So how do you determine where these 16 extreme compass points are located? I'll describe my methodology here, which I worked out while I was still in Qatar planning this trip. As it turns out, it’s pretty simple.
First of all, here's a compass rose. Note that 0 degrees (or 360) is at the top, indicating north and 180 degrees indicates south. Each of the 16 compass points, in fact, corresponds to a specific bearing. For instance:
Direction  Bearing  
North  0  
Northeast  45  
East  90  
Southeast  135  
South  180  
Southwest  235  
West  270  
Northwest  315 
Now here's how to figure out the extreme points:
1. Determining the Extreme Cardinal Points
Most people can easily figure out on a map the extreme cardinal points, like the northernmost point, the southernmost point or the westernmost point of the United States. It’s easy. You can do this either graphically (i.e., by looking at a map) or numerically (i.e., by calculating the longitude and latitude values for each point on the perimeter of the United States).
To calculate an extreme point graphically, all you do is:
 Figure out what point you want to identify (say, the northernmost point of the Lower 48 states),
 Draw a line, either imaginary in your mind or actually on a map, that’s perpendicular to that direction. If you use a map, I recommend using the Mercator projection, since in a Mercator projection, all directions are true. So in this case, you’d draw a line that runs eastwest, since you're trying to determine the northernmost point.
 Now move that eastwest line towards the United States until it touches the first point on the United States perimeter. That’s it: you've determined the northernmost point of the United States.
To calculate this same point numerically, without using any maps or illustrations, all you do is:
 Calculate the longitude and latitude of every point on the perimeter of the United States and put it into a table or spreadsheet (you can use Geographic Information Systems, or GIS, software to quickly do this).
 Look through the table for the point with the highest value of latitude (in the case of the northernmost point). Or you look for the corresponding value, say the lowest value of latitude if you’re trying to determine the southernmost point, or the smallest value of longitude (i.e., largest negative value) if you’re trying to determine the westernmost point.
OK, so that’s simple. Either way, graphical or numerical, it’s easy to find an extreme cardinal point, such as the northernmost point or westernmost point of the contiguous United States.
2. Determining the Extreme Ordinal Points
But what about the ordinal directions, like northeasternmost or southwesternmost point? How do you figure out the “southwesternmost” point of the United States? As it turns out, you use the exact same methodology as above.
In the case of ordinal directions, both directions (in this case of "southwesternmost," the directions of south and west) have exactly the same weight or value. “South” is not more important than “west.” This is true because if you look at a compass, the direction of “southwest” is exactly in between the directions of south and west. The southwestern direction isn’t more “south” than “west,” just as the northeastern direction isn’t more “north” than “east.” Both directions have exactly the same weight.
So to determine an ordinal direction, such as the southwesternmost point of the United States, you use the same methodology as you used to determine the extreme cardinal direction points. So in the case of the southwesternmost point:
 You draw a line that’s perpendicular to that direction. If north is 0 degrees and south is 180 degrees, southwest has a bearing of 225 degrees. So in this case, you’d draw a line that runs from the northwest to the southeast. In other words, the line would have a bearing of 135 degrees (i.e., to the northwest or southeast).
 You move that line towads the United States until it touches the first point on the United States perimeter, just as before. That point is the southwesternmost point of the United States.
You can also do this calculation numerically rather than graphically to determine the extreme ordinal location. In that case, you would add the x/y coordinate values of each point together and the point with the highest combined value is the extreme ordinal point. Before adding them together, though, you might first have to perform a transformation depending on the direction, since latitude gets smaller as you go south and longitude gets smaller (or more precisely, more negative) as you travel west. Therefore,
 If you wanted to determine the extreme southwest point, you would add the coordinates of each point on the perimeter of the United States together and look for the smallest combined value
 If you wanted to determine the extreme northeast point, you would add the coordinates of each point on the perimeter of the United States together and look for the largest combined value
 If you wanted to determine the extreme northwest point, you would first reverse the longitude values (e.g., 124.3 degrees becomes 124.3 degrees) and then add the coordinates of each point together and look for the largest combined value.
 If you wanted to determine the extreme southeastern point, you would first reverse the latitude values (e.g., 36 degrees becomes 36), add the two coordinates, and the smallest combined value (i.e., the largest negative value) is the southeasternmost point.
So my “point” (heh, heh) is that you can calculate the ordinal extreme points either graphically or numerically, just as with the cardinal directions. And you should theoretically come up with the same point as being the most extreme regardless of which way you calculate it. If you come up with different points using the two methods, then something’s wrong.
Right: Many people mistakenly think that San Diego is the southwesternmost point in the United States. To determine the southwesternmost point, 1). Draw a line that's perpendicular to that direction (i.e., bearing 135 degrees, or in other words, from northwest to southeast), then 2). Move that line towards the U.S. until it touches the first point. If you draw a line bearing 135 through San Diego (blue point and blue line), you'll see that there are many other places in California farther southwest than San Diego, including Santa Barbara. In fact, the southwesternmost point in the U.S. is an unnamed point near Lompoc, California (red line and red point). You can use this same approach to calculate any extreme compass point in the U.S.

3. Determining the Extreme Interordinal Points
You use the same methodology to determine the extreme interordinal points, such as the southsoutheasternmost point or the westsouthwesternmost point. Again, you can do this either graphically or numerically and both ways should yield the exact same location. To summarize:
 To do it graphically, you draw a line perpendicular to the point you’re looking for, just as before. So for instance, if you’re trying to determine the northnortheasternmost point (northnortheast is an angle of 22.5 degrees), you would draw a line that has a bearing of 112.5 degrees/292.5 degrees, which is perperpendicular to your bearing of 22.5 degrees. Then move this line towards the United States until it touches the first point of the United States perimeter. That’s your extreme point.
 To do it numerically, you would add the coordinates of each point to determine the point with the largest value. But in this case, since you’re trying the determine the northnortheasternmost point, you would first double the value of each latitude coordinate and then add it to the corresponding value of longitude for that point, and repeat this for every point on the perimeter of the United States. If you were trying to determine, say, the westsouthwesternmost point numerically, you would double the value of each longitude value and then add them to the latitude value.
But again, you should come up with the exact same interordinal extreme point regardless of whether you’re doing it graphically or numerically.
Using my GIS software, I performed these calculations in Qatar  both numerically and graphically  as I was planning my trip was thus able to determine the 16 extreme compass points of the United States. For each of the 16 points, both my numerical calculations and the graphic method yielded the same exact location.
The term "southwesternmost" or "eastnortheasternmost" may be a bit fuzzy or nebulous to some folks. But to me, the southwesternmost or eastnortheasternmost points of the United States are just as clearlydefined, exact and identifiable as the northernmost or westernmost points. The exact same method is used to identify all 16 extreme points.