Tiêu đề 116 Closed Traverse.pdf ✅

MSTC 116: Closed Traverse 1. Latitude and Departure Usually, a line in a closed traverse is defined by a length and bearing. Given that the bearing angle of a line is θ and its length is L, then Latitude = L cos θ Departure = L sin θ However, signs of latitudes and departures vary for bearing angles located in different quadrants. Direction Latitude Departure Azimuth from North N _ E + + θ S _ E - + 180° − θ S _ W - - 360° + θ N _ W + - −θ Note that azimuths can be helpful when using the complex mode in the calculator to show the signs of the latitude and departure automatically. For example, a 5-m line with direction S 84° E: Here, −0.52264 is the latitude, while 4.9726 is the departure. 2. Error of Closure The error of closure of a closed traverse is the discrepancy between a survey’s results and the condition for a closed traverse. This is usually expressed as the distance between the starting point of the traverse and its apparent end. elat = ∑Latitude eDep = ∑Dep
The magnitude of the length of the error of closure is e = √(eDep) 2 + (eLat) 2 While its bearing angle is θ = arctan eDep eLat 3. Corrections Errors in the latitude and departure are distributed using either the compass or the transit rule. 3.1. Compass Rule The compass rule proportionally distributes the error in the latitude and departure based on the length of each line. Therefore, the corrected latitude and departure of a specific line is Dep′ = Dep − L ∑ L ⋅ eDep Lat ′ = Lat − L ∑ L ⋅ eLat 3.2. Transit Rule The transit rule balances a traverse when angular measurements are more precise than linear measurements. The errors are proportionally distributed based on the departures (for departure corrections) and latitudes (for latitude corrections). Dep ′ = Dep − Dep ∑|Dep| ⋅ eDep Lat ′ = Lat − Lat ∑|Lat| ⋅ eLat 4. Missing Data Assuming that errors in closed traverse are negligible, omitted measurements may be determined using geometry and some concepts in balancing closed traverse. 4.1. Length and Bearing of One Side Unknown To determine the length and bearing of an unknown side, since ∑ Dep = ∑ Lat = 0, then the departure and latitude of the unknown line is Lat = −∑(Latothers) Dep = −∑(Depothers)
4.2. Omitted Measurements involving two adjacent sides A more specific method may be used if two adjacent sides have unknown measurements. 4.2.1. Length of one side and bearing of another side unknown Use these steps to solve for the unknown length and bearing: Step 1: Compute the closing line (similar to the error of closure) of a closed traverse connecting all sides with known measurements. Step 2: Using sine law on the triangle formed by the two sides with unknown measurements and the closing line, the unknown measurements can be determined. 4.2.2. The lengths of the two sides are unknown Use these steps to solve for the two unknown lengths: Step 1: Compute the closing line of a closed traverse connecting all sides with known measurements. Step 2: Use the sine law to solve for the unknown sides. Alternatively, there is a method wherein the calculator is helpful. Step 1: Set x and y as the unknown sides. Step 2: Use the 2-equation-2-unknown function in the calculator to solve for x and y, where the equations are based on ∑ Lat = 0 and ∑ Dep = 0. 4.2.3. Bearing of two sides unknown Use these steps to solve for the unknown bearings: Step 1: Compute the closing line of a closed traverse connecting all sides with known measurements. Step 2: Using the cosine law to solve for the unknown interior angles of the triangle formed by the two sides with unknown measurements and the closing line, the interior angles, and eventually the unknown bearings, will be solvable. 4.3. Omitted Measurements involving two non-adjoining sides 4.3.1. Length of one side and bearing of another side unknown Step 1: Move one of the sides in the traverse so that the two sides with unknown measurements become adjacent. Succeeding Steps: Refer to 4.2.1 4.3.2.The lengths of the two sides are unknown The method is the same as 4.2.2.
4.3.3. Bearing of two sides unknown Step 1: Move one of the sides in the traverse so that the two sides with unknown measurements become adjacent. Succeeding Steps: Refer to 4.2.3 4.4. Area Computation 4.4.1. Double Meridian Distance (DMD) Method The DMD method involves the departures of a traverse. The following are the rules: Rule 1: The DMD of the first line is equal to the departure of the line. DMD1 = Dep1 Rule 2: The DMD of any other line is the sum of the DMD of the prior line, the departure of the preceding line, and the departure of the line itself. DMDn = DMDn−1 + Depn−1 + Depn Rule 3: The DMD of the last line is the additive inverse of the departure of the end line. That is, DMDlast = −Deplast. The area is A = 1 2 ∑(Lat × DMD) 4.4.2. Double Parallel Distance (DPD) Method The DPD method involves the latitudes of a traverse. The following are the rules: Rule 1: The DPD of the first line is equal to the latitude of the line. DPD1 = Lat1 Rule 2: The DPD of any other line is the sum of the DPD of the prior line, the latitude of the preceding line, and the latitude of the line itself. DPDn = DPDn−1 + Latn−1 + Latn Rule 3: The DPD of the last line is the additive inverse of the departure of the end line. That is, DPDlast = −LatLast. The area is A = 1 2 ∑(Dep × DPD)