Measurements of Distances from Compound Projection for GIS

Measurements of Distances from Compound Projection for GIS Mohammed Sabri Akresh Staff member at University of Tripoli –Faculty of engineering [email protected] [email protected] ABSTRCT 3 METHODOLOGY The revolution of technologies in the field of geodesy is important for the coordinates system used in surveying work and geographic information system”GIS”, this paper presents a system of coordinates by harmonic equations projection “the projections of united”, who have five projections (Mercator, Lambert, Russell, Lagrange, and the compound of the projection) in one zone coordinate system. The theory of the projections by a harmonic equations as well as Lagrange projection have eight direct algorithms defined Professor Vladimir podshivolev 1998 with a difficulty and very complicated method. This paper has added news direct algorithms for compound projection, as well as presents distortions scale factor for measured distances for Malaysia also smart local system for cities. The methodologies in news maps projections have standard parallels for any zones, whichever; from proposal scientist Grave Shipeshiv 1845, proposed creates a news projection by two projections with two news scales factor k1+k2 =1. This research uses projections of Lambert and Mercator for create news algorithms from them, and other stapes same methods in four projections by harmonic equations and gives name compound projection; it has spatial property for distortion scale factor, where all projections (Mercator, Lambert, Russell and Lagrange) haven’t these properties. Compound projection algorithms creates from direct algorithms Lambert and Mercator and any one has a new scale factor, follows these algorithms [ 1,2.3]. First: direct algorithms of Mercator projection KEYWORDS System, compound projection, algorithms, harmonic equations. coordinates, C2   C1.sin B0 , C1  m0.c.cos B0 , V 2 2B . cos C 0 (tan2 B V 2 ), C3  1 0 6 1 INTRODUCTION The theory of united projections was introduced by prof. Vladimir Podshivalov in 1998, it was aimed for special cases (constructions system coordinates for GIS for countries by 12*12 degrees long and width of zone); In 2009-2012 Dr Akresh found the general law for indirect algorithms for five projections, general law for direct algorithms of Russell projection and also for Lagrange projection. The theory of united projections has local system for big cities; the local system has an advantage in decreasing of distances distortion and very easy to go back to the main of coordinate system. ISBN: 978-0-9891305-2-3 ©2013 SDIWC . . . . . . . . . C1 sin B 0 cos B 0  2702765  479001600 2 2 2  17460701 tan B 0  189410408 tan B 0  10 C12   tan 10 B 0  16889786 tan  11272037  44281 tan 8  41248981 2 4 B0   517812174 2 B 0  2819266 tan 2  285183772 tan 2 8 tan B0  6 tan 6 4 B0  B0  (1) B 0 ); Second: direct algorithms of Lambert projection Cj  C1  1( j 1) sin B0 ( j 1) , , , J  1,2,...,n. j! 317 C1  C1 C3  , C2   C1 sin 2 B0 6 C1 sin B0 , 2 C , C 4   1 sin 3 B0 ,... (2) 24 Uses compound projection two scale factor and must to be equal one ; if k1= 0.5 , k2 =0.5 created projection of Russell; if using other news two scale factor created different geometric figures “new kind for compound projection”; for chose two scale factor use method adjustment by least square method observation, first contracted of equations and following[4,5] a b Minimum distortion distance or scale factor =1 1000000.00 m  0  k1  1  k 2  1  1  X 12   Y12      m0  k m  1  k1   2 2  2  2 2 m R m R 0 0  0 0    2  X 2   Y22      m0  m  1  k1   k 2 2  2   2m0 R0   2m0 R0   X n2   Yn2      m0 m  1  k1   k  2 2  2  m R 2 2 m R 0 0  0 0    Second step uses method of observation and following Qe  AQAT , , , , , Q  1   1  2  X 1 A   2 m 0 R0  2  Y1  2m R 0 0  1 X 22  2m0 R0 Y22  2m0 R0 F  m0 1 X n2  2 m 0 R0 Yn2  2 m 0 R0 ... ... ... K  Qe1 F m0 ... 1 , V  AT K V  m k1 k2  .   0  1  1   , (4) This research presents some news figures for scale factor and following: C d Figure1, Some compound Projection 4 CASE STUDY The geographic position of Malaysia near of equator, and also it is oblique by 55 degrees, for this reason will be used new coordinates system by compound projection by geometric ellipsoid figure comparison universal transverse Mercator UTM;here givenk1=0.62329276, k2=0.37670724, and scale factor m0=0.9998416, standard parallel 4° N, center meridian 102° E and100000.00m for coordinate y in center meridian edges of zone(20°X20°) by using WGS84 and gives name main coordinates system for Malaysia see figure (1-c); as well as local coordinate system for any city in Malaysia by High accuracy equal to global Positioning system GPS. This paper will studies some cities in Malaysia “Kuala Lumpur, Pulou Pinang ,Ipoh and Johor Bahru”; All studies results listed in tables( 1,2,3,4 ), follows. The relationship between Local and maim system can be determined by equation dx local mlocal  dx main m main X local  X 0  dx local X main  X 0  dx main ISBN: 978-0-9891305-2-3 ©2013 SDIWC (5) 318 Table 1, Kuala Lumpur city – comparison between UTM and Compound projection Ellipsoid parameters WGS 84 a= 6378137.00m, b=6356752.314m φ= 3° 08' N, λ=101° 36' E, X0=442304.3119 projections UTM –Mercator Pro. of compound standard parallel B 0= 4° 00' N zone 47 , L=99°00' E Center meridian L= 102° 00'E Scale factor 0.9996 Main 0.9998416 Local 0.99993512 x 346689.5255 346492.709 346483.747 y 788984.5475 955543.242 955539.084 Scale factor, point 1.00063391 0.9998996 0.9999932 Curves of parallels 358.583 9.365 9.366 φ= 3° 05' N, λ=101° 40' E, X0=442304.3119 projections UTM –Mercator Pro. of compound standard parallel B 0= 4° 00' N zone 47 , L=99°00' E Center meridian L= 102° 00'E Scale factor 0.9996 Main 0.9998416 Local 0.99993512 X meters 335642.8479 335433.204 335423.207 Y meters 796427.4203 962948.912 962945.446 Scale factor, point 1.00068786 0.9999054 0.9999989 Curves of parallels 365.231 6.388 6.387 UTM -Mercator Pro. of compound, Local Distance directly from coordinates, meters 13320.114±8.796 13311.264±0.054 Distance by geodetic problems, meters 13311.318 Table 2, Pulou Pinang city – comparison between UTM and Compound projection Ellipsoid parameters WGS 84 a= 6378137.00m, b=6356752.314m φ= 2° 50' N, λ=100° 50' E, X0=442304.3119 projections UTM –Mercator Pro. of compound standard parallel B 0= 4° 00' N zone 47 , L=99°00' E Center meridian L= 102° 00'E Scale factor --Main 0.9998416 Local 0.99964582 x 589670.4547 589954.478 589925.567 y 647741.7726 815259.896 815296.07 Scale factor, point 0.99987018 1.0002062 1.0000103 Curves of parallels 159.808 226.306 226.262 φ= 2° 52' N, λ=100° 52' E, X0=442304.3119 projections UTM –Mercator Pro. of compound standard parallel B 0= 4° 00' N zone 47 , L=99°00' E Center meridian L= 102° 00'E Scale factor --Main 0.9998416 Local 0.99964582 X meters 598905.4947 599149.441 599118.729 Y meters 656956.0943 824519.954 824554.315 Scale factor, point 0.99990493 1.0001935 0.9999976 Curves of parallels 183.199 206.399 206.359 UTM -Mercator Pro. of compound, Local Distance directly from coordinates, meters 13045.677±1.469 13047.196±0.050 Distance by geodetic problems, meters 13047.146 Table3, Ipoh city – comparison between UTM and Compound projection Ellipsoid parameters WGS 84 a= 6378137.00m, b=6356752.314m φ= 4° 35' N, λ=101° 03' E, X0=442304.3119 projections UTM –Mercator Pro. of compound standard parallel B0= 4° 00' N zone 47, L=99°00' E Center meridian L= 102° 00'E Scale factor --Main 0.9998416 Local 0.9999076 x 501401.747 501337.538 501341.435 y 727453.031 894587.133 894580.175 Scale factor, point 1.00024042 0.9999435 1.0000095 Curves of parallels 321.718 66.020 66.024 φ= 4° 39' N, λ=101° 07' E, X0=442304.3119 projections UTM –Mercator Pro. of compound standard parallel B 0= 4° 00' N ISBN: 978-0-9891305-2-3 ©2013 SDIWC 319 Scale factor X meters Y meters Scale factor, point Curves of parallels zone 47, L=99°00' E --514327.7769 734814.9892 1.00028255 351.746 Distance directly from coordinates, meters Distance by geodetic problems, meters Center meridian L= 102° 00'E Main 0.9998416 Local 0.9999076 514228.921 514233.668 902000.054 901993.585 0.9999398 1.0000058 58.026 58.029 UTM -Mercator Pro. of compound, Local 14875.506±3.887 14871.729±0.11 14871.619 Table 4, Johor Bahru city – comparison between UTM and Compound projection Ellipsoid parameters WGS 84 a= 6378137.00m, b=6356752.314m φ= 1° 57' N, λ=103° 41' E, X0=442304.3119 projections UTM –Mercator Pro. of compound standard parallel B 0= 4° 00' N zone 48,L=105°00' E Center meridian L= 102° 00'E Scale factor --Main 0.9998416 Local 0.99937778 x 160311.505 160458.674 160589.42 y 353521.603 1187384.771 1187297.845 Scale factor, point 0.99986561 1.0004826 1.0000184 Curves of parallels 42.595 115.729 115.729 φ= 1° 33' N, λ=103° 44' E, X0=442304.3119 projections UTM –Mercator Pro. of compound standard parallel B 0= 4° 00' N zone 48,L=105°00' E Center meridian L= 102° 00'E Scale factor --Main 0.9998416 Local 0.99937778 X meters 171364.127 171528.507 171654.118 Y meters 359091.402 1192937.443 1192847.94 Scale factor, point 0.99984579 1.0004704 1.0000063 Curves of parallels 38.875 152.924 152.924 UTM -Mercator Pro. of compound, Local Distance directly from coordinates, meters 12376.717 ±1.787 12378.655±0.151 Distance by geodetic problems, meters 12378.504 All tables shows the distortions of distances measured by rectangular coordinates used UTM universal transverse Mercator and compound projections “main, local” comparison by measurement distances from geodetic problems, and best results for local compound projection.  Very easy transformation rectangular coordinate from local to main compound projection. 6 REFERENCES [1] M.S Akresh, Development of scientific and technical 5 CONCLUSION The coordinate system by compound projection with locals systems better than of old coordinates systems by UTM for Malaysia, and results follows.  Minimums distortions of distances in local compound projection better than of UTM;  Errors in local compound projection for shorts distances 0.00- 200000.00 m ± 0.00 0.150 m; while in UTM ±0.00 – 10.00 m; ISBN: 978-0-9891305-2-3 ©2013 SDIWC [2] [3] [4] [5] foundations and technology of forming a coordinate system for geographic information systems in the Libya, Ph.D dissertation Dept. applied geodesy, Polotsk State Univ., Novopolotsk, Belarus, p. 131, 2010. M. S Akresh, Advance geodesy and cartographic for GIS, 1st ed., Aalmustakbe, Libya , pp. 23-185,2012. V. Morozov, Course spheroid geodesy, 2nd ed., Nedra, Ministry of Education, Moscow, pp.213-253, 1979. U. Padshyvalau, The theoretical basis for forming coordinate environment for GIS, 1st ed., PSU, Novopolotsk, pp. 8-52, 1998. H. Yury, Automated design of coordinate system for long linear objects, scanGIS 2007, sweeden, 2007. 320