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We’ve been working on a way to calculate the distance to a tree situated blocks from our viewing point. Flying raptors, such as our beloved bald eagle, wouldn’t find this in the least bit challenging. They’re able to accurately judge distances due to their special shape-shifting eye lenses which are capable of actually changing curvature spontaneously. But human physiology isn’t equipped for this task, so we’ll have to employ other methods. Today we’ll see how a branch of mathematics known as trigonometry comes into play.
Referring to the illustration, we’d like to calculate the distance, r. You’ll note that a right triangle is formed by Line of Sight A, Path AB, and Line of Sight B. As mentioned in our last blog, right triangles are special because the relationship that exists between their sides and their internal angles is well defined within mathematics. In fact, we can calculate r by using this trigonometric formula, r = d × tan(θ) where d is the length of Path AB, θ is the angle between Line of Sight B and Path AB, and tan(θ) is the trigonometric function known as the tangent. Tangent, and other trigonometric functions like sine and cosine, relate the angles in a right triangle to the ratios of the lengths of the sides of the triangle. If we know two of the variables present in the equation presented above, we can determine the third, and the fact that there’s a right triangle present makes that task so much easier. As things stand now we have two unknowns, d and θ. As pointed out last week, the distance, d, is short, so we’ll use a tape measure to determine its length. Let’s say it measures out to be three feet. Now we need to solve for the angle θ that’s formed between Path AB and Line of Sight B. That’s a bit more challenging. There are a number of devices that can be used to measure θ, including a handheld magnetic compass. However, using a compass often yields inaccurate results, thereby increasing the likelihood of mistakes. A more accurate device to use would be an optical rangefinder, as shown below.
The optical rangefinder is a device that’s often used in the military to measure long distances by using the principle of parallax. It functions much like binoculars do, but with a twist, literally, as we’ll see next time. ____________________________________
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Posts Tagged ‘distance measurement’
Parallax and Trigonometry
Monday, January 26th, 2015
Using Parallax to Measure Distance
Monday, January 19th, 2015|
Last time we introduced an optical effect known as the principle of parallax. Today we’ll get a step closer to seeing how it’s instrumental in measuring distances. For our example we’ll be viewing a point on a tree several blocks away.
Since we can’t measure the distance, r, to the tree directly by using a tape measure, we’ll have to use another approach that is made available to us through the principle of parallax. The first step to doing this is to establish two different lines of sight to the red dot on the tree. For our example these will originate at Points A and B, which are both viewing the same spot. Line of Sight A extends straight from Point A to the red spot, while Point B, which is situated only a few feet to the right of A, will provide us with a slanted line of sight to it. We’ll refer to the path between Points A and B as Path AB, which is represented by a black dashed line and has a length, d. You may have noticed from the illustration that Line of Sight B, because it lies on a slant relative to Line of Sight A, provides us with an internal angle, θ, relative to Path AB. This relationship is in fact required to exist if we are to use the principle of parallax, and you’ll see why later. Another requirement is that Path AB must be at a right angle to Line of Sight A, which it is. The illustration bears this out by the fact that a right triangle is formed when Lines of Sight A, B, and Path AB intersect. Right triangles are generally regarded as special within mathematics, but they are particularly special when applying the principle of parallax. The presence of a right triangle enables us to use a branch of mathematics known as trigonometry to correlate the triangle’s angles to the ratios of the lengths of its sides. In other words, we can calculate r by using trigonometry if we are able to measure d and θ. We’ll see how that calculation is done next time. ____________________________________
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