1.14.6. Computation of Included Angles from Bearings

In a closed traverse, bearings of lines may be calculated if bearing of one of the line and the included horizontal angles between various lines are known, using the relationship:

Bearing of a line = given bearing + included horizontal angle (1.5)

This chunk introduces the concept of included angles in a closed traverse. Here, a closed traverse refers to a path where the starting and ending points are the same, forming a closed shape (like a polygon). The bearings of different lines (the directions of the lines with respect to a reference direction) play a critical role in navigating this closed path. The key formula provided allows surveyors to calculate the bearing of any line by knowing one bearing and the angles included between that line and other lines. Essentially, if you know where one line is pointing, you can find the direction of another line relative to that one using their included angles.

Imagine you're at a park with multiple pathways. If you know the direction of one path (like going north) and you have measured an angle towards another path (like turning 30 degrees to the east), you can clearly understand and define the direction of that second path. This way, knowing one path's direction allows you to calculate the others around it.

If at any point, bearings of any two lines are known, the included angle between these two lines can easily be found by drawing a sketch, and then taking the difference of angles.

In this chunk, the method to find the included angle between two lines based on their bearings is described. When surveyors know the bearings (angles) of two lines, they can visualize these lines on a sketch. By measuring the angles and subtracting one from the other, they find the included angle, which is the angle formed between the two lines. This is a practical skill in surveying, as it helps in determining precise angles needed for construction and mapping.

Think of a game of darts. If you know the angle at which you're standing (let's say 30 degrees) and the angle at which the dartboard is set (say 90 degrees), you can easily find out how to adjust your aim to hit the center by finding the included angle between where you're aiming and the dartboard. In surveying, this skill is crucial for aligning structures accurately.

If the lines are on the same side of the meridian and in the same quadrant (Figure 1.17a), the included angle ∠AOB = the difference of the reduced bearings of OA and OB. If the lines are on the same side of the meridian but in different quadrants (Figure 1.17b), the included angle ∠AOB = 180° – sum of the reduced bearings of OA and OB. If the lines are not on the same side of the meridian but they are in the adjacent quadrants (Figure 1.17c), the included angle ∠AOB = sum of the reduced bearings of OA and OB. If the lines are not on the same side of the meridian and also not in the opposite quadrants (Figure 1.17d), the included angle ∠AOB = 360° – difference of the whole circle bearings of OA and OB.

This chunk discusses specific conditions under which the included angles are calculated based on their bearings. It breaks down various scenarios regarding the positioning of lines in relation to a reference meridian. For instance, if both lines lie in the same quadrant and on the same side of the meridian, simply subtracting their reduced bearings will yield the included angle. Conversely, if they are in different quadrants, more complex calculation rules apply. These different cases emphasize the importance of understanding the spatial relationships between lines to make accurate calculations.

Imagine you’re driving in a city with quadrants divided by main roads (like a grid). If you’re at a corner (like a line on the meridian) and want to calculate the best way to turn onto another street (the included angle), knowing whether you’re turning into the same block or crossing into the next block (quadrant) impacts how sharply you need to turn. Similarly, in surveying, recognizing the alignment of lines relative to a reference direction ensures accurate routes are plotted.

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