Terrestrial Distance Measurement Help
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.
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