Computer Program for the Inverse Transformation of the Winkel Projection

Transcription

1 omputer Program for the Inverse Transformation of the Winkel Projection engizhan Ipbuker 1 and I. Oztug Bildirici Abstract: The map projection problem involves transforming the graticule of meridians and parallels of a sphere onto a plane using a specified mathematical method according to certain conditions. Map projection transformations are a research field dealing with the method of transforming one kind of map projection coordinates to another. The conversion from geographical to plane coordinates is the normal practice in cartography, which is called forward transformation. The inverse transformation, which yields geographical coordinates from map coordinates, is a more recent development due to the need for transformation between different map projections, especially in Geographic Information Systems GIS. The direct inverse equations for most of the map projections are already in existence, but for the projections, which have complex functions for forward transformation, defining the inverse projection is not easy. This paper describes an iteration algorithm to derive the inverse equations of the Winkel tripel projection using the Newton Raphson iteration method. DOI: / ASE :4 15 E Database subject headings: omputer programming; Mapping; artography; Surveys; Geographic information systems. Introduction The artesian coordinates X,Y of a point on a map are calculated from latitude and longitude using the functions X = f x, Y = f y, 1 1 Istanbul Technical Univ., Faculty of ivil Engineering, Division of artography, Maslak Istanbul, Turkey. buker@itu.edu.tr Selcuk Univ., Faculty of Engineering and Architecture, Dept. of Geodesy and Photogrammetry, Kampüs, Konya, Turkey. bildirici@selcuk.edu.tr Note. Discussion open until April 1, 006. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASE Managing Editor. The manuscript for this paper was submitted for review and possible publication on November 7, 00; approved on December 30, 004. This paper is part of the Journal of Surveying Engineering, Vol. 131, No. 4, November 1, 005. ASE, ISSN /005/ / $5.00. The X-axis denotes the equator, positive to the east, and the Y-axis denotes the central meridian, positive to the north. The functions or equations define a map projection in general. This conversion or transformation from geographical to plane coordinates is called the forward transformation and is the normal practice in cartography. A frequently faced problem in cartography is calculating the geographical coordinates from the forward projection equations. This process is commonly called inverse mapping. Although the inverse equations for many projections are already known, in some cases they must be developed Yang et al However, developing the inverse equations has sometimes proven difficult due to the complex projection equations. In some cases where projection parameters are not exactly known, numerical methods can be employed Bildirici 003. The Winkel tripel projection s inverse equations cannot be derived easily. In this study, an iterative algorithm is described for the inverse solution of the Winkel tripel projection using partial derivatives. A computer program is also developed based on this algorithm, and the test results are also presented. Winkel Tripel Projection The Winkel tripel projection was developed by Oswald Winkel in 191 by averaging the cylindrical equidistant equirectangular and Aitoff projections Winkel 191. Winkel himself applied the German term tripel in English, triple, because he considered it a compromise of the properties of three elements area, angle, and distance which resulted in a lower distortion distributed uniformly overall Kessler 000. artographers have suggested that it is suitable for whole-world applications due to the distortion characteristics Francula 1971; Ozgen and Ucar 198; apek 001. The Winkel tripel projection is a modified azimuthal projection that is neither conformal nor of equal area, like the Winkel I and II. However, by using L. P. Lee s definitions, the tripel can also be classified as a polyconic Lee The functions for the Winkel tripel projection are presented as follows: where f 1 i, i = 1 D 1/ cos i sin i + i cos 0 X =0 R f i, i = 1 D 1/ sin i + i Y R =0 D = arccos cos cos =1 cos cos 5 and 0 =standard parallel chosen by Winkel as 50 8 in the equidistant cylindrical component of the projection Winkel 191; 3 4 JOURNAL OF SURVEYING ENGINEERING ASE / NOVEMBER 005 / 15

2 Francula 1971; Bugayevsky and Snyder 1995; Kessler 000. The partial derivatives for these functions are Francula 1971; Ucar and Ipbuker 1998; Oztan et al. 001; Ipbuker 00 = sin i sin i D 4 3/ sin i sin i 6 = 1 cos i sin i f = 1 sin i cos + D 3/ cos i cos i sin i + cos 0 i 8 f = 1 sin i sin + D 3/ 1 cos i cos i +1 i 7 8 D 3/ sin i cos i sin i 9 The latitude and longitude values can be calculated using the coordinates of an arbitrary point selected on a map produced in the Winkel tripel projection by using the Newton Raphson iteration method as follows Ipbuker 00; Ipbuker and Bildirici 00 i = f i, i f 1 i, i f f f 13 Newton s iteration in Eqs. 10 and 11 needs an initial guess s, s, which is composed of initial latitude and longitude approximating the given X and Y through the forward projection equations. The initial guess is based upon the functions f 1 and f as defined by Eqs. and 3. These functions are used to examine the change in X i and Y i,tox and Y, respectively. The absolute values of Eqs. 1 and 13 are compared to an accuracy level as follows 14a 14b It can be taken as If the condition being defined with Eqs. 14a and 14b is fulfilled, the iteration will stop. This means X i and Y i are sufficiently close to the selected coordinates X,Y at this iteration step Ipbuker 00; Ipbuker and Bildirici 00. The points on the equator, the poles, and the central meridian must be handled separately. Since analytical solutions for these points are possible, no iteration is needed. If these points are included in the iteration process, iteration may not stop. Actually, it is determined that the iteration does not stop for the points on the equator and the poles. For the points on the equator i.e., Y =0, the geographical coordinates can be calculated as follows i+1 = s i 10 =0 where i+1 = s i 11 X = R 1 + cos 0 15 i = f 1 i, i f f i, i f f 1 For the points on the poles i.e., Y =±R /, the geographical coordinates = Y R Table 1. Part of the Points Used in Test 1 Point number X Y after iteration step after iteration step 16 / JOURNAL OF SURVEYING ENGINEERING ASE / NOVEMBER 005

3 Table. Steps at Point 10 in Table 1 angle unit is radians step X = R cos 0 For the points on the central meridian i.e., X=0 16 = Y R 17 =0 onsidering these cases, the initial values of the geographical coordinates for the iteration can be taken as omputer Program s = Y R X s = R 1 + cos 0 18 In order to determine that the method described above can be used properly, a computer program was developed with the FORTRAN 90 programming language. The program is capable of forward and inverse transformation of Winkel tripel projection. We call it WPROJ. It processes DXF or text files, which must be formatted accordingly. The subroutines for forward and inverse transformation can be found in the Appendix. Some of the points and their coordinates are shown in Table 1. In the table the iteration steps are given for longitudes and latitudes. Since the iteration steps are at acceptable levels, and the steps for longitudes and latitudes do not differ significantly, it can be assumed that the initial guess in Eq. 18 is suitable. For one of the points, the iteration steps are given in detail in Table. The plane coordinates and iterative geographical coordinates were calculated in the same program session, hence the test above looks a bit artificial. Therefore a second test was done. In the Test, a DXF file that contains worldwide country data was transformed to Winkel plane coordinates, which is shown in Fig. 1. From these coordinates, the geographical coordinates were obtained with iteration. Some coordinates of test points are shown in Table 3. The test characteristics and results are as follows: R=6370 km Earth radius ; X and Y coordinates are rounded to 1 decimal places in DXF file ; =10 1 precision ; Number of points= 46,67; Maximum iteration step= 94; and Minimum iteration step= 4. Test of the Program and Method In order to examine if both the method and the program work properly, we performed two tests. In Test 1 we created a 5 5 degree graticule of points. We first calculated the plane coordinates of these points, and then we recalculated the geographical coordinates using the iterative method presented here. The test characteristics and results are as follows: R=1 Earth radius ; X and Y coordinates are not rounded; =10 1 precision ; Number of points= 703; Maximum iteration step=49 =85, =180 ; and Minimum iteration step= 7. Fig. 1. Worldwide country data in Winkel tripel projection data set of test JOURNAL OF SURVEYING ENGINEERING ASE / NOVEMBER 005 / 17

4 Table 3. Sample oordinates of Test X km Y km Step Step In Table 3 the iteration steps are also given. From the table it can be concluded that the iteration steps for latitudes increase toward the equator. It is also significant that the longitudes are calculated with less iteration in this direction. In Test iteration steps are also at acceptable levels as in Test 1, so the initial guess in Eq. 18 seems to be suitable. Since Test includes a large amount of points with homogenous distribution, this initial guess can be used for the whole world. We can conclude from the tests above that the iteration always stops at reasonable steps. With a computer program 94 steps do not cause a large amount of PU time. For example, the whole PU time for the Test transformation of 46,67 points is about 0 s with a P III 550 processor. This time also includes reading from an input file and writing to an output file. Since each DXF file input file and output file contains 661,106 lines, the processing time should be considered reasonable. Appendix. Subroutines for Winkel Projection onclusion The forward projection functions for the Winkel projection are complex. Therefore, to derive the inverse projection functions, a specific method is required. The algorithm used here is a Newton Raphson iteration with partial derivatives. Since the manual calculation using this algorithm is not easy, a computer program was developed. With such a program, data capture from analog maps produced using Winkel tripel projection can be possible; captured data can also be easily integrated into any GIS system. After tests we performed with our program WPROJ we showed that the iteration actually stopped at reasonable steps. The total PU time was also relatively short. We also showed that some special points e.g., points on equator could be analytically transformed, which had to be actually excluded from the iteration due to their negative effect. They cause large numbers of iteration steps, or iteration does not stop. Finally, we suggest that the algorithm presented here can be efficiently used for the inverse transformation of Winkel tripel projection. 18 / JOURNAL OF SURVEYING ENGINEERING ASE / NOVEMBER 005

5 X coordinate axis denotes the equator positive to the east ; Y coordinate axis denotes the central meridian positive to the north ; accuracy level taken as 10 1 ; geographical longitude; geographical latitude; 0 latitude of the standard parallel chosen by Winkel as 50 8 ; and s, s initial latitude and longitude values for the iteration. References Notation The following symbols are used in this paper: R radius of Earth curvature; Bildirici, I. O Numerical inverse transformation for map projections. omput. Geosci., 9, Bugayevski, L., and Snyder, J. P Map projections: A reference manual, Taylor and Francis, London. apek, R Which is the best projection for the world map? Proc., 0th Int. artographic onf., Beijing, hina, Vol. 5, Francula, N Die vorteilhaftesten Abbildungen in der Atlaskartographie. Dissertation, Uniersitaet Bonn. Ipbuker,. 00. An inverse solution to the Winkel tripel projection using partial derivatives. artogr. Geogr. Inf. Syst., 9 1, Ipbuker,., and Bildirici, I. O. 00. A general algorithm for the inverse transformation of map projections using Jacobian matrices. Proc., 3rd Int. onf. on Mathematical & omputational Applications, Konya, Turkey, Kessler, F A visual basic algorithm for the Winkel tripel projection. artogr. Geogr. Inf. Syst., 7, Lee, L. P The nomenclature and classification of map projections. Empire Survey Review, 7 51, Oztan, O., Ipbuker,., and Ulugtekin, N A numerical approach to pseudo-projections on example Franz Mayr projection. J. General ommand of Mapping, 15, in Turkish. Ozgen, M. G., and Ucar, D Investigation of suitable projections for mapping the whole world. J.. General ommand of Mapping, 88, 1 11 in Turkish. Ucar, D., and Ipbuker, Graphic visualisation of deformation ellipses in cartographic projections. J. General ommand of Mapping, 119, in Turkish. Winkel, O Neue gradnetzkombinationen. Petermanns Mitteilungen, 6, Yang, Q., Snyder, J. P., and Tobler, W Map projection transformation: Principles and applications, Taylor and Francis, London. JOURNAL OF SURVEYING ENGINEERING ASE / NOVEMBER 005 / 19