Question 2 of 8: Computations of Positions on the Ellipsoid

Question 2: Computations of Positions on the Ellipsoid (20 marks)

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Given. Relative positioning of a new point from a known point, carried out either two-dimensionally on the reference ellipsoid or three-dimensionally in the geocentric Cartesian frame; the two are claimed to be equivalent when done rigorously.

Find. (a) the “given / observed / wanted” of the direct problem in each case; (b) the required observation reductions, each classified as physical or geometrical; (c) a comparison of the computational complexity.

(a) The direct problem. In the 2-D (ellipsoidal) direct problem the given is the geodetic latitude and longitude \((\varphi_1,\lambda_1)\) of the known point plus the ellipsoid parameters \((a,f)\); the observed quantities are the geodesic (ellipsoidal) distance \(s\) and the forward geodetic azimuth \(\alpha_{12}\); the wanted is the position \((\varphi_2,\lambda_2)\) of the new point and the back-azimuth \(\alpha_{21}\). In the 3-D (spatial) direct problem the given is the geocentric Cartesian position \((X_1,Y_1,Z_1)\) of the known point; the observed are the spatial (slope) distance, the astronomic azimuth and the vertical/zenith angle — equivalently the baseline components \((\Delta X,\Delta Y,\Delta Z)\); the wanted is \((X_2,Y_2,Z_2)\), afterward converted to \((\varphi_2,\lambda_2,h_2)\). The two are equivalent because ellipsoidal and Cartesian coordinates are related by an exact, invertible transformation.

(b) Required reductions and their type. For the 2-D problem the raw terrain observations must be reduced onto the ellipsoid: (i) correcting azimuths/directions for the deflection of the vertical and applying the Laplace correction — physical (they depend on the gravity field); (ii) correcting the measured distance for atmospheric refraction — physical; (iii) the height (sea-level) reduction of the distance to the ellipsoid and the chord-to-geodesic reduction — geometrical; (iv) the skew-normal and height-of-target corrections — geometrical. For the 3-D problem the same raw observations are used, but the vertical is handled explicitly: the zenith angle is corrected for atmospheric refraction and the direction for the deflection of the vertical (both physical), after which the baseline vector is formed by pure geometrical vector algebra. Both problems therefore share the same physical (gravity + atmosphere) corrections; the 2-D problem carries extra geometrical reductions to force the data onto a surface.

(c) Complexity. The 2-D direct problem requires solving the direct geodetic problem on the ellipsoid — propagating \((\varphi,\lambda,\alpha)\) along a geodesic using series or iterative formulae (Bessel, Rainsford, Vincenty, or the Gauss mid-latitude formula). This is analytically intricate but propagates only three quantities. The 3-D problem is conceptually simpler once the observations are reduced: positioning is the vector addition \(\mathbf{X}_2=\mathbf{X}_1+\Delta\mathbf{X}\) in Cartesian space, followed by a routine \((X,Y,Z)\to(\varphi,\lambda,h)\) conversion. The price of that algebraic simplicity is that the 3-D route demands the full gravity-field information (deflections, geoid) to reduce the vertical dimension, whereas the 2-D route hides the height dimension but pays for it with elaborate geodesic mathematics. That is precisely why they are equivalent when done correctly: the same corrected information, organized two different ways.