CHAPTER 6
TRIANGULATION
Triangulation is a method of conventional survey used when the traverse method is impractical or impossible. This method is ideally suited to rough or mountainous terrain. The triangulation method employs oblique triangular figures and enables the surveyor to cross obstacles and long distances. This method is timeconsuming and requires careful planning and extensive reconnaissance.
Section I
METHODS AND OPERATIONS
The triangulation method of survey uses triangular figures to determine survey data. If the values of certain elements of a triangle are known, the values of other elements of the triangle can be computed.
61. SURVEY METHODS USING TRIANGLES
a. In FA surveys, the term triangulation is restricted to operations that involve the measurement of all angles within a triangle. (See Figure 61.) Other methods of survey, however, use the triangular figure, but the procedures used in the fieldwork and in the computations differ somewhat from the methods used in triangulation. Two of these methods used in FA survey are intersection and resection.
(1) In the intersection method, two angles are measured. If the length and azimuth of one side and two angles are known, the intersection can be computed. (See Figure 62.)
(2) In resection, the coordinates of an unknown point are obtained by determining the horizontal angles at the unknown point between three unoccupied points of known coordinates. (See Figure 63.)
b. Survey methods using the triangular figure may be used at all levels of FA survey to establish the locations of survey control points. Generally, it is better to use some method of survey that employs these procedures if the distance or terrain involved makes traverse difficult or impossible.
c. Triangulation involves single triangles (Figure 61) and chains or schemes of triangles (Figure 64). Whether a particular triangle is a single triangle or part of a scheme of triangles, the angles in the triangle are determined in the same manner, and the unknown elements of the triangle are computed in the same manner. In a single triangle, the base (known side) is measured with electronic distancemeasuring devices or a 30meter steel tape, or it is computed from known coordinates of the required accuracy. In a chain of triangles, the base of the first triangle is determined in the same manner as a single triangle. The base for the second triangle is the side of the first triangle that is common to both triangles. This side is computed, which establishes a base distance for computation of the second triangle. The same procedure is used to determine the base for each triangle in a triangulation scheme.
62. TRIANGULATION PARTIES
a. Triangulation operations are not confined to one area at any one time. Several operations are involved in triangulation. The personnel involved in each phase are usually separated from the personnel performing other phases of the survey. Since different lengths of time are required for the various operations, no standard division of duties can be made. There are general operations that must be performed in triangulation, and each of the survey personnel is assigned to one or more of these functions. The general operations are as follows:
 Reconnaissance and planning.
 Determination of angles.
 Base measurement.
 Computation.
b. The recon party normally consists of the survey officer and/or the chief surveyor and such other personnel as deemed necessary. The recon party selects and marks each station to be used in the triangulation scheme. It also may erect a target over each station. If considerable clearing of trees and underbrush is required near the station, clearing may be performed by either the recon party or the anglemeasuring personnel. Additional personnel may be required in the party that does the clearing. (Usually, the time and personnel available to perform artillery surveys preclude extensive clearing of trees and underbrush.)
c. The anglemeasuring party consists of the instrument operator, the recorder, and (when necessary) personnel for clearing.
d. When the base is measured with the SEDMEMR, a team of two men is required. This team consists of one instrument operatorrecorder and one man to set up the reflector prism at the end of the base. If taping teams are used, the base is double taped to the specified comparative accuracy. If only one taping team is available, it must make at least two independent measurements of the base to the specified comparative accuracy. A tension handle with a 25pound pull should be used, and horizontal alignment must be maintained to ensure the required accuracy of a taped base measurement. If SEDMEMR devices are used to measure the base, the base will be measured three times and the mean reading used to compute the base. After completing all required measurements, these personnel may be used as directed by the chief of party or the survey and recon officer.
e. The computing party consists of two computers (one has a dual role of computerrecorder). The computers make independent computations and compare their results. They make their computations at any convenient location specified by the party chief or the survey officer. Data from the various points in the survey are reported to the computer by some prearranged method (for example, by radiotelephone).
63. TRIANGULATION FIELD NOTES
a. As the bases are measured, each tapeman of the two taping teams records all the base measurements in the taping notebook. As soon as practical, these recorded distances are transferred to the field notes kept by the recorder. When the base length is measured with the SEDMEMR, the recorder at the end of the base at which the SEDME is located enters the distance in the field notebook.
b. The field notes kept by the recorder in an anglemeasuring party are discussed in Chapter 4. Figures 47 through 49 are examples of notes kept by a recorder for 1:1,000 (T16 theodolite) and 1:3,000 (T2 theodolite) triangulation.
64. STANDARDS AND SPECIFICATIONS
In triangulation, fieldwork and computations adhere to certain standards and specifications to produce surveys of the desired accuracy. These standards and specifications are described in Appendix B.
Section II
TRIANGULATION COMPUTATION
A triangle is defined as a closed threesided geometric figure containing three interior angles the sum of which is 3,200 mils. Each triangle is solved separately whether it is a single triangle or a triangle in a scheme. The only type of triangulation problem (excluding intersection and resection) involved in artillery surveys is the solution of a triangle in which the values of all three angles and the length and azimuth of one side are known. This type of problem is solved by using DA Form 5592R (Computation of Plane Triangle Coordinates and Height From One Side, Three Angles, and Vertical Angles (BUCS)). (A reproducible copy of this form is included in the Blank Forms section of this book.)
Note. If BUCS or form is not available, the triangle can be solved by using the law of sines. For an explanation on the law of sines, see paragraph 616. 
65. SURVEY APPLICATION OF BASIC TRIANGLE
To determine the coordinates and height of a point from another point of known coordinates and height requires three items of informationvertical angle, azimuth, and distance. To associate the basic triangle in Figure 65 with these objectives, assume that Point A of this triangle is a point at which the coordinates and height are to be determined by triangulation. To do this, take the steps discussed below.
a. Select two other points, B and C, intervisible to each other and Point A. The coordinates and height of at least one of these two points must be known.
b. Measure the horizontal interior angles and vertical angles at Points A, B, and C with an instrument. With one more item, a known side (the base), the triangle can be solved. The specifications for angle measurement for fourth and fifthorder accuracies are shown in Appendix B.
c. The length of the base can be obtained in either of two waysby direct measurement or by computation (see (1) below). As mentioned earlier, the coordinates and height of Point B or C must be known. The specifications for baseline measurement for fourth and fifthorder accuracies are shown in Appendix B.
(1) If the coordinates and height of both B and C are known, the azimuth and length of the base (the line joining B and C) can be computed on DA Form 5590R. These known coordinates and heights must be of an equal or higher accuracy than that of the survey being performed. (If a fourthorder survey is being conducted, the coordinates used to compute the azimuth and length of the base should be of thirdorder accuracy or higher but must be at least fourthorder accuracy.)
(2) If the coordinates of only one of the points are known, the base length must be taped or measured electronically. The azimuth must be obtained by astronomic observation, by gyroscopic means, or by sighting on a point of known azimuth (azimuth mark).
Note. If the distance is determined electronically, it must be converted to horizontal distance before computation. To convert a slope to horizontal distance, see paragraph 223d. 
d. When the three horizontal interior angles, vertical angles, azimuth of the base (thus, the azimuths of all three sides), and length of the base have been determined, the triangle can be solved to determine coordinates and height of A. The decision as to which side to compute will be based on distance angles.
66. DISTANCE ANGLES OF A SINGLE TRIANGLE
a. Only strong figures should be used in triangulation to minimize the effects of small measurement errors. The ideal figure is an equilateral triangle. However, field conditions generally make the use of the ideal figures impractical, and often figures must be selected that only approximate the ideal.
b. Computing the length of a side in a triangle involves two of the three angles in the triangle. The angles involved in the computation of the length of a side are called distance angles. Distance angles are defined as those angles in a triangle opposite the known side (base) and the required side (side common to an adjacent triangle). Since in a single triangle there is no specific required side, the distance angles in a single triangle are the angles opposite the known side and the stronger (closest to 1,600 mils) of the two remaining angles. (See Figure 65.)
c. The difference between the sines of angles near 0 mils or 3,200 mils is quite large for very small differences between angles. Because of this, a small error in the value of an angle near 0 mils or 3,200 mils will cause a relatively large error in the sine of the angle and a corresponding error in the computed length of the side opposite the angle. Therefore, for best results, distance angles must be at least 400 mils and preferably 533 mils. For this reason, in the case of a single triangle, the side opposite the stronger angle is the side computed.
d. Vertical angles at each end of the side of a triangle should be measured reciprocally to the height of instrument. Often, because of distances involved, the instrument operator must measure the vertical angle to a target erected and plumbed over the forward station. When the triangulation stations are greater than 1,000 meters, the vertical angle is measured reciprocally. In any triangulation scheme, the coordinates and height of at least one station must be known. This station is used as the starting point to obtain the height of the next station. The height control is extended along the forward line of each triangle in the scheme. In Figure 66, use the height of Point Dave as a starting point, since the height of Point John must be computed along the forward line (DaveJohn). Using the height of Point John, compute the height of Point Bill and the height of Point Mike. If side BillMike had been the forward line, then the computation would have been from Point Bill to Point Mike.
Section III
TRIANGULATION SCHEMES
A chain (scheme) of triangles is a series of single triangles connected by common sides. (See Figure 64.) In a chain of triangles, only the length of the first, or original, base and the length of each check base are measured. The lengths of all other sides are computed.
67. DESCRIPTION, SOLUTION, AND CHECKS OF A CHAIN OF TRIANGLES
a. In Figure 67, side JohnBill is common to triangles JohnBillJoe and MikeBillJohn. Each of these triangles may be solved individually if one side and the interior angles are known. In Figure 67, assume there is a requirement to locate an additional point, Mike, outside the single triangle JohnBillJoe. While the interior and vertical angles at John and Bill are being measured, the angles for the new triangle would also be measured. Then the horizontal and vertical angles at Point Mike would be determined. Since side JoeBill is a known side, all the information needed to compute both triangles would then be available. Side JohnBill would be the required side (side JohnBill is common to both triangles), and the angle at Point Joe would be its distance angle, regardless of its value. A distance angle is an angle in a triangle opposite the known side and the angle opposite the required side (side common to an adjacent triangle). The required side is also known as the forward line or forward base, because it will become the base for the next triangle in the scheme. In a chain of triangles, the last triangle in the scheme is computed as a single triangle, and its distance angles are determined as discussed in paragraph 66b. For example, in Figure 67, the distance angles in triangle JohnBillJoe are at points John (angle opposite known side JoeBill) and Joe (angle opposite the required side). For triangle MikeBillJohn, the distance angles are at Point Mike (angle opposite known side JohnBill, which has been solved in triangle JohnBillJoe) and Point Bill (the stronger of angles at John and Bill).
b. The size of the distance angles in a triangle is used as the measure of relative strength of the figure. The strength factor of a triangle is determined by use of a strength of figure factor table. (See Table 61.) The distance angles of the triangle serve as arguments for entering the table, the smaller distance angle dictates the column; the larger distance angle, the row. The smaller the factor, the greater the relative strength of the triangle.
c. When the sum of the strength factors of a chain of triangles exceeds 200 or at every fifth triangle, a check base (the required side) and azimuth must be determined. If the difference between the computed length and azimuth and the measured length and azimuth is within prescribed tolerances, the scheme may be continued with the measured data. For fifthorder surveys, the computed length of the check base must agree with the measured length within 1:1,000 (comparative accuracy), and the computed azimuth must agree with the astronomic (astro) or gyroscopic (gyro) azimuth within 0.1 times the number of main scheme angles used to carry the azimuth to the check base. For fourthorder surveys, the check comparisons are 1:3,000 (comparative accuracy) for the base and 0.04 x N, where N is the number of main scheme angles used to carry the azimuth to the check base.
Note. The accuracy of the check azimuth must be considered. 
d. A chain of triangles does not provide enough internal checks for an estimation of the accuracy of the work performed. As a check, the length of the last computed side of the final triangle is measured and the computed and measured lengths are compared. The results of this comparison must produce a comparative accuracy as shown in c above. The azimuth of the last computed side must be determined by astro or gyro observation as soon as possible. Error in azimuth is determined by comparing the astro or gyro azimuth with the azimuth computed through the scheme as described in c above. If the scheme closes on a known point, an accuracy ratio must be determined. The total length used for computing the accuracy ratio is the sum of the lengths of the sides of the triangles used to compute coordinates in the scheme from starting station to closing station. The azimuth is verified by turning a closing angle to an azimuth mark. The sum of the closing angle and the azimuth of the base must agree with the known azimuth within the accuracies stated in c above. (See Figures 68 and 69.)
68. TRIANGLE CLOSURE
When interior angles of a triangle are measured in the field, the sum of the angles may vary from 3,200 mils by a small amount. The term used to describe this variance is triangle closure. If the variance is within tolerance (Appendix B), the angles are adjusted to equal 3,200. The BUCS will automatically do this by distributing the closure variance equally among the three angles. The BUCS will display the closure. (See Figures 610, 611, and 612 for triangulation computations.)
69. DA FORM 5592R
a. DA Form 5592R is used for computing triangulation schemes. The front of the form is designed for the solution of one triangle and for entering the field data for a second triangle.
b. DA Form 5592R has five major sections used for computations. These sections are described in (1) through (5) below in the sequence in which they should be computed.
(1) The SKETCH: block is provided for the user to draw a sketch of the triangulation scheme or the single triangle. The sketch should be properly labeled with the station angles, station names and numbers, the starting base, and all the required sides for determining data.
(a) Label triangle 1 as follows: Station A is always located at the first unknown point (station opposite the starting base), and then stations B and C are labeled in a clockwise manner. In Figure 610, station Sue is labeled "A"; station Rob, "B;" and station Ben, "C." For succeeding triangles, the station that is opposite the side common to both triangles (see paragraph 67a) is labeled "A," and so forth.
(b) Before starting computations, the stations must be numbered. Station 1 is located at one end of the starting base (BC). This is the station where the known side and required side meet. Station 2 is the first unknown point (Station A), the succeeding unknown stations will be labeled in order of computation. In Figure 610, Station Ben is number one because it is the point where the known and required side meet. Station Sue is labeled "# 2," since it is the first unknown point. The next unknown station (Bill) is labeled "# 3."
(2) Steps 1 through 8 of the DATA RECORD section are used to record the data for the starting known point (B or C), to include the azimuth and distance of the starting base. If there is more than one triangle to be computed, data from subsequent triangles are also recorded in this section. (See Figure 611.)
Note. When you are entering the base distance, BUCS will prompt for grid or horizontal distance. If the base is measured, the base is a horizontal base. If the base is computed by using known coordinates, it is a grid base. If the base distance was determined by SEDMEMR, it must be converted to a horizontal distance before entering the data into the BUCS. 
(3) Steps 9 through 12 of the DATA RECORD section are provided for the field data as determined by the anglemeasuring party (horizontal and vertical angles).
Note. Specify whether the vertical angle is reciprocal or nonreciprocal. Refer to the note in paragraph 65d. 
(4) Steps 1 through 13 of the DATA RECORD section on the reverse of the form are used when the triangulation scheme is closed on a known point. The known data of the point must be provided so the BUCS can compute the azimuth error, height error, radial error and the accuracy ratio for the triangulation scheme. Refer to paragraph 67d.
(5) Steps 1 through 6 of the DATA RECORD section on the bottom reverse of the form are used when the triangulation scheme requires a check base. For guidance on when to compute a check base, refer to paragraphs 67c and d.
Section IV
INTERSECTION
Intersection is a method of survey used to locate an unknown point by determining azimuths from two or more known points. This method of survey is used as a means of establishing control to desired positions and of checking the locations of points established by other survey methods. A point established by the intersection method should be observed from at least two known stations of equal or higher order of survey than the survey being conducted. One of the points is designated as 01. The height of 01 must also be known. The location and height of the unknown are computed from 01.
610. LIMITATIONS
Limitations for the apex angle are the same as the limitations for distance angles discussed in paragraph 66c. (The exception to this is when intersection is used in target area survey. In this case, the apex angle must be at least 150 mils and preferably 300 mils.)
611. POINT VISIBILITY
The known points can be either intervisible or nonintervisible.
a. If the points are intervisible (Figure 613), measure a horizontal angle from each point to the unknown point, using the other point as the rear station. Measure a vertical angle from 01 to the unknown point. Compute the azimuth between the two points by using DA Form 5590R. Then separately add each angle to the azimuth or backazimuth to determine the azimuths from 01 and 02 to the unknown point.
b. Determine if the apex angle meets the requirements (see paragraph 610) by doing the procedure below.
(1) Determine the backazimuths from the unknown point back to the known points by adding or subtracting 3,200 mils.
(2) Imagine yourself standing at the unknown point looking back at the known points. (See Figure 613.) Point 02 is located to the left side and Point 01 to the right. Subtract the azimuths from left to right.
c. If the points are nonintervisible (Figure 614), do the procedure below. (See the example below.)
(1) Measure a horizontal angle from both points to the unknown point by using a point with a known azimuth as a rear station. Each angle is then added separately to the known azimuth to determine the azimuths from 01 and 02 to the unknown point.
(2) Determine if the apex angle meets the requirements as described in paragraph 611b.
612. INTERSECTION COMPUTATIONS
Intersection computations are done on DA Form 5604R (Computation of Coordinates and Height by Intersection (BUCS)). (A reproducible copy of this form is in the Blank Forms section of this book.) (See Figure 615.) (Table 65 gives the instructions for computing DA Form 5604R.)
613. INTERSECTION ACCURACY
To determine the accuracy of an intersection, do the procedure below.
a. Compute another intersection to the unknown (unk) point by using a different Point 01, Point 02, or Points 01 and 02.
b. Compute a DA Form 5590R from the points designated as 01 to the unknown point.
c. Then compute a DA Form 5590R from the computed coordinates of the first intersection to the computed coordinates of the second intersection. (This will be the radial error.)
d. Divide the radial error into the shorter of the two distances (dis), 01A to the unknown point or 01B to the unknown point. This will be the accuracy ratio of the intersection. The accuracy ratio must agree within the prescribed accuracy limits for the type of survey being performed (1:1,000 for fifth order, 1:3,000 for fourth order).
AR = 1/ 
Distance 01A to unk or 01B to unk 
e. Once the accuracy ratio has been computed and meets specifications described in d above, determine the mean coordinates and elevation for the unknown point. (See Figure 615, REMARKS: block.)
Section V
THREEPOINT RESECTION
Threepoint resection is a method of survey used to obtain control for an unknown point on the basis of three inaccessible known points. However, before the fieldwork is begun, several factors must be considered. In Figure 616, Stations A, B, and C are the known points and Station P is the occupied station for which coordinate are to be determined. All points must be selected so that Angles P1, P2, C, and B (Figure 616) are at least 400 mils. The preferred value for the angles is 533 mils. Also, if the sum of the Angles P1, P2, and A1 is between 2,845 mils and 3,555 mils, no valid solution is possible. This simply means that Station P lies on or near a circle passing through Stations A, B, and C. To eliminate the possibility of this occurring, a map reconnaissance must be made. The fieldwork consists of measuring horizontal Angles P1 and P2 and the vertical angle to the center station. Resection field notes are recorded in the field notebook in the same basic format as triangulation field notes except that the height of target (known or estimated) and the height of instrument (measured to the nearest 0.1 meter) are recorded in the remarks section. A threepoint resection may be considered a closed survey if a fourth known point is used to compute a second resection, the solutions are compared, and the accuracy ratio meets the specified position closure requirement. The points used for threepoint resection should be fourthorder or better. For an example of the field notes of a threepoint resection, see Figure 411 in Chapter 4.
614. DA FORM 5593R
a. DA Form 5593R (Computation of Coordinates and Height by ThreePoint Resection (BUCS)) is shown in Figure 617. (A reproducible copy of this form is included in the Blank Forms section of this book.) The top of the form is for recording administrative data, These data include the following:
 Computer's name.
 Checker's name.
 Area in which survey was performed.
 Notebook reference.
 Date the computations were performed.
 Identification of the sheet number.
b. The next part of the form provides notes for specific operations of the program and other notes necessary for completion of this form.
c. The form is divided into two major sections. The section on the left provides instructions for the computer. The right section (DATA RECORD) is for recording databoth field and computed. Two resections can be computed on each form. The left section is divided into three columnsSTEP, PROMPT, and PROCEDURE. The STEP column is the numerical sequence the operator uses as he proceeds down the form. The PROMPT column tells the operator what will appear on the BUCS display at each particular step. The PROCEDURE column tells the computer the action he will take at each particular step or prompt.
d. Entries required are the coordinates of Points A, B, and C; the horizontal and vertical angles measured at Point P; and the height of Point A. The points used for threepoint resection should be of fourthorder or higher accuracy.
615. CHECKS AND CLOSURE
When possible, the checks below are made to verify data obtained by threepoint resection.
a. A horizontal angle to a fourth known station is measured. This will give another threepoint resection from which the coordinates of Point P1 (a second location of P) may be determined. (See Figure 618.) An accuracy ratio is computed by the following formula:
AR = 1/ 
Distance P to CENTER or P1 to CENTER 1 
(1) Compute DA Form 5590R by using the coordinates of the established point to the center station of the first resection (PCENTER). (See Figure 617, REMARKS: block.)
(2) Then compute DA Form 5590R from the computed coordinates of the second resection to the center station (P1CENTER 1).
(3) Compute DA Form 5590R from the computed coordinates of the first resection to the computed coordinates of the second resection (PP1). The distance will be the radial error.
(4) Divide the radial error into the shorter of the two distancesPCENTER or P1CENTER 1. The result is the accuracy ratio of the intersection. The accuracy ratio must agree within the prescribed accuracy limits for the type of survey being performed (1:1,000 for fifth order, 1:3,000 for fourth order).
b. An astronomic azimuth or gyroscopic azimuth is determined to check the azimuth.
c. If a fourth point (a above) is not known, then Point P must be verified by some other method of survey before it can be used to extend control.
616. LAW OF SINES
As shown in Figure 619, the law of sines is a straight proportiontype formula which states that the sine of the angle at A is to its opposite side as the sine of the angle at B is to its opposite side or the sine of the angle at C is to its opposite side. If the value of each of the interior angles of the triangle were known and the length of side a were known, the law of sines would be transposed as shown in b and c below.
a. Ensure the BUCS is operating in degrees. (Type DEGREES, and press the END LINE key.)
b. To determine the length of side b, use the following formula:
side b = side a x sin Angle B
sin Angle A
The formula when using the BUCS is as follows:
side b = side a * SIN(Angle B * .05625)/SIN(Angle A * .05625) END LINE
c. To determine the length of side c, use the following formula:
side c = side a x sin Angle C
sin Angle A
The formula when using the BUCS is as follows:
side c = side a * SIN(Angle C * .05625)/SIN(Angle A * .05625) END LINE.
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