Geometric Geodesic Ellipsoid Materials

INDIVIDUAL ASSIGNMENT: GEOMETRIC GEODESIC ELLIPSOID MATERIALS Question 1. The position of point C will be determined by its geodetic coordinates as in the picture (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 Measured : β = 116°21’33” 𝑑𝑎𝑐 = 77.324 𝑚 𝑚𝑎𝑐 = 10°10’48” Calculate the coordinates of the point C ! 2. Write 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: The geodetic position is known, the horizontal angle is measured, the spatial distance and the tilt angle are measured. Its geodetic position will be determined. Answer : 1. To determine the position of Point C, we use topocentric to geodetic coordinate conversion. Beacuse the known data is the spatial distance(𝑑𝑎𝑐 ) and the tilt angle (𝑚𝑎𝑐), while the azimuth from A to C (𝛼𝑎𝑐) is not yet known, it is necessary to find it first. The A to C azimuth can be found using horizontal angle data (β) and the A to B azimuth (𝛼𝑎𝑏) by means 𝛼𝑎𝑐 = β + 𝛼𝑎𝑏 - 360° or β + (𝛼𝑏𝑎 + 180°) – 360°. By using 2 known geodetic coordinates A and B, we can obtain the azimuth B to A with the equation: Because 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. The conversion of geodetic coordinates to global 3-dimensional Cartesian coordinates can be obtained by Eq: 𝑋 = (𝑁 + ℎ) cos𝜑 cos𝜆 𝑌 = (𝑁 + ℎ)cos 𝜑 sin 𝜆 𝑍 = (𝑁(1 − 𝑒2) + ℎ)sin 𝜑 Main vertical bend radius value (N) and eccentricity (e) can be found with the equation: e² = then the value of N at points A and B is: Because 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 calculations: 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 values 𝑋𝑎 = -1946007.2300258828 m and 𝑋𝑏 = -1945971.0418587273 m 𝑌𝑎 = 6031649.039918526 m and 𝑌𝑏 = 6031712.752662332 m 𝑍𝑎 = -720510.7479456195 m and 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 azimuth A to C: 𝛼𝑎𝑐 = 𝛼𝑎𝑐 = 116°21′33 + (156°32'33.51465294" + 180°) − 360° 𝛼𝑎𝑐 = 92°54′6.514524" 360° After obtaining the azimuth A to C (𝛼𝑎𝑐), direct distance data AC (𝑑𝑎𝑐) and tilt 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. This 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 . This 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 𝑛𝑐 Δx [Δy] = 𝑅−1 [𝑒𝑐 ] = 𝑅𝑇 [𝑒𝑐 ] 𝑢𝑐 𝑢𝑐 Δz From the equation above, it is obtained −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 Because the values 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 After point C is converted into 3-dimensional Cartesian coordinates, point C can be expressed in geodetic coordinates using a closed formula. Before using a closed formula to convert 3-dimensional Cartesian coordinates to geodetic coordinates, what must be found first is: m 𝑧 = −720516.1297970753 m 6.50747132247° So you can get the geodetic coordinates of point C using the following closed formula 6°31’45.01” + 180° = 107°52’55,51” Then the geodetic coordinates of point C adalah (𝛗, 𝝀, 𝒉 ) = (6°31’45.01” , 107°52’55,51”, 776,909789 m). 2. Find point C From the initial data, the position of other points can be searched for geodetic coordinates 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, it is necessary to find its topocentric coordinates so that they can be converted into geodetic coordinates. Because the slant angle of point B to C, the direct distance from point B to C is known, so to determine the topocentric coordinates of the point to be searched, namely point C, you need to find the azimuth from point B to C. 𝛼𝑎𝑐 = β + 𝛼𝑏𝑎 - 360° Because point A and point B have known geodetic coordinates, it is necessary to convert them to 3-dimensional Cartesian coordinates to obtain the values Δx, Δy, and Δz which are used in the equation to find the azimuth of 2 points whose geodetic coordinates are known. The equation for converting from geodetic coordinates to 3dimensional Cartesian is as follows: 𝑋 = (𝑁 + ℎ) cos𝜑 cos𝜆 𝑌 = (𝑁 + ℎ)cos 𝜑 sin 𝜆 𝑍 = (𝑁(1 − 𝑒2) + ℎ)sin 𝜑 The required N is a function of latitude and e is the eccentricity whose values a and b depend on the reference ellipsoid used. For the N value, substitute the latitude coordinates at points A and B to obtain 2 main vertical curve radius values at points A and B e² = After obtaining the values 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 get the values of use the data Δx, Δy, and Δz 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 determined 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 to find the values Δx, Δy, and Δz from the topocentric coordinates of the point C that has been searched. This 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) 𝑅=[ −sin(λa) cos( λa) 0 ] 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 for Δx, Δy, and Δz can be obtained. Because the difference between the 3-dimensional Cartesian coordinates at points B and C has been searched and point B has been expressed in 3-dimensional Cartesian coordinates, we can obtain point C 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 expressed in geodetic coordinates using a closed or iterative formula. In this case, a closed formula is used to convert from 3-dimensional Cartesian coordinates to geodetic coordinates. Before using a closed formula to convert 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: Looking for point D Basically, to find point D, the process 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 calculating it in the problem above. The 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 initial spatial distance data CD (𝑑𝑐𝑑), oblique angle (𝑚𝑐𝑑), and azimuth (𝛼𝑐𝑑 ). Because (𝛼𝑐𝑑) is not yet known, (𝛼𝑐𝑑) is searched first. The two points in geodetic coordinates are converted to 3-dimensional 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 and C to obtain the values of Δx, Δy, and Δz which are used to find the CB azimuth (𝛼𝑐𝑏). (𝛼𝑐𝑑) = β + 𝛼𝑐𝑏 - 360° With, Because the space distance CD (𝑑𝑐𝑑), oblique angle (𝑚𝑐𝑑), and azimuth (𝛼𝑐𝑑) have been obtained, point C in topocentric coordinates can be determined by 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. This formula 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 [Δy] = 𝑅−1 [𝑒𝑐 ] = 𝑅𝑇 [𝑒𝑐 ] 𝑢𝑐 𝑢𝑐 Δz From the equation above, the values for Δx, Δy, and Δz can be obtained. Then we get point D 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 expressed in geodetic coordinates using a closed or iterative formula. In this case, a closed formula is used to convert from 3-dimensional Cartesian coordinates to geodetic coordinates. Before using a closed formula to convert 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: