Terrestrial Distance Measurement Help (page 2)
Terrestrial Distance Measurement—Parallax
Trigonometry is used to determine distances by measuring angles. In some cases, the angles are exceedingly small, requiring observational apparatus of high precision. In other cases, angle measurement is less critical. Trigonometry can also be used in the reverse sense: determining angles (such as headings or bearings) based on known or measured distances.
In order to measure distances using trigonometry, observers rely on a principle of classical physics: rays of light travel in straight lines. This can be taken as “gospel” in surveying and in general astronomy. (There are exceptions to this rule, but they are of concern only to cosmologists and astrophysicists in scenarios where relativistic effects take place.)
Parallax makes it possible to judge distances to objects and to perceive depth. Figure 8-1 shows the basic principle. Nearby objects appear displaced, relative to a distant background, when viewed with the left eye as compared to the view seen through the right eye. The extent of the displacement depends on the proportional difference between the distance to the nearby object and the distant reference scale, and also on the separation between the left eye and the right eye.
Parallax can be used for navigation and guidance. If you are heading toward a point, that point seems stationary while other objects seem to move radially outward from it. You can observe this effect while driving down a flat, straight highway. Signs, trees, and other roadside objects appear to move in straight lines outward from a distant point on the road. Parallax simulation gives 3D video games their realism, and is used in stereoscopic imaging.
The Base Line and Accuracy
The Base Line
The use of parallax in distance measurement involves establishing a base line. This is a line segment connecting two points of observation. Let’s call the observation points P and Q . If the distant object, to which we want to find the distance, is at point R , then we must choose the base line such that Δ PQR comes as near to being a right triangle as we can manage. We want the base line segment PQ to be perpendicular to either line segment PR or line segment QR, as shown in Fig. 8-2.
At first thought, getting the base line oriented properly might seem to be a difficult task. But because the distance we want to measure is almost always much longer than the base line, an approximation is good enough. A hiker’s compass will suffice to set the base line PQ at a right angle to the line segment connecting the observer and the distant object.
In order to measure distances to an object within sight, the base line must be long enough so there is a significant difference in the azimuth of the object (that is, its compass bearing) as seen from opposite ends of the base line. By “significant,” we mean an angular difference well within the ability of the observing apparatus to detect and measure.
The absolute accuracy (in fixed units such as meters) with which the distance to an object can be measured depends on three factors:
- The distance to the object
- The length of the base line
- The precision of the angle-measuring apparatus
As the distance to the object increases, assuming the base line length stays constant, the absolute accuracy of the distance measurement gets worse; that is, the error increases. As the length of the base line increases, the accuracy improves. As the angular resolution, or precision of the angle-measuring equipment, gets better, the absolute accuracy improves, if all other factors are held constant.
Stadimetry can be used to measure the distance to an object when the object’s height or width is known. The angular diameter of the object is determined by observation. The distance is calculated using trigonometry. This scheme works in the same way as the base-line method described above, except that the “base line” is at the opposite end of the triangle from the observer.
Figure 8-4 shows an example of stadimetry as it might be used to measure the distance d, in meters, to a distant person. Suppose the person’s height h, in meters, is known. The vision system determines the angle θ that the person subtends in the field of view. From this information, the distance d is calculated according to the following formula:
d = h/ (tan θ )
In order for stadimetry to be accurate, the linear dimension axis (in this case the axis that depicts the person’s height, h ) must be perpendicular to a line between the observation point and one end of the object. Also, it is important that d and h be expressed in the same units.
Terrestrial Distance Measurement Practice Problems
Suppose we want to determine the distance to an object at the top of a mountain. The base line for the distance measurement is 500.00 meters (which we will call 0.50000 kilometers) long. The angular difference in azimuth is 0.75000° between opposite ends of the base line. How far away is the object?
It helps to draw a diagram of the situation, even though it cannot be conveniently drawn to scale. (The base line must be shown out of proportion to its actual relative length.) Figure 8-3 illustrates this scenario. The base line, which is line segment PQ, is oriented at right angles to the line segment PR connecting one end of the base line and the distant object. The right angle is established approximately, using a hiker’s compass, but for purposes of calculation, it can be assumed exact, so Δ PQR can be considered a right triangle.
We measure the angle θ between a ray parallel to line segment PR and the observation line segment QR, and find this angle to be 0.75000°. The “parallel ray” can be determined either by sighting to an object that is essentially at an infinite distance, or, lacking that, by using an accurate magnetic compass.
One of the fundamental principles of plane geometry states that pairs of alternate interior angles formed by a transversal to parallel lines always have equal measure. In this example, we have line PR and an observation ray parallel to it, while line QR is a transversal to these parallel lines. Because of this, the two angles labeled θ in Fig. 8-3 have equal measure.
We use the triangle model for circular functions to calculate the distance to the object. Let b be the length of the base line (line segment PQ ), and let x be the distance to the object (the length of line segment PR ). Then the following formula holds:
tan θ = b/x
Plugging in known values produces this equation:
The object on top of the mountain is 38.195 kilometers away from point P.
Why can’t we use the length of line segment QR as the distance to the object, rather than the length of line segment PR in the above example?
We can! Observation point Q is just as valid, for determining the distance, as is point P. In this case, the base line is short compared to the distance being measured. In Fig. 8-3, let y be the length of line segment QR. Then the following formula holds:
sin θ = b/y
Plugging in known values, we get this:
The percentage difference between this result and the previous result is small. In some situations, the absolute difference between these two determinations (approximately 3 meters) could be of concern, and a more precise method of distance measurement, such as laser ranging, would be needed. An example of such an application is precise monitoring of the distance between two points at intervals over a period of time, in order to determine minute movements of the earth’s crust along a geological fault line.
Practice problems for these concepts can be found at: Surveying, Navigation, and Astronomy Practice Test
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