.  To get a conversion factor it’s easy to just use θ=1∘θ=1∘. For latitude, r is just the radius of Earth since the distance between lines of latitude does not change.  But for longitude, r=rearthcosθlatr=rearthcos⁡θlatwhere >θlatθlatis the latitude of your reference point. So once you do this calculation you’ll end up with two different values of s, one for latitude and one for longitude.  These are your conversion factors, so let’s call those slatslatand slonslon

Take the relative coordinates that you found in step 2 and multiply each one’s latitude by slatslatand it’s longitude by slonslon, and that will give you the coordinates of each position in your local, planar coordinate system, which can be used by vehicle instruments and planning software and to which you can apply all of your familiar geometric and trigonometric relationships like the distance formula or course and bearing calculations.

Now that we’ve stepped through the process, practice applying this technique in the following exercise:

We’ll return to our shipwreck from Activity 4.2. Assume that to conduct your survey of the site you want to program an AUV to transit along the triangular path defined by the waypoints listed in the following table and return to Waypoint 0 at the end. Use the techniques defined above to fill in the missing data. Remember that you get to choose the reference point, so it can be anything you want that is in or near your operational area. Submit a spreadsheet with at least this completed table and the numbers that you calculated for s_Lat and s_Lon. You are not required to show your work, but the easiest way to complete this assignment is to set up a spreadsheet that will do the required calculations, and that also documents the process that you used so that it is easier to give partial credit and to understand your approach.

The formulas in this activity are taken from Chapter 24 of The American Practical Navigator, which is freely available for download from the National Geospatial Intelligence Agency’s website: https://msi.nga.mil/Publications/APN