Radio signals transmitted by orbiting earth satellites can be used to determine the horizontal coordinates, as well as the Table 7-10. Triangulation: Bearing Computation
Course Bearing Angles
AC - S 55°0000 E 67°2932
152°3513 85°0541
-97°3513 152°3513
179°5960
CE S 82°2447 E 37°5842
-77°4106 39°4224
ED -N 4°4341 W 77°4106
101°4230
-96°5849
179°5960
DB -N 83°0111 W 38°3506
142°0306 56°5537
BA S 59°0155 W 46°3223
65°5805 142°0306
125°0000
180°0000
AC -S 55°0000 E Check
Table 7-11. Triangulation: Computation of Coordinates
Bearings Cosine Coordinates
Station Lengths Sine Latitude Departure
A S 5°50000 E 0.57358 1000.00 1000.00
C 345.70 0.81915 -198.29 +283.18
C S 82°2447 E 0.13203 801.71 1283.18
0 8
E 592.36 0.99124 -78.21 +587.17
E N 4°4341 W 0.99659 723.51 1870.36
D 448.59 0.08243 +447.06 - 36.98
D N 83°0111 W 0.12153 1170.57 1833.38
0
B 459.51 0.99259 + 55.84 -456.11
B S 59°0155 W 0.51456 1226.41 1377.28
A 440.00 0.85746 -226.41 -377.28
A 1000.00 1000.00
Final Coordinates
Station North East
A 1000.00 1000.00
B 1226.41 1377.28
C 801.71 1283.18
D 1170.57 1833.38
E 723.51 1870.36
elevation, of any point on the earth’s surface. A small portable antenna receiver and power source are set up over the survey station to track the radio signals (see Figure 7-30). These devices can operate day or night, under all weather conditions, and a clear or unobstructed line of sight between stations is not required. (A clear view of the sky, though, is needed.) Greater than first-order accuracy in position (1:100,000) can be achieved after about 2 hours of signal observation at a station.
The satellite system now used for geodetic control surveys is called the Navstar Global Positioning System (GPS). The 32 active GPS satellites provide 24-hour receiving capability from any point on the earth. The NGS has already begun to use GPS for upgrading and densifying the national control network; other federal, state, and private surveying organizations are also making use of the system.
The cost of the GPS receiving equipment is high; however, more and more surveyors in private practice are making use of this technology. It is possible to lease the necessary equipment and associated software on a daily basis and to receive training
in its use. Or a consulting firm specializing in GPS can be retained to do the work. The total time for a control survey can be significantly reduced with GPS (days instead of months), and the total cost for a large network may be as little as 25 percent of the total cost by traditional methods.
For remote stations that cannot be reached by car or other vehicle, a compact battery-powered unit can be carried to the site. This instrument can produce second-order accu-racy within 30 minutes of radio signal observation at each station; less-accurate positions can be determined in as little as a few seconds of data acquisition. Data reduction can be done using commonly available software and computers.
GPS offers extraordinary potential for surveying. It is possible that the time and cost of determining the coordi-nates of a point will be reduced to less than the cost of monumenting the point; permanent control monuments will then be unnecessary. Surveyors may be able to locate a point with precision in only a few minutes of time. But they will still have to have a good understanding of control networks and traditional surveying theory.
FIGURE 7-30. Global Positioning System. (Courtesy of Leica Geosystems, Inc.)
Questions for Review
1. What is the purpose of a control survey?
2. What is a traverse? What are the basic types?
3. What is the purpose of witnessing a point? What are two basic methods?
4. List six factors regarding proper witnessing of a point.
5. What is meant by adjusting a traverse?
6. List the basic steps for closing a traverse.
7. For geometric consistency, what should the sum of the adjusted interior angles in a traverse with n sides equal?
8. Briefly define the terms latitude and departure as they pertain to traverse computations.
9. What is the sign convention for latitude and departure?
10. In an adjusted traverse, what should the sum of lati-tudes or departures equal? Why?
11. What are the compass and transit rules used for?
12. What is meant by the term inversing?
13. What is the trapezoidal rule used for?
14. What is meant by the term side shot?
15. A distance–distance type of intersection problem involves the measurement of an angle and/or distance from each of two stations of known position.
16. A bearing–bearing type of intersection problem involves the measurement of an angle and/or distance from each of two stations of known position.
17. What are state plane coordinates?
18. What is meant by coordinate transformation?
19. What is the difference between triangulation and trilat-eration?
20. Briefly describe GPS technology.
Practice Problems
1–4. In each column of the following table is given the field measurements of the interior angles of a loop traverse of 12 sides. First, adjust these angles. The bearing of one side is given. With this bearing and the adjusted interior angles, draw a sketch of the traverse. The alphabetical order of the stations gives the forward, counterclockwise direction around the loop. Looking forward around the loop, the interior angles are on the left. Compute course bearings.
The lengths of the courses have no effect on the results, and so they can be made any convenient lengths.
6. The course bearings and lengths of a traverse follow.
Determine the relative accuracy of the survey.
Interior Angles 5. The course bearings and lengths of a traverse follow.
Determine the relative accuracy of the survey.
Course Length, ft Bearing
Station Traverse Angle Length, ft
A 96°05 AB 560.27
B 95°20 BC 484.18
C 65°15 CD 375.42
D 216°22 DE 311.44
E 67°08 EA 449.83
7–12. The field data and the fixed data follow for each of six traverses. The forward direction is given by the alphabeti-cal order of the station names. Each angle is measured clockwise from the back direction to the forward direction so that they are on the left of the traverse looking forward.
In each problem, draw a sketch, compute the accuracy, and compute the final coordinates. Adjust by the com-pass rule.
7. Loop traverse.
Station Traverse Angle Length, ft
A 91°18 AB 554.09
Coordinate B: N 1000.00, E 1000.00
8. Same as Problem 7, but bearing BC: N 9°17 W.
9. Loop traverse.
Bearing EA: S 10°14 E
Coordinate E: N 1000.00, E 1000.00
10. Same as Problem 9 but bearing EA: N 18°53 E.
12. Same as Problem 11, except coordinate values:
20. Compute the area within the loop traverse of Problem 9 by the coordinate method.
21. Compute the area within the loop traverse of Problem 17 by the coordinate method.
22. Compute the area within the loop traverse of Problem 18 by the coordinate method.
23. Perpendicular offsets are measured at intervals of 50 ft, from a traverse line to a curved boundary. The offset distances are given as follows: 12.5, 27.6, 49.2, 87.5, 123.4, 159.0, 135.7, 102.4, 74.1, 32.5, 13.4, and 6.8 ft. The last interval is 28.5 ft. Compute the area, in acres, between the traverse line and the boundary line.
24. Perpendicular offsets are measured at intervals of 15 m, from a traverse line to a curved boundary. The offset distances are given as follows: 4.1, 8.9, 15.8, 28.4, 39.6, 47.2, 41.5, 31.8, 24.6, 9.1, 4.0, and 2.1 m. The last inter-val is 8.7 m. Compute the area, in hectares, between the traverse line and the boundary line.
25. Determine the area of Lot 15, shown in Figure 7-31, and compute the length of the curved boundary line.
Station Angle
N Coordinate
E
Coordinate Course
Length, ft
Ash 1336.35 1050.47
Fir 86°33 1000.00 1000.00 Fir-G 347.15
G 223°55 G-H 449.82
H 114°48 H-Oak 144.76
Oak 141°36 670.23 1780.27
Pine 945.97 1975.74
13. Same as Problem 7, but adjust by the transit rule.
14. Same as Problem 9, but adjust by the transit rule.
15. Compute the final adjusted bearings and lengths for the courses in Problem 7 by inversing.
16. Compute the final adjusted bearings and lengths for the courses in Problem 9 by inversing.
17. The coordinates of loop-traverse stations follow. Com-pute the bearing and length of each course.
18. The coordinates of loop-traverse stations follow.
Compute the bearing and length of each course.
FIGURE 7-31. Illustration for Problem 25.
26. Determine the area of Lot 25, shown in Figure 7-32, and compute the length of the curved boundary line.
FIGURE 7-32. Illustration for Problem 26.
Station N Coordinate E Coordinate
Ash 1266.05 1211.88
Fir 1000.00 1000.00
Oak 324.28 1510.85
Pine 465.34 1818.00
Station Northing Easting
A 1000.00 1000.00
B 750.00 1750.00
C 1345.00 2255.00
D 1567.00 1345.00
Station Northing Easting
1 2345.67 3456.78
2 1357.91 2000.00
3 1075.31 2255.00
4 1000.00 3945.00
11. Connecting traverse (work to nearest minute).
19. Compute the area within the loop traverse of Problem 7 by the coordinate method.
27. A side shot is taken from traverse station M to M1. The measured horizontal distance is 235.7 ft. A clockwise angle of 123°45 is measured at M, from point L to M1;
the bearing of course LM is N 55°15 W. The coordi-nates of station M are N 1000.00/E 1000.00. Determine the coordinates of point M1.
28. A side shot is taken from traverse station S to S10. The measured horizontal distance is 148.35 m. A clockwise angle of 233°15 is measured at S, from point R to S10;
the azimuth of course RS is 155°45. The coordinates of station S are N 500.00/E 500.00. Compute the coordi-nates of point S10.
29. With reference to Figure 7-20, the coordinates of point A are N 100.00, E 100.00 and the coordinates of point B are N 400.00, E 500.00. The bearing of AC is N 12°30 E, and the bearing of BC is N 45°00 W. Determine the position of station C at the intersection of lines AC and BC.
30. With reference to Figure 7-20, the coordinates of point A are N 500.00, E 500.00 and the coordinates of point B are N 300.00, E 200.00. The bearing of AC is S 12°30 W and the bearing of BC is N 75°00 E. Determine the position of station C at the intersection of lines AC and BC.
31. With reference to Figure 7-21, the coordinates of point A are N 2000.00, E 2000.00, and the coordinates of point B are N 1750.00, E 2750.00. Distances AC and BC are measured to be 468.55 ft and 642.08 ft, respectively.
Determine the coordinates of point C at the intersection of lines AC and BC (north or to the left of line AB).
32. With reference to Figure 7-21, the coordinates of point A are N 1000.00, E 1000.00, and the coordinates of point B are N 1100.00, E 1250.00. Distances AC and BC are measured to be 206.80 and 142.15 m, respectively. Deter-mine the coordinates of point C at the intersection of lines AC and BC (south or to the right of line AB).
33. With reference to Figure 7-23, determine the coordi-nates of intersection point S if the azimuth of PQ is 30°.
34. With reference to Figure 7-23, determine the coordi-nates of intersection point S if the azimuth of PQ is 25°
and the coordinates of station B are N 800, E 1500.
35. With reference to Figure 7-24, determine the coordinates of intersection point S between line AB and the circular arc, if the radius of the circle is 850 instead of 1000.
36. With reference to Figure 7-24, the northing of point S is to be N 1650. The northing of the circle center O is to remain fixed at N 1000. What is the required easting of the center point for an arc with radius 1000?
37. With reference to Figure 7-28, compute distance ST.
38. With reference to Figure 7-28, compute distance RS if the baseline is 113.22 m in length.
39. Adjust the angles and compute the final lengths of the sides of the triangulation network shown in Figure 7-33.
FIGURE 7-33. Illustration for Problem 39. Unadjusted field data.
FIGURE 7-34. Illustration for Problem 40. Unadjusted field data.
40. Adjust the angles and compute the final lengths of the sides of the triangulation network shown in Figure 7-34.
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urveying originated primarily from the need for demarcation of land boundaries in the communi-ties of ancient civilizations. A boundary is a line that identifies and separates adjoining tracts of privately (or publicly) owned land. It is also called a property line, and the term lot line may be used as well (generally with reference to city or suburban land parcels). The need for precise location and demarcation of property lines, of course, is still of great social and economic importance in modern times.Property lines are generally monumented, or marked, on the ground at the points where they intersect; such points are usually called property corners. Surveying operations that are applied to the determination of the length and direction of boundary lines, and the exact position of property corners, may be referred to as either property surveying, boundary surveying, or land surveying.
Specifically, a property survey may be performed to accomplish one or more of the following objectives:
1. To locate and reestablish the boundaries of a land parcel that has already been surveyed and legally described at some time in the past; this is called a resurvey.
2. To determine the area of land enclosed within the boundaries of the parcel, generally in terms of acres or hectares.
3. To determine the position of buildings, driveways, fences, and other constructed facilities situated on the land parcel, in relation to the position of its boundary lines.
4. To prepare an updated legal description (written and/or drawn as a plat) of the land parcel.
5. To partition or subdivide the land into two or more smaller parcels (called lots), and to delineate the posi-tions of new public rights-of-way (e.g., roads), if any, that are to be established within the land subdivision that is formed.
A property survey is generally required whenever a parcel of land (real estate) is transferred in ownership.
Naturally, it is necessary for a new owner to be certain of the exact location, the size, and the shape of the land parcel, as well as the position of any existing constructed facilities. In some states, for real estate transactions that involve bank loans or mortgages, the lending institution almost always insists on a new property survey. Generally,