Undersökning av strikta och iterativa metoder för omvandling från kartesiska till geodetiska koordinater
Sammanfattning: Generally there are three different principles to calculate geodetic coordinates (latitude φ, longitude λ and height h) from Cartesian coordinates (X, Y, Z), namely: closed, iterative and approximate methods. The longitude can easily be calculated from Cartesian coordinates. The formula of the longitude is general and is used in all methods. The main problem is to transform (X, Y, Z) to (φ, h). The methods presented are: Heiskanen and Moritz (1967), Paul (1973), Ozone (1985), Borkowski (1989), Fukushima (1999 and 2006), Sjöberg (1999), Pollard (2002), Vermeille (2002) and Sjöberg (2008). Some methods have various degrees of problems for the calculation of the point positions near the poles, around the equator and at big height intervals. Our study covers the presentation of ten methods with different solutions to the main problem. In assessing the performance of a transformation method has accuracy and calculation time been used as criteria. Seven methods have been tested in a given range. The tested methods in Chapter 4 are: Heiskanen and Moritz (1967), Ozone (1985), Borkowski (1989), Fukushima (1999), Sjöberg (1999), Vermeille (2002) and Sjöberg (2008). The test range includes: -90, 31.5 and 0 to 55 with 5 intervals. The methods have acceptable results for practical application of a precision requirement down to 0.5 mm in the calculation of the point position. The numerical comparison shows that this requirement is fulfilled with good margin, and the methods have in some test intervals the result 0.005 mm or better. This result is also valid for the pole and the equator regions, except for the methods of Ozone (1985) and Borkowski (1989), which gives uncertainty results. The closed method of Sjöberg (2008) is faster than the other closed ones, Ozone (1985), Borkowski (1989) and Vermeille (2002). For the iterative methods of Heiskanen and Moritz (1967) and Sjöberg (1999), the number of iterations increase to achieve the requirement in 0.005 mm, and the calculation time gets longer for the iterative methods than for the closed ones. The iterative method of Fukushima (1999) is clearly the fastest, while the iterative method of Heiskanen and Moritz (1967) is the slowest of all in the test.
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