Geometric Geodesy Assignment Report 05

Geometric Geodesy Assignment Report 05 Question 1. The position of point C will be determined by its geodetic coordinates as shown in the image (WGS84 reference ellipsoid). Given the geodetic positions of points A and B Point Φ Λ h A -6°31’44.88308” +107°52’53.03312” +763.243 m B -6°31’40.73534” +107°52’51.27552” +798.507 m Calculated : β = 116°21’33” 𝑑𝑎𝑐 = 77.324 𝑚 𝑚𝑎𝑐 = 10°10’48” Calculate the coordinates of point C! 2. Write down in detail the calculation procedure for this problem, namely determining the three-dimensional geodetic position (latitude, longitude, height) of two red points based on the available data! With its geodetic position known, its horizontal angle is measured, its spatial distance and its oblique angle are measured. Its geodetic position will be determined. Answer: 1. To determine the position of point C, the application of topocentric to geodetic coordinate conversion is used. Because the known data are the spatial distance (𝑑𝑎𝑐) and the oblique angle (𝑚𝑎𝑐), while the azimuth from A to C (𝛼𝑎𝑐) is not yet known, it needs to be found first Azimuth A to C can be found using horizontal angle data (β) and azimuth A to B (β) by using β = β + β - 360° or β + (β + 180°) – 360°. By using 2 known geodetic coordinate points A and B, azimuth B to A can be obtained using the equation: Since the known points A and B are still in geodetic coordinates and have been defined on the WGS 84 reference ellipsoid, to find Δx, Δy, and Δz, both points are converted into global 3-dimensional Cartesian coordinates. Konversi Conversion of geodetic coordinates to global 3-dimensional Cartesian coordinates can be obtained using the equation: 𝑋 = (𝑁 + ℎ) cos𝜑 cos𝜆 𝑌 = (𝑁 + ℎ)cos 𝜑 sin 𝜆 𝑍 = (𝑁(1 − 𝑒2) + ℎ)sin 𝜑 The value of the main vertical curve radius (N) and eccentricity (e) can be found using the equation: e² = then the value of N at points A and B is Since the main vertical curved section (N) and eccentricity have been obtained, points A and B can be converted to 3-dimensional Cartesian with the following calculation: 𝑋𝑎 = (6378413.049926057 + 763.243) cos(−6°31’44.88308”)cos(107°52’53.03312” 𝑌𝑎 = (6378413.049926057 + 763.243) cos(−6°31’44.88308”) sin(107°52’53.03312”) 𝑍𝑎 = (6378413.049926057 (1 − 0.00669437999) + 763.243) sin(−6°31’44.88308”) And, 𝑋𝑏 = (6378412.952926284 + 798.507) cos(−6°31’40.73534”)cos( 107°52’51.27552”) 𝑌𝑏 = (6378412.952926284 + 798.507) cos(−6°31’40.73534”) sin(107°52’51.27552” 𝑍𝑏 = (6378412.952926284(1 − 0.00669437999) + 798.507) sin(−6°31’40.73534”) So that points A and B in the 3-dimensional Cartesian coordinate system have the values 𝑋𝑎 = -1946007.2300258828 m dan 𝑋𝑏 = -1945971.0418587273 m 𝑌𝑎 = 6031649.039918526 m dan 𝑌𝑏 = 6031712.752662332 m 𝑍𝑎 = -720510.7479456195 m dan 𝑍𝑏 = -720388.1548984607 m Then obtained, Δx = 𝑋𝑎 − 𝑋𝑏 = -36.18816715549 m Δy = 𝑌𝑎 − 𝑌𝑏 = -63.71274380572 m Δz = 𝑍𝑎 − 𝑍𝑏 = -122.59304715879 m So the azimutch of A to Ci 𝛼𝑎𝑐 = 360° 𝛼𝑎𝑐 = 116°21′33 + (156°32'33.51465294" + 180°) − 360° 𝛼𝑎𝑐 = 92°54′6.514524" After obtaining the azimuth A to C (𝛼𝑎𝑐), the direct distance data AC (𝑑𝑎𝑐) and the oblique angle AC (𝑚𝑎𝑐), the topocentric coordinates of point C can be obtained with the following calculations 𝑛𝑐 = 77.324𝑚 cos( 10°10′48")cos (92°54′514524") = -3.85286315741 𝑒𝑐 = 77.324𝑚 cos( 10°10′48")sin( 92°54′514524") = 76.00912911039 𝑢𝑐 = 77.324𝑚 sin( 10°10′48") = 13.66633503797 After the topocentric coordinates of point C are obtained, point C can be expressed in geodetic coordinates. In converting topocentric coordinates to geodetic coordinates, a 3-dimensional rotation matrix is used. The formula can be written as follows: From this equation, the z axis is rotated by (𝜆𝑎 − 180) so that the x axis is in the same plane as the 𝑛𝑐 axis, then rotated again on the y axis by (𝜑𝑎 − 90°) so that the x, y, z axes coincide with the 𝑛𝑐, 𝑒𝑐, 𝑢𝑐 axes . The equation can be written as follows: With 𝑅 = 𝑅𝑦(𝜑𝑎, 𝜆𝑎) −sin(φa) cos (λa) − sin(φa) sin(λa) cos(φa) −sin(λa) cos( λa) 0 ] 𝑅=[ cos(φa) cos (λa) cos(φa)sin(λa) sin(φa) 𝑛𝑐 Δx −1 [Δy] = 𝑅 [𝑒𝑐 ] = 𝑅𝑇 𝑢𝑐 Δz 𝑛𝑐 𝑒 [ 𝑐] 𝑢𝑐 From the equation above, we obtain: −0.03491390277 Δx [Δy] = [ 0.10821563502 0.99351416484 Δz −0.95169414148 −0.30704765278 0 −0.30505619231 0.94552161015 −0.11370841776 −3.85286315741 ] × [ 76.00912911039 ] 13.66633503797 Δx −76.37192451321 [Δy] = [ −10.83354960647 ] Δz −5.38185145582 Since the values of Δx, Δy, and Δz have been obtained, the 3-dimensional Cartesian coordinates of point C can be calculated as follows: 𝑋𝑐 = 𝑋𝑎 + 𝛥𝑥 = −1946007.2300258828 + −76.37192451321 = −1946083.601950396 m 𝑌𝑐 = 𝑌𝑎 + 𝛥𝑦 = 6031649.039918526 + −10.83354960647 = 6031638.206368919 m 𝑍𝑐 = 𝑍𝑎 + 𝛥𝑧 = −720510.7479456195 + −5.38185145582 = −720516.1297970753 m SAfter point C is converted into 3-dimensional Cartesian coordinates, point C can be stated in geodetic coordinates with a closed formula. Before using a closed formula in converting 3-dimensional Cartesian coordinates to geodetic coordinates, what must be sought first is: m 𝑧 = −720516.1297970753 m 6.50747132247° So that the geodetic coordinates of point C can be obtained using the following closed formula. 6°31’45.01” + 180° = 107°52’55,51” Then the geodetic coordinates of point C are (𝛗, 𝝀, 𝒉 ) = (6°31’45.01” , 107°52’55,51”, 776,909789 m). 2. Find Point C From the initial data, the geodetic coordinates of the positions of other points can be searched for in the following order: Points A and B whose geodetic coordinates are known must be defined in the same reference ellipsoid, for example WGS 84. To determine point C, its topocentric coordinates need to be found so that they can be converted into geodetic coordinates. Because the angle of inclination of point B to C, the direct distance of point B to C has been known, so to determine the topocentric coordinates of the point to be searched for, namely point C, it is necessary to find the asimut from point B to C. 𝛼𝑎𝑐 = β + 𝛼𝑏𝑎 - 360° Since point A and point B have known geodetic coordinates, it is necessary to convert them to 3-dimensional Cartesian coordinates to obtain the values of Δx, Δy, and Δz which are used in the equation to find the azimuth of 2 points whose geodetic coordinates are known. The equation to convert from geodetic coordinates to 3dimensional Cartesian coordinates is as follows: 𝑋 = (𝑁 + ℎ) cos𝜑 cos𝜆 𝑌 = (𝑁 + ℎ)cos 𝜑 sin 𝜆 𝑍 = (𝑁(1 − 𝑒2) + ℎ)sin 𝜑 It is required that N is a function of latitude and e is the eccentricity whose values a and b depend on the reference ellipsoid used. For the value of N, substitute the latitude coordinates at points A and B to obtain 2 values of the main vertical curvature radius at points A and B. e² = After obtaining the value of the main vertical curve radius (N) and eccentricity at each point, namely points A and B which are defined on the same reference ellipsoid, then substitute into the equation above to obtain the values of X, Y, Z. In finding the azimuth B to A, the data Δx, Δy, and Δz are used, then use the latitude and longitude coordinates of point B. The equation can be written as follows: With Δx = 𝑎 − 𝑋𝑏 Δy = 𝑌𝑎 − 𝑌𝑏 Δz = 𝑍𝑎 − 𝑍𝑏 Substitute into the equation 𝛼𝑏𝑐 = β + 𝛼𝑏𝑎 - 360° to obtain the azimuth value of point B to C. After obtaining the values of 𝛼𝑏𝑐, 𝑚𝑏𝑐, and 𝑑𝑏𝑐, the topocentric coordinates of point C can be found using the following equation. 𝑛𝑐 = 𝑑𝑎𝑐 cos(𝑚𝑎𝑐)𝑐𝑜𝑠(𝛼𝑎𝑐) 𝑒𝑐 = 𝑑𝑎𝑐𝑐𝑜𝑠 (𝑚𝑎𝑐) sin( 𝛼𝑎𝑐) 𝑢𝑐 = 𝑑𝑎𝑐sin (𝑚𝑎𝑐) After the topocentric coordinates of point C are obtained, point C can be expressed in geodetic coordinates. In converting topocentric coordinates to geodetic coordinates, a 3-dimensional rotation matrix is used. The application of a 3dimensional rotation matrix is used in finding the values of Δx, Δy, and Δz from the topocentric coordinates of point C that have been searched. The formula can be written as follows: From this equation, the z axis is rotated by (𝜆𝑏 − 180) so that the x axis is in the same plane as the 𝑛𝑐 axis, then rotated again on the y axis by (𝜑𝑏 − 90°) so that the x, y, z axes coincide with the 𝑛𝑐, 𝑒𝑐, 𝑢𝑐 axes . This equation can be written as follows: With 𝑅 = 𝑅𝑦(𝜑𝑏, 𝜆𝑏) −sin(φa) cos (λa) − sin(φa) sin(λa) cos(φa) −sin(λa) cos( λa) 0 ] 𝑅=[ cos(φa) cos (λa) cos(φa)sin(λa) sin(φa) 𝑛𝑐 𝑛𝑐 Δx −1 𝑇 [Δy] = 𝑅 [𝑒𝑐 ] = 𝑅 [𝑒𝑐 ] 𝑢𝑐 𝑢𝑐 Δz From the equation above, the values of Δx, Δy, and Δz can be obtained. Because the difference between the 3-dimensional Cartesian coordinates at points B and C has been sought and point B has been expressed in 3-dimensional Cartesian coordinates, point C can be obtained which is expressed in 3-dimensional Cartesian coordinates as follows: 𝑋𝑐 = 𝑋𝑏 + 𝛥𝑥 𝑌𝑐 = 𝑌𝑏 + 𝛥𝑦 𝑍𝑐 = 𝑍𝑏 + 𝛥𝑧 After point C is converted into 3-dimensional Cartesian coordinates, point C can be stated in geodetic coordinates with a closed formula or iteration. In this case, a closed formula is used to convert from 3-dimensional Cartesian coordinates to geodetic coordinates. Before using a closed formula in converting 3 -dimensional Cartesian coordinates to geodetic coordinates, what must be found first is: Then, using a closed formula, the position of point C in geodetic coordinates can be obtained as follows: Finding Point D Basically, to find point D, the flow is the same as finding the coordinates of point C, the difference is that to find point D, point C is used as a reference after being calculated in the problem above. Initial data such as spatial distance, oblique angle, and azimuth used are from point C to point D. The position of point D in topocentric coordinates can be determined using the initial data of spatial distance CD (𝑑𝑐𝑑), oblique angle (𝑚𝑐𝑑), and azimuth (𝛼𝑐𝑑). Because (𝛼𝑐𝑑) is not yet known, (𝛼𝑐𝑑) is searched for first. The two points in geodetic coordinates are converted to 3dimensional Cartesian with the following equation: 𝑋 = (𝑁 + ℎ) cos𝜑 cos𝜆 𝑌 = (𝑁 + ℎ)cos 𝜑 sin 𝜆 𝑍 = (𝑁(1 − 𝑒2) + ℎ)sin 𝜑 After obtaining points B and C in Cartesian coordinates, then differentiate points B from C to obtain the values of Δx, Δy, and Δz which are used in finding the azimuth CB (𝛼𝑐𝑏). (𝛼𝑐𝑑) = β + 𝛼𝑐𝑏 - 360° With Since the CD spatial distance (𝑑𝑐𝑑), oblique angle (𝑚𝑐𝑑), and azimuth (𝛼𝑐𝑑) have been obtained, point C in topocentric coordinates can be determined using the equation 𝑛𝑑 = 𝑑𝑐𝑑 cos(𝑚𝑐𝑑)𝑐𝑜𝑠(𝛼𝑐𝑑) 𝑒𝑑 = 𝑑𝑐𝑑𝑐𝑜𝑠 (𝑚𝑐𝑑) sin( 𝛼𝑐𝑑) 𝑢𝑑 = 𝑑𝑐𝑑sin (𝑚𝑐𝑑) After the topocentric coordinates of point D are obtained, point D can be expressed in geodetic coordinates. In converting topocentric coordinates to geodetic coordinates, a 3-dimensional rotation matrix is used. The formula can be written as follows: With 𝑅 = 𝑅𝑦(𝜑𝑏, 𝜆𝑏) −sin(φa) cos (λa) − sin(φa) sin(λa) cos(φa) 𝑅=[ −sin(λa) cos( λa) 0 ] cos(φa) cos (λa) cos(φa)sin(λa) sin(φa) 𝑛𝑐 Δx −1 [Δy] = 𝑅 [𝑒𝑐 ] = 𝑅𝑇 𝑢𝑐 Δz 𝑛𝑐 [ 𝑒𝑐 ] 𝑢𝑐 From the equation above, the values of Δx, Δy, and Δz can be obtained. Then point D is obtained which is expressed in 3-dimensional Cartesian coordinates as follows: 𝑋𝑑 = 𝑋𝑐 + 𝛥𝑥 𝑌𝑑 = 𝑌𝑐 + 𝛥𝑦 𝑍𝑑 = 𝑍𝑐 + 𝛥𝑧 After point D is converted into 3-dimensional Cartesian coordinates, point D can be stated in geodetic coordinates with a closed formula or iteration. In this case, a closed formula is used to convert from 3-dimensional Cartesian coordinates to geodetic coordinates. Before using a closed formula in converting 3-dimensional Cartesian coordinates to geodetic coordinates, what must be found first is: Then, using a closed formula, the position of point D in geodetic coordinates can be obtained as follows::