In this post, I show a variation on the theme that allows a person to assess his or her position from a single LOP and a range.
So, first I have to say something about range. While the bearing is the azimuth (or angle from north) to an object, the range is the distance to that object. It makes sense that if you have an LOP from the object and you have a range, then you have effectively determined your position.
Let's talk about ways of determining range. Keeping in the theme of on-the-go navigation, you have a limited set of tools available to determine range. Probably the simplest is the use of a hand at the end of an outstretched arm. You may have seen some sketch of a landscape painter in front of an easel with a thumb raised at the end of an arm. This trick is simply a way of measuring the angular width of a feature a feature that he or she is painting to get the relative dimensions correct.
Let's say you're looking at a house from some distance. This is illustrated in the figure below. You hold up your hand at an arm's length and what you see is that the angular height of the lower two stories of the house (i.e. discounting the roof) spans the same width in your visual field as the angular width of your finger, which for me (and most people) is about 1.4 degrees.
The view of a house in the distance and your finger at the end of an outstretched arm in the foreground.
Now, if we know one more piece of information - the height of the house - we can figure out our distance to the house. How? Well, as illustrated below the triangle formed by the eye and the width of the finger is similar to the triangle formed by the eye and the first two stories of the house.
Similar triangles formed by eye-finger width, and eye-house height. Click on the figure to enlarge.
A detail of the eye-finger width triangle is shown below:
Eye-finger width triangle.
Since the triangles are similar, this means that the angles are the same and the ratios of the lengths of the sides are the same. What's more for small angles, there's something called the small angle approximation. The small angle approximation means that you can do away with trigonometry and just deal with simple ratios, which makes it easy to figure distances. What is a small angle? Well, generally 10 degrees or less, but you can extend it up to 20 degrees and no suffer too much error.
There is a nominal conversion factor that you can remember easily:
100 feet at 1 mile spans 1 degree.
To see what I mean, refer to the example of the house above. The 100 feet in the above statement refers to the physical size of an object at one mile. The one degree is its angular width. By holding up the finger at arm's length, we get a kind of calibration of angle.
How does a difference in heights, distances or angles alter the above statement?
1.) If the object is taller than 100 feet, it spans a wider angular width at 1 mile, smaller, it spans a smaller angular width.
2.) Objects have a smaller angular width at larger distances and a larger angular width at smaller distances.
We can convert this into some nominal equation:
Range (in miles) = [Height (ft)/100] *[1/angle (degrees)]
Now, for the view of the house above, we know something helpful - each story in a house is about 10 feet high, give or take. Since the finger is covering two stories, the angular width of 1.4 degrees is covering 20 feet. So, 20/100 is 1/5th. Multiply this by 1/1.4 (degrees), and you get 1/7th of a mile, or about 700 feet.
This may sound painful as a calculation, but I usually can do this in my head when I'm out hiking or backpacking - it's something that you can practice on just about any place you see an object in the distance. The fact that the above formula is just ratios reflects the small angle approximation. I don't have to throw in tangents or anything as long as the angle is less than 10 degrees, but even up to 20 degrees the equation works pretty well.
Below is a selection of angles associated with different configurations of my fingers and hand at the end of my outstretched arm.
Angular widths of combinations of fingers and hand at the end of my outstretched arm.
Now, you might be wondering if you can get away with the above figure for your own arm and fingers. The answer is "yes", since there is not a lot of variation among individuals. A lot of things cancel out in the eye-to-hand triangle. Take the differences between men and women. On average, women may have narrower fingers, but they also have shorter arms, so the angular width of a woman's finger and a man's finger are very nearly the same, and the variations among individuals may be larger. In testing a large number of students, I've found little spread from the above chart. If, however, you really want to find out your own angles, you can mark off distances on a chalk or a white board and stand some distance away and figure out the combinations by measuring them yourself.
So, that's the angle measuring trick to find range, but it's helpful to know the heights of a hill or structure you might be sighting in the distance. Already, the 10 feet = 1 story in a house is helpful. On charts or topographic maps, the heights of hills are well marked. The heights of steeples and light houses are also marked, so this gives some easy heights to find.
You might also be able to use the width of an island that you can measure directly off the map as a way of establishing the physical size of an object in the distance.
There are also some nominal guesses one can make. Lighthouses are rarely shorter than 50 feet, and rarely taller than 200 feet, so if you saw a lighthouse and guessed 100 ft, you'd get a range that's probably correct to within a factor of 2. Water towers are also in the 100-200 ft range, and steeples for churches are in the 50-100 ft. range.
Now, we're ready for an exercise where we combine a line of position with range to find a location. Let's say you're out paddling in Boston Harbor and you see the scene below. It's the Boston Lighthouse, which is 100 feet tall. You hold up your finger and see the scene in the figure. Let's say you also take a bearing to the lighthouse with your compass and it reads 350 degrees (magnetic). Where are you?
Scene of finger being help up, looking at the Boston Lighthouse.
The first step in using this information is establishing a line of position. Since the bearing to the lighthouse is 350 degrees, the back-bearing is 170 degrees. This is shown graphically on the figure below, using the compass to draw a line of the back-bearing, which is the line of position. You lie somewhere along that line.
Drawing a line-position using the back-bearing from 350 degrees.
Next we need to figure the distance. The figure above with the finger and the lighthouse shows that there are some judgement calls involved and the technique has its imprecisions, but as long as we're aware of the uncertainties, we can cope. On the chart, the height of the lighthouse is given as 102 feet (this can be seen on the figure below if you blow it up by clicking on it). The height is typically the height above the high water mark. Since my finger is 1.4 degrees wide, I'm going to call the angular height of the lighthouse a little over 2 degrees. From the formula above, this gives a distance of 1/2 of a mile from the observer and the lighthouse.
If we go back to our line of position, we can then measure off half a nautical mile (1.15 statute miles), and plot this along our back-bearing line. This is shown in the figure below.
Mark off the range of 1/2 mile from the lighthouse along the line of position to find your position, noted by the arrow (click on figure to enlarge).
From the figure above you now have your position using your range to the lighthouse and the single line-of-position. Note that your position in this case is just to the north of a major shipping channel, which might give you reason to move a bit toward the lighthouse to make sure you're clear of traffic.
In the next post, I deal with the question of whether one quotes bearings in true or magnetic and how to make the conversions.
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