Today’s essay is going to be the first of a series. We’ll start by looking at and discussing the basic fundamentals of navigation principles: how to look at the world around you and use landmarks you can see to determine exactly where on a map your position is. We’re then going to dig into what you do with positions, and maybe even detour into some of the history and application of modern navigation systems.
The first concept I need to introduce is simple: a position fix, or just fix for short. A fix represents a location; a single point on your chart which signifies where you are. We obtain our fixes by determining where we are in relation to one or more landmarks, which can be whatever’s handy and visually distinct: a mountain peak, a tall radio tower, a lighthouse, a buoy in the harbor.
The second concept is a bearing. A bearing is a measure between yourself and the landmark you’re using to obtain your fix, represented as an angle within circle of three hundred and sixty degrees with you in the center. For any point where you might possibly be on earth, a bearing of 0° will represent precisely north, a bearing of 180° will be precisely south, a bearing of 90° will be east, and a bearing of 270° will be west. (This holds true unless you’re on one of the poles; you can’t go north or east or any other direction but south from the north pole, and you can’t go anywhere but north from the south pole, so bearings break at that point.)
Bearings are similar to the “o’clock” method, where you envision yourself at the center of the face of a clock, with 12:00 being north or ahead of you, 6:00 being south or behind you, and so forth. In fact, it’s not just similar, it is precisely the same system; each hour is 30 degrees of arc. An object at your one o’clock is at a bearing of 30 degrees; an object at your six is at a bearing of 180 degrees.
Third is the concept of a reciprocal bearing. This is exactly the same as a bearing, but it’s the opposite direction. If a landmark is due west of you, which is to say it’s at a bearing of 270° from you, then you know that you are due east of it at a reciprocal bearing of 90°. You can express this mathematically by either adding 180° (if the bearing is less than 180°) or subtracting 180° (if the bearing is 180° or more) from the bearing. A bearing of 208° yields a reciprocal bearing of 28°. A bearing of 19° yields a reciprocal bearing of 199°.
Okay. So we have one or more landmarks, and we have the concept of bearings. How do we turn them into a fix? There are three methods, and which one you use will be determined by the information you have available to you regarding those landmarks.
The first and simplest method is the bearing/range method. If you’re standing on the roof of an unknown building and know only that: a) a tower is due east of you at a bearing of 90°, and b) it’s precisely one kilometer, then congratulations! Having a range and a bearing to a known position is a fix all on its own, so you already know your position and there’s nothing more to do here.
Usually it’s more complicated than this, though; we often don’t have ranges available in navigation until after we’ve got our fix, at which point we determine the range by measuring the distance between our position and the known position of the landmark.
The second method, which is far more common, uses bearings but no ranges. You need two or more landmarks instead of one, and you need to measure the bearing to each, such as with a compass. That radio tower at 90° is the first; you can draw that yellow line beginning at that radio tower and going directly west (remember that reciprocal bearing? The reciprocal bearing of 90°, or east, is 270°, or west, so that’s the direction we need to go to find our own position from the tower.) Then you take a bearing to a different landmark - an especially large and prominent tree somewhere to our northwest, say - and determine that the tree is at a bearing of 318°; draw a line on your map beginning at the tree and proceeding at a reciprocal bearing of 138°. Where your two bearing lines cross, X marks the spot… and now you know where that saying comes from. Two landmarks is enough to make the X, but usually we use three or more for greater precision, especially when the bearings are close together instead of spaced widely.
The third method uses only ranges and no bearings at all, and because of that it’s used even more rarely than the first case (with one extremely important exception which we will certainly discuss later.)
We know from geometry class that a circle is defined as all possible points on a two-dimensional plane which exist at a given distance from a center point. So envision your radio tower, which you know from previously is exactly one kilometer away from you, as the center point of a circle whose radius is one kilometer. Draw that circle on your mental map. Then take that tree, which it turns out is 0.4 kilometers away, and draw a second circle around that tree with a 0.4km radius. These two circles must intersect, because if they don’t, then they do not accurately describe your position, which means one or both of your two given ranges is invalid.
It’s possible that they will intersect at exactly and precisely one point, in which case you’re done and you’ve got your fix, but far more likely is that they’ll overlap somewhat, like a Venn diagram, and intersect at two points.
So how do we figure out which of those two points is our position? With a third landmark and range, of course: the mast of a ship in the harbor, it turns out, is also exactly 1km away from us. Once again, the third range circle must have exactly one point where it intersects with both others, or the ranges you have are invalid for your position.
There you have it. Not one, not two, but three methods for obtaining a position fix.
However, everything we’ve discussed so far is essentially working on a two-dimensional plane. The ship, the tower, and the tree are all represented as one specific point with no dimension of height. That works for this simple case, where we use a flat two-dimensional map that we’re also on the surface of. There’s one problem with that approach: What if you’re not on the surface? What if you’re aboard an aircraft?
We could add additional angles to see how high a landmark is above the horizon, or how far below us it is, and in fact that’s one of the measurements an Age of Sail navigator would take with his sextant in celestial navigation. But then we have to start doing a whole bunch of fiddly math with those angles, and the amount of time required becomes problematic because your aircraft probably isn’t a hovering helicopter. If it’s moving, the longer you take to get your fix, the more out of date each of the angles are by the time you’ve finished recording them, and your fix is degraded by the time you have it.
Remember how I said that there was one important exception to the technique where you use only ranges and no bearings to calculate your fix? What if you had radio beacons which transmitted extremely precise timing signals? Based on the amount of time it took you to receive that signal, and the known speeds of RF energy propagation through air (or space), you could figure out exactly how far away those beacons were, and use the locations of the beacon emitters to set up your fix.
In the range-only version of two-dimensional navigation, we started out with one circle (all possible points at a specified range from a specified point on a 2D plane); added a second circle to narrow it down to two points; then finally added a third circle to narrow down that two points to one point. We can do the same thing in 3D space: start out with a sphere (all possible points at a specified range from a specified point within 3D space); add a second sphere to narrow it down to the circle where those two spheres intersect, add a third sphere to narrow down that circle to two points, and then a fourth sphere to eliminate one of the two points, confirming the other as our position.
But where can we get a pile of radio beacons transmitting extremely precise time signals from known locations in three-dimensional space?
From the United States military, of course.
That story, while fascinating, is a whole side story of its own and a major tangent from this discussion of basic navigation principles. If people are interested, I may go more in-depth on that later.
To be continued…