We left off with an artillery soldier spotting an enemy tank in the distance. Luckily, he’s got an optical rangefinder at his disposal to measure the distance between them and crank out an accurate shot. His first action is to peer through the rangefinder’s eyepieces, rotating the adjustable mirror on the right eyepiece until the tank comes into focus. The two lines of sight provided by the left and right eyepieces of the rangefinder form a right triangle due to the parallax effect. One of the angles in this triangle is designated on the illustration by the angle θ, and that’s the angle we’ll be working with. The fact that a right triangle exists means the distance, r, to the tank can be easily measured using principles found in trigonometry, a branch of mathematics that addresses the properties of triangles, hence, the prefix tri in its name. Tangent, and other trigonometric functions like sine and cosine, relate the angles of a right triangle to the ratios of the lengths of the sides of the triangle. In our case we’re concerned with the tangent, which is simply the value that’s derived by dividing the length of the side opposite to the angle θ by the length of its adjacent side. This value can be found in most trigonometry textbook tables, but is most easily found by using a calculator. Simply enter the angle θ‘s value, then press the TAN button. So how does the artillery soldier determine θ‘s value? With the tank in clear focus, it’s easily measured with an indicator gauge attached to the adjustable mirror near the right eyepiece on an optical rangefinder. Our soldier reads the gauge and determines that θ is equal to 89.935°, so the tangent of θ is equal to: tan(θ) = tan(89.935°) = 881.473 Now that we’ve determined the values for d and the tangent of the angle θ, we can plug those numbers into our equation to determine r, the distance to the enemy tank using the equation, r = d × tan(θ) Plugging in numerical values, the equation becomes, r = 3 feet × 881.473 = 2644.419 feet The distance to the tank is 2644.419 feet. Next time we’ll see how the peculiarities of the tangent function place limitations on the accuracy of optical rangefinders over extremely long distances. ____________________________________

Archive for February, 2015
How An Optical Rangefinder Uses Trigonometry
Monday, February 23rd, 2015What’s Inside an Optical Rangefinder
Friday, February 13th, 2015
Last time we covered the mirrors found inside an optical rangefinder. Today we’ll discuss its other components, magnifying lenses E and F and the indicator gauge, G. See Figure 1. Figure 1 Magnifying lenses E and F are positioned in such a way that the two horizontal lines of sight reflecting off mirrors C and D pass through them on their way to mirrors A and B. Lenses E and F are adjustable, allowing the viewer to manipulate them until the distant image being viewed is seen in clear focus through both eyepieces, just as would happen with binoculars. To see the rangefinder in action let’s go out to the battlefield, where it’s routinely used. A soldier notices an enemy tank off in the distance. To make an accurate hit with his artillery he must determine the distance r from his position to the tank. The soldier in charge of this task observes the tank through the rangefinder’s eyepieces and manually adjusts lenses E and F until it comes into focus. He then adjusts the rangefinder’s mirror B until the line of sight extending from it converges on the tank along with the fixed line of sight from mirror A as shown in Figure 2. When the two images converge a right triangle is formed, and as we learned in a previous article, this fact will enable us to calculate r by using trigonometry with this formula: r = d × tan(θ) We already know that the value of d, the distance between mirrors A and B, is three feet. All that is left to be determined is the angle θ that’s formed between adjustable mirror B‘s forward line of sight out the tube’s opening and the horizontal line of sight between D and B. The last component within an optical rangefinder to be discussed is the indicator gauge, notated as G in Figure 1. This gauge is attached to mirror B inside the tube and is used to measure the angle θ that’s produced as the mirror is adjusted. When the lines of sight from mirrors A and B converge and the tank comes into clear view through the eyepieces, the value of θ can be read on the gauge’s numerical graduated scale, which is located on the exterior of the tube. Next time we’ll see how trigonometry and the study of right triangles prove indispensable when measuring distances with an optical rangefinder. ____________________________________

The Mirrors Inside an Optical Rangefinder
Friday, February 6th, 2015
Last time we touched on the fact that humans require instruments to facilitate the optical measurement of distance to faraway objects, such as is represented by r in Figure 1. Figure 1 Today we’ll take a look at the mirrors inside an optical rangefinder, one of the devices that’s commonly used to measure great distances. The internal workings of an optical rangefinder are shown in Figure 2. Figure 2 The rangefinder’s body is a straight tube containing fixed mirrors, A, C, and D, and an adjustable mirror, B. Mirrors A and B in Figure 2 are analogous to Points A and B in Figure 1. For the sake of our example, we’ll say that the distance d in Figure 2 is equal to the distance d in Figure 1. This distance is three feet. The rangefinder functions similarly to a pair of binoculars, but with a twist — literally. Its four internal mirrors redirect the straightahead perspective that’s provided by the eyepieces into sideways orientations. Sight is deflected by mirrors C and D off to the left and right, respectively, as represented by red dashed lines. Finally, the lines of sight are once again aligned into straightahead orientations as they reflect off mirrors A and B and are guided out the openings at either end of the tube. Adjustable mirror B‘s line of sight is positioned at an angle that’s represented by the Greek letter θ, a symbol commonly used to represent angles in mathematics, engineering, and science. This angle is formed between mirror B’s line of sight and the horizontal line of sight traversing the expanse between mirrors D and B. The reason mirror B is an adjustable mirror is so that the line of sight extending from the opening at its location can converge into the line of sight provided by mirror A. Mirror B is manually manipulated until the image seen through it becomes one with A‘s image. Until this convergence takes place two separate images are seen through A and B. Next week we’ll see what happens when the images converge, and we’ll cover the remaining components within an optical rangefinder, items E, F, and G. ____________________________________
