1Rujukan Kami: JUPEM 18/7/2.148 Jld. 3 ( 22 ) Tarikh: 5 Jun 2009 Semua Pengarah Ukur dan Pemetaan Negeri PEKELILING KETUA PENGARAH UKUR DAN PEMETAAN BILANGAN 3 TAHUN 2009 GARIS PANDUAN MENGENAI PENUKARAN KOORDINAT, TRANSFORMASI DATUM DAN UNJURAN PETA UNTUK TUJUAN UKUR DAN PEMETAAN 1. TUJUAN Pekeliling ini bertujuan untuk memberikan garis panduan mengenai kaedah-kaedah penukaran koordinat, transformasi datum dan unjuran peta bagi kegunaan kerja-kerja ukur dan pemetaan. 2. LATAR BELAKANG 2.1 Jabatan Ukur dan Pemetaan Malaysia (JUPEM) bertanggungjawab melaksanakan kerja-kerja ukur hakmilik tanah dan pemetaan asas. Dalam memenuhi tanggungjawab tersebut, JUPEM telah menyediakan produk dan perkhidmatan yang berasaskan koordinat bagi memenuhi keperluan semasa pelanggan. 2.2 Mutakhir ini teknologi satelit Global Navigation Satellite System (GNSS) semakin giat digunakan di dalam melaksanakan kerja- kerja ukur dan pemetaan bagi mendapatkan koordinat dengan cepat dan tepat di Malaysia. 2 2.3 Bagaimanapun, kaedah penentududukan dan penentuan koordinat berasaskan teknologi satelit GNSS telah menggunakan datum geodetik yang berlainan daripada datum-datum geodetik yang sedia ada di Malaysia. Ketidakharmonian di antara koordinat yang berpunca daripada perbezaan datum geodetik ini, jika sekiranya tidak diuruskan dengan baik, boleh menimbulkan kesilapan di dalam kerja-kerja pengukuran serta produk ukur dan pemetaan. 2.4 Oleh itu, maklumat mengenai kaedah penukaran koordinat, transformasi datum dan unjuran peta yang sedia ada wajar dikemaskini dan didokumentasikan supaya ianya sentiasa relevan dengan kehendak dan tuntutan semasa serta dapat dijadikan panduan dan rujukan kepada para pengguna dalam menguruskan kerja-kerja yang mempunyai kaitan dengan koordinat. Sehubungan dengan itu, Pekeliling ini diharapkan akan dapat membantu pengguna-pengguna produk-produk pemetaan, kadaster, utiliti dan sebagainya untuk memahami kaedah-kaedah tersebut dan seterusnya dapat menggunakannya dengan cara yang betul di dalam urusan kerja mereka, terutamanya di dalam era penggunaan teknologi GNSS yang semakin meluas di Malaysia. 3. PERKEMBANGAN PEMBANGUNAN INFRASTRUKTUR GEODETIK DI MALAYSIA 3.1 Sejak penubuhan Jabatan Ukur dan Pemetaan Malaysia (JUPEM) lebih daripada 120 tahun yang lalu, datum rujukan geodetik yang digunapakai untuk kegunaan ukur dan pemetaan telah mengalami pelbagai perubahan. Dalam hal ini, sebelum tahun 1990an, Datum Kertau telah pun menjadi tulang belakang kepada Malayan Revised Triangulation 1968 (MRT68) di Semenanjung Malaysia, manakala bagi Sabah, Sarawak dan 3 Labuan pula, Borneo Triangulation 1968 (BT68) merujuk kepada Datum Timbalai sebagai asasnya. 3.2 Dengan perkembangan teknologi GNSS yang meluas sekitar tahun 1980-an, JUPEM telah membangunkan rangkaian kawalan ukur geodetik yang baru dengan menggunakan teknologi tersebut untuk menentukan koordinat bagi stesen-stesen kawalan. Bagi Semenanjung Malaysia, rangkaian ini telah ditubuhkan dalam tahun 1994 dan dikenali sebagai Peninsular Malaysia Geodetic Scientific Network 1994 (PMGSN94). Manakala di Sabah, Sarawak dan Labuan pula, rangkaian kawalan ukur geodetiknya adalah East Malaysia Geodetic Scientific Network 1997 (EMGSN97), yang ditubuhkan pada tahun 1997. 3.3 Antara tahun 1998 dan 2001 pula, JUPEM telah membangunkan rangkaian Malaysia Active GPS System (MASS) dan ini telah diikuti dengan Malaysia Real-Time Kinematic GNSS Network (MyRTKnet) antara tahun 2002 dan 2008. Dalam pada itu, pada 26 Ogos 2003, JUPEM telah melancarkan Geocentric Datum of Malaysia (GDM2000), iaitu datum rujukan geodetik baru yang seragam bagi kerja-kerja ukur dan pemetaan di Malaysia. 3.4 Bagaimanapun, sistem rujukan GDM2000 telah mengalami anjakan yang signifikan, lanjutan daripada berlakunya beberapa siri gempabumi besar di Indonesia, terutamanya pada tahun 2004 dan 2005. Oleh hal yang demikian, datum ini telah disemak dan dihitung semula bagi menghasilkan GDM2000 (2009). 3.5 Maklumat lebih lanjut mengenai sistem rujukan koordinat terkandung di dalam Pekeliling Ketua Pengarah Ukur dan Pemetaan Malaysia bilangan 1/2009 bertarikh 25 Mei 2009 bertajuk ‘Garis Panduan Mengenai Sistem Rujukan Koordinat Di dalam Penggunaan Global Navigation Satellite System (GNSS) Bagi Tujuan Ukur dan Pemetaan’. Ia menyenaraikan semua jenis 4 sistem rujukan koordinat yang telah dibangunkan oleh JUPEM di Malaysia disamping memberi maklumat-maklumat teknikal mengenai sistem tersebut dan datum geodetik. 4. GARIS PANDUAN PENUKARAN KOORDINAT, TRANSFORMASI DATUM DAN UNJURAN PETA UNTUK TUJUAN UKUR DAN PEMETAAN Penerangan lebih lanjut tentang amalan penggunaan penukaran koordinat, transfomasi datum dan unjuran peta terkandung di dalam dokumen Technical Guide to the Coordinate Conversion, Datum Transformation and Map Projection seperti di Lampiran ‘A’ yang disertakan. Intisari garis panduan tersebut adalah seperti berikut: Perenggan Perkara 1. INTRODUCTION 2. COORDINATE CONVERSION 2.1 GEOGRAPHICAL AND CARTESIAN COORDINATES 2.2 CONVERSION BETWEEN GEOGRAPHICAL COORDINATES AND CARTESIAN COORDINATES 2.3 TEST EXAMPLE 3. DATUM TRANSFORMATION 3.2 INTRODUCTION 3.2 BURSA-WOLF DATUM TRANSFORMATION FORMULAE 3.3 MULTIPLE REGRESSION MODEL 3.4 TEST EXAMPLES 4. MAP PROJECTION 4.1 RECTIFIED SKEW ORTHOMORPHIC PROJECTION (RSO) 4.2 CASSINI-SOLDNER PROJECTION 4.3 POLYNOMIAL FUNCTION 5 4.4 RE-DEFINITION OF STATE ORIGINS IN GDM2000 AND GDM2000 (2009) 4.5 TEST EXAMPLES 5. CONCLUSION 5. TARIKH BERKUATKUASA Pekeliling ini adalah berkuatkuasa mulai tarikh ianya dikeluarkan. Sekian, terima kasih. “BERKHIDMAT UNTUK NEGARA” ( DATUK HAMID BIN ALI ) Ketua Pengarah Ukur dan Pemetaan Malaysia Salinan kepada: Timbalan Ketua Pengarah Ukur dan Pemetaan Pengarah Ukur Bahagian (Pemetaan) Pengarah Ukur Bahagian (Kadaster) Pengarah Bahagian Geospatial Pertahanan Setiausaha Bahagian (Tanah, Ukur dan Pemetaan) Kementerian Sumber Asli dan Alam Sekitar Pengarah Institut Tanah dan Ukur Negara (INSTUN) Kementerian Sumber Asli dan Alam Sekitar 6 Pengarah Pusat Infrastuktur Data Geospatial Negara (MaCGDI) Kementerian Sumber Asli dan Alam Sekitar Ketua Penolong Pengarah Unit Ukur Tanah, Cawangan Pengkalan Udara dan Maritim Ibu Pejabat Jabatan Kerja Raya Malaysia Penolong Pengarah Unit Ukur Tanah, Bahagian Kejuruteraan Awam Ibu Pejabat Jabatan Perumahan Negara Setiausaha Lembaga Jurukur Tanah Semenanjung Malaysia Setiausaha Lembaga Jurukur Tanah Sabah Setiausaha Lembaga Jurukur Tanah Sarawak Technical Guide to the Coordinate Conversion, Datum Transformation and Map Projection JABATAN UKUR DAN PEMETAAN MALAYSIA 2009 Lampiran ‘A’ i TABLE OF CONTENTS 1. INTRODUCTION 1 2. COORDINATE CONVERSION 5 2.1 Geographical and Cartesian Coordinates 5 2.2 Conversion between Geographical Coordinates and Cartesian Coordinates 5 2.3 Test Example 8 3. DATUM TRANSFORMATION 9 3.1 Introduction 9 3.2 Bursa-Wolf Datum Transformation Formulae 10 3.3 Multiple Regression Model 12 3.4 Test Examples 15 4. MAP PROJECTION 20 4.1 Introduction 20 4.2 Rectified Skew Orthomorphic Projection (RSO) 20 4.3 Cassini-Soldner Map Projection 30 4.4 Polynomial Function 33 4.5 Redefinition of State Origins in GDM2000 and GDM2000 (2009) 34 4.6 Test Examples 35 5. CONCLUSION 39 REFERENCES 1 1. INTRODUCTION 1.1 The Department of Survey and Mapping Malaysia (JUPEM) defines and maintains the Coordinate Reference System (CRS) and the Vertical Reference System (VRS) for the whole country. It establishes and manages these geodetic infrastructures for the purpose of cadastral survey, mapping, engineering and scientific research. The coordinate reference systems that have been introduced and used since the late 1800s in Malaysia are listed in Table 1. Table 1: Coordinate Reference Systems in Malaysia No. Coordinate Reference System Coordinate System Geodetic Datum 1. Malayan Revised Triangulation 1968 (MRT68) KERTAU Ellipsoid: Modified Everest 2. Borneo Triangulation 1968 (BT68) TIMBALAI Ellipsoid: Modified Everest 3. Peninsular Malaysia Geodetic Scientific Network 1994 (PMGSN94) WGS84 Ellipsoid: WGS84 Reference Frame: WGS84 Epoch: 1987.0 4. East Malaysia Geodetic Scientific Network 1997 (EMGSN97) WGS84 Ellipsoid: WGS84 Reference Frame: WGS84 (G783) Epoch: 1997.0 5. Malaysia Active GPS System (MASS) GDM2000 Ellipsoid: GRS80 Reference Frame: ITRF2000 Epoch: 2000.0 6. Malaysia Primary Geodetic Network 2000 (MPGN2000) GDM2000 Ellipsoid: GRS80 Reference Frame: ITRF2000 Epoch: 2000.0 GDM2000 (2009) Ellipsoid: GRS80 Reference Frame: ITRF2000 Epoch: 2000.0 2 Table 1: (continued) No. Coordinate Reference System Coordinate System Geodetic Datum 7. Malaysia Real-Time Kinematic GNSS Network (MyRTKnet) GDM2000 Ellipsoid: GRS80 Reference Frame: ITRF2000 Epoch: 2000.0 GDM2000 (2009) Ellipsoid: GRS80 Reference Frame: ITRF2000 Epoch: 2000.0 1.2 Figure 1 represents a schematic diagram to assist users in navigating between the different types of coordinates. All Coordinate Reference Systems (CRS) available in Malaysia are shown; some are on the same geodetic datum and others, on different ones. Coordinate conversions which do not involve a change of datum are shown as dashed lines. Datum transformations, on the other hand, involve a change of datum and are shown as thick solid lines and map projection as thin solid lines. 1.3 This technical guide is produced to assist users in understanding the concept and procedures involved in the process of coordinate conversion, datum transformation and map projection as practiced in Malaysia. 3 Figure 1: Relationship between various coordinate reference system Coordinate conversion Datum transformation Map projection BOX 2 BOX 2 BOX 2 BOX 3 BOX 7 GRS80 El l i psoi d GDM2000 ( 2009) BOX 1 o ì h X Y Z BOX 1 o ì h X Y Z GDM2000 Modi f i ed Ever est El l i psoi d MRT68/ BT68 BOX 1 o ì h X Y Z N E N E CASSI NI MRSO/ BRSO BOX 4 WGS84 El l i psoi d PMGSN94/ EMGSN97 o ì h X Y Z BOX 1 Geocent r i c MRSO/ BRSO Geocent r i c CASSI NI N E N E BOX 5 BOX 6 4 Notes to Figure 1: Box 1: Coordinate conversion between geographical coordinate and cartesian coordinate Box 2: Transformation between various datums using Bursa-Wolf formulae Box 3: Map projection of MRT68 / BT68 geographical coordinate to Rectified Skew Orthomorphic (RSO) plane coordinates Box 4: Coordinate transformation from RSO to Cassini using polynomial function Box 5: Map projection of GDM2000 geographical coordinate to Geocentric Cassini plane coordinates Box 6: Map projection of GDM2000 geographical coordinate to Geocentric RSO plane coordinates Box 7: Datum transformation from GDM2000 to GDM2000 (2009) using multiple regression model 5 2. COORDINATE CONVERSION 2.1 Geographical and Cartesian Coordinates 2.1.1 Three-dimensional geographical coordinates can be defined with respect to an ellipsoid as follows: Latitude: the angle north or south from the equatorial plane Longitude: the angle east or west from the prime meridian Height: the distance above the surface of the ellipsoid. 2.1.2 A set of cartesian coordinates is defined with the three axes at the origin at the center of the ellipsoid, such that: Z-axis: is aligned with the minor (or polar) axis of the ellipsoid X-axis: is in the equatorial plane and aligned with the prime meridian Y-axis: forms a right-handed system 2.1.3 In this regard, positions in geographical coordinates of latitude, longitude and height (o, ì, h) can be converted into cartesian coordinates (X, Y, Z) and vice-versa. 2.2 Conversion between Geographical Coordinates and Cartesian Coordinates 2.2.1 The conversion of three-dimensional coordinates from geographical to cartesian or vice versa can be carried out through the knowledge of the parameters of an adopted reference ellipsoid (Figure 2). The Geographical Coordinates Cartesian Coordinates Latitude, Longitude and Height (o, ì, h) X Y Z 6 forward conversion from geodetic coordinates ( h , , ì o ) to cartesian coordinate ( Z Y X , , ) is as follows: Figure 2: Geographical and cartesian coordinates ì o cos cos ) ( h N X + = ì o sin cos ) ( h N Y + = o sin | | . | \ | + = h N a b Z 2 2 where the prime vertical radius of curvature (N) is: 2 1 2 2 2 2 ) sin cos ( o o b a a N + = with: P Z Y X h a b N ì o 7 a : the semi-major axis of the reference ellipsoid; b : the semi-minor axis of the reference ellipsoid; 2.2.2 The non-iterative reverse conversion from cartesian coordinates ( Z Y X , , ) to geodetic coordinates ( h , , ì o ) is as follows: ÷ + = u a e P u b Z 3 2 3 2 cos sin arctan c o = X Y arctan ì o o o 2 2 1 sin sin cos e a Z P h ÷ ÷ + = with: = bP aZ u arctan 2 2 Y X P + = 2 2 1 e e ÷ = c 2 2 2 2 a b a e ÷ = where, u : the parametric latitude; e : the first eccentricity of the reference ellipsoid; c : the second eccentricity of the reference ellipsoid. 8 2.3 Test Example Geographical Coordinate Latitude = N 06 27' 00.56909" Longitude = E 100 16' 47.05076" Ellipsoidal Height (h) = 18.078 m BOX 1 3D-Cartesian Coordinate (ECEF) X-Axis = -1131051.654 m Y-Axis = 6236311.800 m Z-Axis = 711748.112 m Coordinate Conversion using GRS80 Ellipsoid 9 3. DATUM TRANSFORMATION 3.1 Introduction 3.1.1 Datum transformation is a computational process of converting a position given in one coordinate reference system into the corresponding position in another coordinate reference system. It requires and uses the parameters of the transformation and the ellipsoids associated with the source and target coordinate reference systems. For example, the source can be coordinate reference system 1 and the target can be coordinate reference system 2: No. Coordinate Reference System 1 Coordinate Reference System 2 1 Geocentric Datum of Malaysia (GDM2000) Geocentric Datum of Malaysia GDM2000 (2009) Peninsular Malaysia Geodetic Scientific Network 1994 (PMGSN94) East Malaysia Geodetic Scientific Network 1997 (EMGSN97) Malayan Revised Triangulation 1968 (MRT68) Borneo Triangulation 1968 (BT68) (Sabah) Borneo Triangulation 1968 (BT68) (Sarawak) 2 Peninsular Malaysia Geodetic Scientific Network 1994 (PMGSN94) Malayan Revised Triangulation 1968 (MRT68) 3 East Malaysia Geodetic Scientific Network 1997 (EMGSN97) Borneo Triangulation 1968 (BT68) (Sabah) Borneo Triangulation 1968 (BT68) (Sarawak) 10 3.1.2 The transformation parameter values associated with the transformation can be determined empirically from a measurement or a calculation process. The parameters are computed based on coordinates of control stations which are common to different datums. They are generated through least square analysis using various models accepted by the global geodetic community. 3.1.3 Datum transformation can be accomplished by many different methods. A simple three parameter conversion can be accomplished by conversion through Earth-Centred Earth Fixed (ECEF) cartesian coordinates from one reference datum to another by three origin offsets that approximate differences in rotation, translation and scale. A complete datum conversion is usually based on seven parameter transformations, which include three translation parameters, three rotation parameters and a scale. 3.2 Bursa-Wolf Datum Transformation Formulae 3.2.1 Bursa-Wolf formulae is a seven-parameter model for transforming three-dimensional cartesian coordinates between two datums (see Figure 3). This transformation model is more suitable for satellite datums on a global scale (Krakwisky and Thomson, 1974). The transformation involves three geocentric datum shift parameters ( Z Y X A A A , , ), three rotation elements ( Z Y X R R R , , ) and a scale factor ( L A + 1 ). 3.2.2 The model in its matrix-vector form could be written as ( Burford 1985): A + ÷ A + ÷ ÷ A + + A A A = MRT MRT MRT X Y X Z Y Z WGS WGS WGS Z Y X L R R R L R R R L Z Y X Z Y X 1 1 1 84 84 84 11 Figure 5: Bursa-Wolf 3-D model transformation where; 84 84 84 , , WGS WGS WGS Z Y X : are the global datum (WGS84) cartesian coordinates; MRT MRT MRT Z Y X , , : are the local datum (MRT) cartesian coordinates. In order to convert the cartesian coordinates of XYZ to the geographical coordinates of o ì h, ellipsoid properties for the respective datum as listed in Table 2 below are used: 12 Table 2: Ellipsoid Properties No. Ellipsoid semi-major axis, a (m) flattening, f (m) 1 Geodetic Reference System 1980 (GRS80) 6378137.000 298.2572221 2 World Geodetic System 1984 (WGS84) 6378137.000 298.2572236 3 Modified Everest (Peninsular Malaysia) 6377304.063 300.8017 4 Modified Everest (East Malaysia) 6377298.556 300.8017 3.3 Multiple Regression Model (i) Displacement Computation The computation and modelling of differences in the coordinate between the two systems, i.e. GDM2000 and GDM2000 (2009) were carried out using their respective geographical coordinates in the format of (o, ì and h). These differences are then converted to the local geodetic horizon to avoid mathematical errors for some very small values. The conversion from geographical system to local geodetic system used the following factor, i.e. 1”= 30 meter. The differences in the three components are computed separately by using the following formulae: ANorth (N) = (o ” GDM2000 - o ” GDM2000 (2009) ) x 30 AEast (E) = (ì ” GDM2000 - ì ” GDM2000 (2009) ) x 30 AHeight (U) = (h GDM2000 - h GDM2000 (2009) ) 13 (ii) Displacement Modelling The differences in coordinate of every GPS station are similarly computed between the two systems, i.e. GDM2000 and GDM2000 (2009) using their respective coordinates in the format of (E and N). The coordinate differences are then gridded to derive the Regression Coefficient. The gridding method used is the polynomial regression with the power to the second order. The surface definition used the bi-linear saddle regression coefficient with the following formulae: Z(E,N,U) = A 00 + A 01 N + A 10 E + A 11 EN where, Z = Value of regression coefficient or the value of displacement correction for each component (i.e. East, North and Up) E = East coordinate in decimal degree N = North coordinate in decimal degree (iii) Basic Formulae To convert the coordinate in GDM2000 to GDM2000 (2009), the following formulae shall be used: GDM2000 (2009) = GDM2000 + correction o ” GDM2000 (2009) = o ” GDM2000 + (Z N / 30) ì ” GDM2000 (2009) = ì ” GDM2000 + (Z E / 30) h GDM2000 (2009) = h GDM2000 + Z U 14 where, o ” = Latitude in second of arc ì ” = Longitude in second of arc h ” = Ellipsoidal height in metre Z N = Displacement correction in northing Z E = Displacement correction in easting Z U = Displacement correction in height where, Z(E,N, U) = A 00 + A 01 N + A 10 E + A 11 EN and A 00, A 01, A 10 and A 11 are the coefficients of the multiple regression model. 15 3.4 Test Examples (i) GDM2000 to GDM2000 (2009) (ii) GDM2000 to PMGSN94 GDM2000 Coordinate Latitude = N 01° 51' 27.04635" Longitude = E 102° 56' 31.75616" Ellipsoidal Height (h) = 7.724 m BOX 7 GDM2000 (2009) Coordinate Latitude = N 01° 51' 27.04737" Longitude = E 102° 56' 31.75374" Ellipsoidal Height= 7.757 m Multiple Regression PMGSN 94 Coordinate Latitude = N 01° 51 ' 27.08807 " Longitude = E 102 ° 56' 31.72178 " Ellipsoidal Height = 7.118 m GDM2000 Coordinate Set Latitude = N 01° 51 ' 27.04635 " Longitude = E 102 ° 56' 31.75616 " Ellipsoidal Height (h) = 7.724 m BOX 2 Bursa -Wolf 7- Parameter Transformation 16 (iii) GDM2000 / GDM2000 (2009) to EMGSN97 (iv) GDM2000 to MRT68 GDM2000 Coordinate Latitude = N 01° 51' 27.04635 " Longitude = E 102° 56' 31.75616" Ellipsoidal Height = 7.724 m MRT Coordinate Latitude = N 01° 51' 27.38567 " Longitude = E 102° 56' 37.52660" Ellipsoidal Height (h) = 2.338 m BOX 2 Bursa-Wolf 7-Parameter Transformation 17 (v) GDM2000 / GDM2000 (2009) to BT68 (Sabah) BOX 2 Bursa-Wolf 7-Parameter Transformation BT68 (Sabah) Coordinate Latitude = N 06 52' 46.04669" Longitude = E 116 50' 37.60565" Ellipsoidal Height (h) = 1.693 m GDM2000/GSM2000(2009) Coordinate Latitude = N 06 52' 42.45022" Longitude = E 116 50' 47.58835" Ellipsoidal Height (h) = 52.620 m (vi) GDM2000 / GDM2000 (2009) to BT68 (Sarawak) 54’ 41.51240” ‘ 49.44225” 18 (vii) PMGSN94 to MRT68 (viii) EMGSN97 to BT68 (Sabah) BOX 2 Bursa-Wolf 6-Parameter Transformation PMGSN94 Coordinate Latitude = N 01° 51' 27.08807 " Longitude = E 102° 56' 31.72178" Ellipsoidal Height = 7.118 m MRT Coordinate Latitude = N 01° 51' 27.38567 " Longitude = E 102° 56' 37.52660" Ellipsoidal Height (h) = 2.338 m 19 (ix) EMGSN97 to BT68 (Sarawak) BT68 Coordinate (Sarawak) Latitude = N 01 51' 41.51179" Longitude = E 112 51' 49.44453" Ellipsoidal Height (h) = 933.784 m EMGSN94 Coordinate Latitude = N 01 54' 39.30200" Longitude = E 112 52' 01.25200" Ellipsoidal Height (h) = 977.000 m BOX 2 Bursa-Wolf 7-Parameter Transformation 20 4. MAP PROJECTION 4.1 Introduction 4.1.1 Typically, there are many different methods for projecting longitude and latitude in coordinate reference system 1 onto a flat map in the same system: No. Coordinate Reference System 1 Map Projection System 1 1 Geocentric Datum of Malaysia GDM2000 (2009) Geocentric Datum of Malaysia (GDM2000) Geocentric Cassini-Soldner 2 Geocentric Datum of Malaysia GDM2000 (2009) Geocentric Datum of Malaysia (GDM2000) Geocentric Malayan Rectified Skew Orthomorhic (Peninsular Malaysia) 3 Geocentric Datum of Malaysia GDM2000 (2009) Geocentric Datum of Malaysia (GDM2000) Geocentric Borneo Rectified Skew Orthomorhic (Sabah and Sarawak) 4 Malayan Revised Triangulation 1968 (MRT68) Malayan Rectified Skew Orthomorhic (Peninsular Malaysia) 5 Rectified Skew Orthomorhic (Peninsular Malaysia) Cassini-Soldner Old 6 Borneo Triangulation 1968 (BT68) Borneo Rectified Skew Orthomorhic (Sabah and Sarawak) 4.2 Rectified Skew Orthomorphic (RSO) Map Projection 4.2.1 The rectified skew orthomorphic (RSO) map projection (Figure 4) is an oblique Mercator projection developed by Hotine in 1947 (Snyder, 1984). This projection is orthomorphic (conformal) and cylindrical. All meridians and parallel are complex curves. Scale is approximately 21 true along a chosen central line (exactly true along a great circle in its spherical form). It is thus a suitable projection for an area like Switzerland, Italy, New Zealand, Madagascar and Malaysia as well. 4.2.2 The RSO provides an optimum solution in the sense of minimizing distortion whilst remaining conformal for Malaysia. Table 3 tabulates the new geocentric RSO parameters for Peninsular Malaysia and East Malaysia. Figure 4: Hotine, 1947 (Snyder, 1984), Oblique Mercator (Source: EPSG) 22 Table 3: The New Geocentric RSO Projection Parameters Peninsular RSO East Malaysia BRSO Ellipsoid Parameters Ellipsoid GRS 80 GRS 80 Major axis, a 6378137.000 Meters 6378137.000 Meters Flattening, 1/f 298.2572221 298.2572221 Defined Parameters. Latitude of Origin, o o Longitude of Origin, ì Rectified to Skew Grid, ¸ o Azimuth of Central Line, o c Scale factor, k False Origin (Easting) False Origin (Northing) D e f i n e d p r o j e c t i o n p a r a m e t e r s c a n b e o b t a i n e d f r o m J U P E M 4.2.3 The notation adopted for use in this section is as follows: o c = latitude of center of the projection. ì c = longitude of center of the projection. o c = azimuth (true) of the center line passing through the center of the projection. ¸ c = rectified bearing of the center line. k c = scale factor at the center of the projection. o = geographical latitude ì = geographical longitude t o = isometric latitude for o= 4° t o = o o u o t u c c c e e k sin sin log tan log ÷ + ÷ | | . | \ | + 1 1 2 2 4 or Semi 23 t = isometric latitude ìo = basic longitude a = semi major axis of ellipsoid b = semi minor axis of ellipsoid f = flattening of ellipsoid ( ) a b a f / ÷ = e = eccentricity of ellipsoid e 2 =(a 2 – b 2 )/a 2 e 1 2 =(a 2 – b 2 )/b 2 p = radius of curvature in the meridian ( ) ( ) 2 3 2 2 2 1 1 / . / o p Sin e e a ÷ ÷ = v = radius of curvature in the prime vertical ( ) o p v 2 2 1 1 Cos e . + = m = scale factor o m = scale factor at the origin ¸ = skew convergence at meridians p = distance from polar axis, m ucos = p R ¸ =rectified convergence of meridians " 6314 . 11 ' 52 36° + = ¸ ¸ R u = skew coordinate parallel to initial line v = skew coordinate at right angles to initial line N = Northing map coordinate E = East map coordinate FE = False Easting at the natural origin. FN = False Northing at the natural origin. 24 4.2.4 The Constants of the Projection (Table 4) are given as follows: ( ) 2 1 4 2 1 1 c e B o cos . ÷ = or ( ) 2 1 o o B A v p = ' o o Bt p A C ÷ | . | \ | = ÷1 cosh To avoid problems with computation of F, if D<1, make D 2 = 1. 6 . 0 sin ÷ = o ¸ Or Basic Longitude: 25 Table 4: Constant of Projection Constant of Projection Peninsular Malaysia East Malaysia Parameter A Parameter A' Parameter B Parameter C Basic Longitude. ì o D e f i n e d p r o j e c t i o n p a r a m e t e r s c a n b e o b t a i n e d f r o m J U P E M 4.2.5 The projection formulae are as follows: (a) Conversion of Geographical to Rectangular and vice versa Forward Case: To compute (E, N) from a given (o, ì): For the Hotine Oblique Mercator (where the FE and FN values have been specified with respect to the origin of the (u, v) axes): 26 The rectified skew coordinates are then derived from: Reverse Case: Compute (o, ì) from a given (E, N): For the Hotine Oblique Mercator: Then the other parameters can be calculated. 27 (b) Convergence of Map Meridians The convergence of the map meridians is defined as the angle measured clockwise from True North to the Rectified Grid North, and is denoted R ¸ (Figure 5): ( ) 6 . 0 sin 1 ÷ ÷ = ÷ ¸ ¸ R " 6314 . 11 ' 52 36° + =¸ where ( ) ( ) ( ) ( ) C Bt B C Bt B o o o + ÷ + ÷ ÷ = cosh cos sinh sin tan tan ì ì ì ì ¸ ¸ o o o o m A Bu m A Bv m A Bu m A Bv ' cosh ' cos tan ' sinh ' sin ¸ + = R ¸ TN RGN Figure 5: Convergence of Map Meridians 28 (c) Scale Factor at any Point The formulas giving the scale factor “ m ” at any point in terms of isometric latitude and longitude and coordinate (v,u) are: ( ) C Bt m A Bu p m A m o o + = cosh ' cosh ' = ( ) ì ì ÷ o o o B m A Bv p m A cos ' cos ' It can be shown that the initial line of the projection has a scale factor that is nearly constant throughout its length. (d) Scale Factor for a Line The Scale factor for a line can be computed from the formula: ( ) 2 3 1 4 6 1 m m m m + + = where 2 1 , m m are the scale factors at the ends of the line and 3 m the scale factor at its mid-point. The scale factor for a line also may be evaluated from the following formula: ( ) ( ) + + + + = 2 2 2 1 2 1 2 2 2 6 1 u u u u m A B C B v m A m o m m m o ' cosh cos ' m m where m m and m v are evaluated for the mid-latitude of the line 1 u and 2 u are the u - coordinate of the points: 29 (e) Arc-to-Chord Correction In Figure 6, if o is the true azimuth of a line, R | is the rectified grid bearing, then u ¸ | o + = + R R where u is given in seconds by the formula ( ) ) ` ¹ ¹ ´ ¦ + ÷ = 3 2 2 1 2 2 1 1 2 u u m A B v v m A B o o ' tanh " sin ' u ( ) ( ) 2 1 2 3 2 1 ì ì o o o v p ÷ ÷ + o o o o k sin sin sin where, ( ) 2 1 2 3 1 3 o o o + = and ( ) 2 1 ì ì ÷ is measured in seconds. For a line not exceeding 113 km (70 miles) in length, the maximum value of the second term of the formula is 0.007”; it can therefore be safely neglected. R ¸ TN RGN R | o u Figure 8: Arc-to-Chord 30 4.3 Cassini-Soldner Map Projection 4.3.1 There are nine state origins used in the coordinate projection in the cadastral system of Peninsular Malaysia. The Cassini-Soldner map projection has been used for over one hundred years and shall continue to be used for cadastral surveys in the new geodetic frame. 4.3.2 The mapping equations are as given in Richardus and Adler, 1974 and the formulas to derive projected Easting and Northing coordinate are as follows: (a) Forward Computation E = FE + v[A – T(A3/6) - (8 –T + 8C)T * A5/120] N = FN + M – Mo + v*tano[A2/2 + (5 – T + 6C)A4/24] where, N, E = Computed Cassini coordinate FE, FN = Cassini State origin coordinate A = (ì - ìo)Coso where, ì = Longitude of computation point ìo = Longitude of state origin o = Latitude of computation point T = Tan2o C = o 2 2 2 1 Cos e e ) ( ÷ V = Radius of curvature in prime vertical = 2 1 2 2 1 / ) ( o Sin e a ÷ 31 M = Meridianal arc distance = a[1 – e 2 /4 – 3e 4 /64 – 5e 6 /256-..)o - (3e2/8 + 3e4/32 + 45e6/1024 + …)sin2o + (15e4/256 + 45e6/1024 +…)sin4o - (35e6/3072 + …)sin6o ..] with o is in radians. Mo = the value of M calculated for the latitude of the chosen origin. (b) Reverse Computation + ÷ ÷ = 24 3 1 2 4 2 1 1 1 1 1 D T D Tan v ) ( p o o o ) 1 1 1 1 0 15 3 1 3 5 3 o ì ì Cos D T T D T D / ) ( + + ÷ + = where, o1 = u 1 + (3e 1 /2 – 27e 1 3 /32+ ..)Sin2u 1 +(21e 1 2 /16 - 55e14/32 + ..)Sin4u 1 + (151e 1 3 /96 + …)Sin6u 1 + (1097e14/512 - …)Sin8u 1 + … p1 = 2 3 1 2 2 2 1 1 / ) ( ) ( o Sin e e a ÷ ÷ v1 = 2 1 1 2 2 1 / ) ( o Sin e a ÷ where, e1 = 2 1 2 2 1 2 1 1 1 1 / / ) ( ) ( e e ÷ + ÷ ÷ u 1 = ...) / / / ( ÷ ÷ ÷ ÷ 256 5 64 3 4 1 6 4 2 1 e e e a M 32 M1 = M o + (N – FN) = M o is the value of M calculated for the latitude of the origin T1 = tan 2 o 1 D = (E – FE)/v1 (c) Scale and Arc-to-Chord Correction for Cassini Projection Figure 7: Scale and Arc-to-Chord Refer to Figure 7, oAB = (t – T)" = ((N b – N a )(E b + 2E a ))/(6R 2 .Sin1") where, Na, Nb, Ea, Eb = Cassini Coordinate (t – T)" = Arc-to-Chord Bearing correction:- (| - o)" = - ((Sino o .Coso o )/(6R 2 .Sin1"))E 2 u B' B D = D' A = A' s' s o AB | o o = T o = t 33 where, E2u = (E2a + EaEb + E2b) Linear correction:- s' = s + [((Cos 2 o o )/(6R 2 ))E 2 u ]s Projected Coordinate on Cassini:- E b = E a + s'.Sin| N b = N a + s'.Cos| 4.4 Polynomial Function The relationship between the MRSO coordinate and the Cassini Soldner Old is defined by a series of polynomial function which make use of the coordinate of the origin of both projections for each state in Peninsular Malaysia. The following are the formulae used: N(MRSO)= ∆N(CAS) + N(OMRSO) + (R1+XA1+YA2+XYA3+X2A4+Y2A5) E(MRSO)= ∆E(CAS) + E(OMRSO) + (R2+XB1+YB2+XYB3+X2B4+Y2B5) N(CAS)= ∆N(MRSO) + N(OCAS) - (R1+XA1+YA2+XYA3+X2A4+Y2A5) E(CAS)= ∆E(MRSO) + E(OCAS) - (R2+XB1+YB2+XYB3+X2B4+Y2B5) where N(MRSO) : MRSO Coordinate in North Component E(MRSO) : MRSO Coordinate in East Component N(CAS) : Cassini Coordinate in North Component E(CAS) : Cassini Coordinate in East Component ∆N(CAS) : Cassini Coordinate - Cassini Coordinate of Origin (North) ∆E(CAS) : Cassini Coordinate - Cassini Coordinate of Origin (East) ∆N(MRSO) : RSO Coordinate - RSO Coordinate of Origin (North) ∆E(MRSO) : RSO Coordinate - RSO Coordinate of Origin (East) X : ∆N(CAS)/10000 or ∆N(MRSO)/10000 Y : ∆E(CAS)/10000 or ∆E(MRSO)/10000 All values of the coordinates must be in unit chains (use 0.11678249 as the multiplying factor to convert from metres) 34 4.5 Re-definition of State Origins in GDM2000 and GDM2000 (2009) In order to prepare the geodetic infrastructure fully for the implementation of e-Kadaster, the re-definition of the nine (9) state cadastral origins need to be carried out. In this regard, JUPEM had successfully carried out GPS observations at all the origins in August and October of 2005 and the results are given in Table 5. Table 5: Re-definition of State Cadastral Origins in GDM2000 and GDM2000 (2009) Coordinates State Station Location GDM2000 (2009) Cassini-Soldner GDM2000 Latitude (North) Longitude (East) Northing Easting Johor Gunung Belumut 2° 02’ 33.20279” 103° 33’ 39.83599” 0.000 0.000 2° 02’ 33.20196” 103° 33’ 39.83730” Negeri Sembilan Melaka Gun Hill 2° 42’ 43.63412” 101° 56’ 22.92628” 0.000 0.000 2° 42’ 43.63383” 101° 56’ 22.92969” Pahang Gunung Sinyum 3° 42’ 38.69308” 102° 26’ 04.60447” 0.000 0.000 3° 42’ 38.69263” 102° 26’ 04.60772” Selangor Bukit Asa 3° 40’ 48.37751” 101° 30’ 24.48130” 0.000 0.000 3° 40’ 48.37778” 101° 30’ 24.48581” Terengganu Gunung Gajah Trom 4° 56’ 44.97144” 102° 53’ 37.00068” 0.000 0.000 4° 56’ 44.97184” 102° 53’ 37.00496” Pulau Pinang Seberang Perai Fort Cornwallis 5° 25’ 15.20204” 100° 20’ 40.75188” 0.000 0.000 5° 25’ 15.20433” 100° 20’ 40.76024” Kedah Perlis Gunung Perak 5° 57’ 52.81981” 100° 38’ 10.93028” 0.000 0.000 5° 57’ 52.82155” 100° 38’ 10.93860” Perak Gunung Hijau Larut 4° 51’ 32.64361” 100° 48’ 55.46334” 0.000 0.000 4° 51’ 32.64488” 100° 48’ 55.47038” Kelantan Bukit Panau (Baru) 5° 53’ 37.07908” 102° 10’ 32.24004” 0.000 0.000 5° 53’ 37.07975” 102° 10’ 32.24529” 35 4.6 Test Examples (i) GDM2000 (2009) to Geocentric Cassini (2009) (ii) GDM2000 to Geocentric Cassini BOX 6 Cassini-Soldner Projection GDM2000 (2009) Coordinate Latitude = N 06° 08' 22.98682" Longitude = E 100° 23' 6.56827" Ellipsoidal Height = -10 .183 m Geocentric Cassini (2009) Coordinate (Kedah/Perlis) Northing = 19364.3084 m Easting = -27805.4141 m GDM2000 Coordinate Latitude = N 06° 08' 22.98892" Longitude = E 100° 23' 6.57684" Geocentric Cassini Coordinate (Kedah/Perlis) Northing = 19364.3195 m Easting = -27805.4063 m BOX 6 Cassini-Soldner Projection 36 (iii) GDM2000 (2009) to Geocentric MRSO (2009) (iv) GDM2000 to Geocentric MRSO BOX 6 Oblique Mercator Projection GDM2000 (2009) Coordinate Latitude = N 06° 8' 22.98682" Longitude = E 100° 23' 6.56827" Ellipsoidal Height = -10.183 m Geocentric RSO (2009) Coordinate Northing = 679690.850 m Easting = 266843.634 m GDM2000 Coordinate Latitude = N 06° 8' 22.98892" Longitude = E 100° 23' 6.57684" Geocentric RSO Coordinate Northing = 679690.914 m Easting = 266843.898 m BOX 6 Oblique Mercator Projection 37 (v) GDM2000 / GDM2000 (2009) to BRSO Oblique Mercator Projection BOX 6 GDM2000 Coordinate Latitude = N 06 52' 42.45022" Longitude = E 116 50' 47.58835" Geocentric RSO Coordinate Northing = 762081.047 m Easting = 793704.631 m (vi) MRT68 to MRSO MRT Coordinate Latitude = N 06° 8' 24.50917" Longitude = E 100° 23' 11.20955" MRSO Coordinate Northing = 679686.405 m Easting = 267038.511 m Oblique Mercator Projection BOX 6 38 (vii) MRSO to Cassini Soldner (viii) BT68 to BRSO BT68 Coordinate Latitude = N 06 52' 46.04669" Longitude = E 116 50' 37.60565" BRSO Coordinate Northing = 762132.514 m Easting = 793326.383 m Oblique Mercator Projection BOX 6 Cassini-Soldner Coordinate (Kedah/Perlis) Northing = 19364.884 m Easting = -27807.589 m MRSO Coordinate Northing = 679686.405 m Easting = 267038.511 m Polynomial Transformation Coefficient BOX 4 39 5 CONCLUSION 5.1 The determination of a position requires the choice of a coordinate reference system. Until recently, an in-depth knowledge of coordinates and datums was generally confined to users with specialised knowledge in surveying. However, the development of techniques such as GPS has made available entirely new method of acquiring accurate coordinates. A situation now exists where it is common for a user acquiring data to be using a set of coordinates that is completely different to the one in which the data will be ultimately required. 5.2 For this reason and with the availability of various systems in Malaysia, JUPEM has produced various sets of datum transformation and map projection parameters to relate the different types of coordinate. This technical guide is designed for those dealing with coordinates as a practical solution to the problems that may be encountered with datums and map projections. 5.3 The parameter values relating to different coordinate reference systems are derived from standard coordinate conversion formulae, Bursa-Wolf transformation tormulae and multiple regression model. In addressing the effects of plate tectonic motion as well as natural disasters particularly earthquakes, JUPEM will continue to monitor the geodetic infrastructures for any significant displacement and produce up-dated parameters to relate the various coordinates accurately and timely. 40 REFERENCES Bowring, B.R. (1985), The accuracy of geodetic latitude and height equations, Survey Review 28(218): 202-206. Burford, B.J. (1985), A further examination of datum transformation parameters in Australia The Australian Surveyor, vol.32 no. 7, pp. 536-558. Bursa, M. (1962), The theory of the determination of the non-parallelism of the minor axis of the reference ellipsoid and the inertial polar axis of the Earth, and the planes of the initial astronomic and geodetic meridians from observations of artificial Earth Satellites, Studia Geophysica et Geodetica, no. 6, pp. 209-214. Defence Mapping Agency (DMA), (1987), Department of defence World Geodetic System 1984: its definition and relationship with local geodetic systems (second edition). Technical report no. 8350.2, Defence Mapping Agency, Washington. Heiskanen and Moritz (1967), Physical Geodesy, W. H Freeman and Company, San Francisco, USA. Richardus, Peter, and Adler, R. K., (1974), Map Projections for Geodesists, Cartographers and Geographers: Amsterdam, North-Holland Pub. Co. Snyder, J.P. (1984), Map projections used by the U.S. Geological Survey. Geological Survey Bulletin 1532, U.S. Geological Survey. Wolf, H. (1963), Geometric connection and re-orientation of three-dimensional triangulation nets. Bulletin Geodesique. No, 68, pp165-169. Rujukan Kami: JUPEM 18/7/2.148 Jld. 3 ( 22 ) Tarikh: 5 Jun 2009 Semua Pengarah Ukur dan Pemetaan Negeri PEKELILING KETUA PENGARAH UKUR DAN PEMETAAN BILANGAN 3 TAHUN 2009 GARIS PANDUAN MENGENAI PENUKARAN KOORDINAT, TRANSFORMASI DATUM DAN UNJURAN PETA UNTUK TUJUAN UKUR DAN PEMETAAN 1. TUJUAN Pekeliling ini bertujuan untuk memberikan garis panduan mengenai kaedah-kaedah penukaran koordinat, transformasi datum dan unjuran peta bagi kegunaan kerja-kerja ukur dan pemetaan. 2. LATAR BELAKANG 2.1 Jabatan Ukur dan Pemetaan Malaysia (JUPEM) bertanggungjawab melaksanakan kerja-kerja ukur hakmilik tanah dan pemetaan asas. Dalam memenuhi tanggungjawab tersebut, JUPEM telah menyediakan produk dan perkhidmatan yang berasaskan pelanggan. 2.2 Mutakhir ini teknologi satelit Global Navigation Satellite System (GNSS) semakin giat digunakan di dalam melaksanakan kerjakerja ukur dan pemetaan bagi mendapatkan koordinat dengan cepat dan tepat di Malaysia. koordinat bagi memenuhi keperluan semasa 1 2.3 Bagaimanapun, kaedah penentududukan dan penentuan koordinat berasaskan teknologi satelit GNSS telah menggunakan datum geodetik yang berlainan daripada datum-datum geodetik yang sedia ada di Malaysia. Ketidakharmonian di antara koordinat yang berpunca daripada perbezaan datum geodetik ini, jika sekiranya tidak diuruskan dengan baik, boleh menimbulkan kesilapan di dalam kerja-kerja pengukuran serta produk ukur dan pemetaan. 2.4 Oleh itu, maklumat mengenai kaedah penukaran koordinat, transformasi datum dan unjuran peta yang sedia ada wajar dikemaskini dan didokumentasikan supaya ianya sentiasa relevan dengan kehendak dan tuntutan semasa serta dapat dijadikan panduan dan rujukan kepada para pengguna dalam menguruskan kerja-kerja yang mempunyai kaitan dengan koordinat. Sehubungan dengan itu, Pekeliling ini diharapkan akan dapat membantu pengguna-pengguna produk-produk pemetaan, kadaster, utiliti dan sebagainya untuk memahami kaedah-kaedah tersebut dan seterusnya dapat menggunakannya dengan cara yang betul di dalam urusan kerja mereka, terutamanya di dalam era penggunaan teknologi GNSS yang semakin meluas di Malaysia. 3. PERKEMBANGAN PEMBANGUNAN INFRASTRUKTUR GEODETIK DI MALAYSIA 3.1 Sejak penubuhan Jabatan Ukur dan Pemetaan Malaysia (JUPEM) lebih daripada 120 tahun yang lalu, datum rujukan geodetik yang digunapakai untuk kegunaan ukur dan pemetaan telah mengalami pelbagai perubahan. Dalam hal ini, sebelum tahun 1990an, Datum Kertau telah pun menjadi tulang belakang kepada Malayan Revised Triangulation 1968 (MRT68) di Semenanjung Malaysia, manakala bagi Sabah, Sarawak dan 2 Labuan pula, Borneo Triangulation 1968 (BT68) merujuk kepada Datum Timbalai sebagai asasnya. 3.2 Dengan perkembangan teknologi GNSS yang meluas sekitar tahun 1980-an, JUPEM telah membangunkan rangkaian kawalan ukur geodetik yang baru dengan menggunakan teknologi tersebut untuk menentukan koordinat bagi stesen-stesen kawalan. Bagi Semenanjung Malaysia, rangkaian ini telah ditubuhkan dalam tahun 1994 dan dikenali sebagai Peninsular Malaysia Geodetic Scientific Network 1994 (PMGSN94). Manakala di Sabah, Sarawak dan Labuan pula, rangkaian kawalan ukur geodetiknya adalah East Malaysia Geodetic Scientific Network 1997 (EMGSN97), yang ditubuhkan pada tahun 1997. 3.3 Antara tahun 1998 dan 2001 pula, JUPEM telah membangunkan rangkaian Malaysia Active GPS System (MASS) dan ini telah diikuti dengan Malaysia Real-Time Kinematic GNSS Network (MyRTKnet) antara tahun 2002 dan 2008. Dalam pada itu, pada 26 Ogos 2003, JUPEM telah melancarkan Geocentric Datum of Malaysia (GDM2000), iaitu datum rujukan geodetik baru yang seragam bagi kerja-kerja ukur dan pemetaan di Malaysia. 3.4 Bagaimanapun, sistem rujukan GDM2000 telah mengalami anjakan yang signifikan, lanjutan daripada berlakunya beberapa siri gempabumi besar di Indonesia, terutamanya pada tahun 2004 dan 2005. Oleh hal yang demikian, datum ini telah disemak dan dihitung semula bagi menghasilkan GDM2000 (2009). 3.5 Maklumat lebih lanjut mengenai sistem rujukan koordinat terkandung di dalam Pekeliling Ketua Pengarah Ukur dan Pemetaan Malaysia bilangan 1/2009 bertarikh 25 Mei 2009 bertajuk ‘Garis Panduan Mengenai Sistem Rujukan Koordinat Di dalam Penggunaan Global Navigation Satellite System (GNSS) Bagi Tujuan Ukur dan Pemetaan’. Ia menyenaraikan semua jenis 3 sistem rujukan koordinat yang telah dibangunkan oleh JUPEM di Malaysia disamping memberi maklumat-maklumat teknikal mengenai sistem tersebut dan datum geodetik. 4. GARIS PANDUAN PENUKARAN KOORDINAT, TRANSFORMASI DATUM DAN UNJURAN PETA UNTUK TUJUAN UKUR DAN PEMETAAN Penerangan lebih lanjut tentang amalan penggunaan penukaran koordinat, transfomasi datum dan unjuran peta terkandung di dalam dokumen Technical Guide to the Coordinate Conversion, Datum Transformation and Map Projection seperti di Lampiran ‘A’ yang disertakan. Intisari garis panduan tersebut adalah seperti berikut: Perenggan 1. 2. Perkara INTRODUCTION COORDINATE CONVERSION 2.1 2.2 GEOGRAPHICAL AND CARTESIAN COORDINATES CONVERSION BETWEEN GEOGRAPHICAL COORDINATES AND CARTESIAN COORDINATES 2.3 TEST EXAMPLE 3. DATUM TRANSFORMATION 3.2 3.2 3.3 3.4 INTRODUCTION BURSA-WOLF DATUM TRANSFORMATION FORMULAE MULTIPLE REGRESSION MODEL TEST EXAMPLES 4. MAP PROJECTION 4.1 4.2 4.3 RECTIFIED SKEW ORTHOMORPHIC PROJECTION (RSO) CASSINI-SOLDNER PROJECTION POLYNOMIAL FUNCTION 4 4 RE-DEFINITION OF STATE ORIGINS IN GDM2000 AND GDM2000 (2009) 4. TEST EXAMPLES CONCLUSION 5. terima kasih. TARIKH BERKUATKUASA Pekeliling ini adalah berkuatkuasa mulai tarikh ianya dikeluarkan.4. Ukur dan Pemetaan) Kementerian Sumber Asli dan Alam Sekitar Pengarah Institut Tanah dan Ukur Negara (INSTUN) Kementerian Sumber Asli dan Alam Sekitar 5 . Sekian. “BERKHIDMAT UNTUK NEGARA” ( DATUK HAMID BIN ALI ) Ketua Pengarah Ukur dan Pemetaan Malaysia Salinan kepada: Timbalan Ketua Pengarah Ukur dan Pemetaan Pengarah Ukur Bahagian (Pemetaan) Pengarah Ukur Bahagian (Kadaster) Pengarah Bahagian Geospatial Pertahanan Setiausaha Bahagian (Tanah.5 5. Pengarah Pusat Infrastuktur Data Geospatial Negara (MaCGDI) Kementerian Sumber Asli dan Alam Sekitar Ketua Penolong Pengarah Unit Ukur Tanah. Bahagian Kejuruteraan Awam Ibu Pejabat Jabatan Perumahan Negara Setiausaha Lembaga Jurukur Tanah Semenanjung Malaysia Setiausaha Lembaga Jurukur Tanah Sabah Setiausaha Lembaga Jurukur Tanah Sarawak 6 . Cawangan Pengkalan Udara dan Maritim Ibu Pejabat Jabatan Kerja Raya Malaysia Penolong Pengarah Unit Ukur Tanah. . Lampiran ‘A’ Technical Guide to the Coordinate Conversion. Datum Transformation and Map Projection JABATAN UKUR DAN PEMETAAN MALAYSIA 2009 . 2 4.3 Test Example 1 5 5 5 8 9 9 10 12 15 20 20 20 30 33 34 35 39 3.2 Geographical and Cartesian Coordinates Conversion between Geographical Coordinates and Cartesian Coordinates 2. MAP PROJECTION 4.3 3.1 3. DATUM TRANSFORMATION 3.6 Introduction Rectified Skew Orthomorphic Projection (RSO) Cassini-Soldner Map Projection Polynomial Function Redefinition of State Origins in GDM2000 and GDM2000 (2009) Test Examples 5.5 4.3 4.1 2.4 Introduction Bursa-Wolf Datum Transformation Formulae Multiple Regression Model Test Examples 4.2 3.4 4. CONCLUSION REFERENCES i . COORDINATE CONVERSION 2.1 4. INTRODUCTION 2.TABLE OF CONTENTS 1. 0 3.0 GDM2000 Ellipsoid: GRS80 Reference Frame: ITRF2000 Epoch: 2000. engineering and scientific research. Table 1: Coordinate Reference Systems in Malaysia No. Malaysia Primary Geodetic Network 2000 (MPGN2000) 1 .0 GDM2000 Ellipsoid: GRS80 Reference Frame: ITRF2000 Epoch: 2000.0 GDM2000 (2009) Ellipsoid: GRS80 Reference Frame: ITRF2000 Epoch: 2000. Malaysia Active GPS System (MASS) 6.0 WGS84 Ellipsoid: WGS84 Reference Frame: WGS84 (G783) Epoch: 1997. mapping. Coordinate Reference System Coordinate System 1. It establishes and manages these geodetic infrastructures for the purpose of cadastral survey.1 The Department of Survey and Mapping Malaysia (JUPEM) defines and maintains the Coordinate Reference System (CRS) and the Vertical Reference System (VRS) for the whole country.1. INTRODUCTION 1. East Malaysia Geodetic Scientific Network 1997 (EMGSN97) 5. The coordinate reference systems that have been introduced and used since the late 1800s in Malaysia are listed in Table 1. Peninsular Malaysia Geodetic Scientific Network 1994 (PMGSN94) 4. Malayan Revised Triangulation 1968 (MRT68) Borneo Triangulation 1968 (BT68) Geodetic Datum KERTAU Ellipsoid: Modified Everest TIMBALAI Ellipsoid: Modified Everest WGS84 Ellipsoid: WGS84 Reference Frame: WGS84 Epoch: 1987. 2. Table 1: (continued) Coordinate Reference System Coordinate System Geodetic Datum GDM2000 Ellipsoid: GRS80 Reference Frame: ITRF2000 Epoch: 2000. Coordinate conversions which do not involve a change of datum are shown as dashed lines. 7.2 Figure 1 represents a schematic diagram to assist users in navigating between the different types of coordinates.0 GDM2000 (2009) Ellipsoid: GRS80 Reference Frame: ITRF2000 Epoch: 2000. 2 . 1. datum transformation and map projection as practiced in Malaysia. Malaysia Real-Time Kinematic GNSS Network (MyRTKnet) 1. on the other hand.3 This technical guide is produced to assist users in understanding the concept and procedures involved in the process of coordinate conversion. some are on the same geodetic datum and others. All Coordinate Reference Systems (CRS) available in Malaysia are shown. Datum transformations.0 No. on different ones. involve a change of datum and are shown as thick solid lines and map projection as thin solid lines. GRS80 Ellipsoid WGS84 Ellipsoid GDM2000 PMGSN94/ EMGSN97 Modified Everest Ellipsoid GDM2000 (2009) Geocentric CASSINI N E MRT68/BT68 CASSINI h h h h N E BOX 5 BOX 1 BOX 6 BOX 1 BOX 1 BOX 1 BOX 4 X Y Z N E X Y Z X Y Z X Y Z N E Geocentric MRSO/BRSO MRSO/BRSO BOX 7 BOX 2 BOX 2 BOX 3 BOX 2 Figure 1: Relationship between various coordinate reference system Coordinate conversion Datum transformation Map projection 3 . Notes to Figure 1: Box 1: Coordinate conversion between geographical coordinate and cartesian coordinate Box 2: Transformation between various datums using Bursa-Wolf formulae Box 3: Map projection of MRT68 / BT68 geographical coordinate to Rectified Skew Orthomorphic (RSO) plane coordinates Box 4: Coordinate transformation from RSO to Cassini using polynomial function Box 5: Map projection of GDM2000 geographical Geocentric Cassini plane coordinates Box 6: Map projection of GDM2000 geographical Geocentric RSO plane coordinates coordinate coordinate to to Box 7: Datum transformation from GDM2000 to GDM2000 (2009) using multiple regression model 4 . 1 Three-dimensional geographical coordinates can be defined with respect to an ellipsoid as follows: Latitude: Longitude: Height: the angle north or south from the equatorial plane the angle east or west from the prime meridian the distance above the surface of the ellipsoid. The 5 . positions in geographical coordinates of latitude. h) can be converted into cartesian coordinates (X. such that: Z-axis: X-axis: Y-axis: is aligned with the minor (or polar) axis of the ellipsoid is in the equatorial plane and aligned with the prime meridian forms a right-handed system 2. COORDINATE CONVERSION 2.1.1 Geographical and Cartesian Coordinates 2. . 2.1.1 The conversion of three-dimensional coordinates from geographical to cartesian or vice versa can be carried out through the knowledge of the parameters of an adopted reference ellipsoid (Figure 2).2. Longitude and Height (. Y. longitude and height (.1.2 A set of cartesian coordinates is defined with the three axes at the origin at the center of the ellipsoid.3 In this regard.2 Conversion between Geographical Coordinates and Cartesian Coordinates 2. Geographical Coordinates Latitude. . h) Cartesian Coordinates X Y Z 2. Z) and vice-versa.2. Z ) is as follows: Z P h b N Y a X Figure 2: Geographical and cartesian coordinates X ( N h) cos cos Y ( N h) cos sin b2 Z 2 N h sin a where the prime vertical radius of curvature (N) is: N 2 a2 (a cos b sin 2 2 1 2 ) with: 6 . h ) to cartesian coordinate ( X . Y .forward conversion from geodetic coordinates ( . . . h ) is as follows: Z 2b sin 3 u arctan 2 3 P e a cos u arctan Y X h P cos Z sin a 1 e 2 sin 2 with: aZ u arctan bP P X2 Y2 e2 e2 1 e2 a2 b2 a2 where.a b : the semi-major axis of the reference ellipsoid.2.2 The non-iterative reverse conversion from cartesian coordinates ( X . 7 . Y . Z ) to geodetic coordinates ( . u e : the parametric latitude. : the semi-minor axis of the reference ellipsoid. : the first eccentricity of the reference ellipsoid. : the second eccentricity of the reference ellipsoid. 2. 654 m = 6236311.05076" Ellipsoidal Height (h) = 18.078 m BOX 1 Coordinate Conversion using GRS80 Ellipsoid 3D-Cartesian Coordinate (ECEF) X-Axis Y-Axis Z-Axis = -1131051.112 m 8 .800 m = 711748.3 Test Example Geographical Coordinate Latitude = N 06 27' 00.56909" Longitude = E 100 16' 47.2. For example. It requires and uses the parameters of the transformation and the ellipsoids associated with the source and target coordinate reference systems.3.1 Introduction 3. DATUM TRANSFORMATION 3.1. the source can be coordinate reference system 1 and the target can be coordinate reference system 2: No. Coordinate Reference System 1 Coordinate Reference System 2 Geocentric Datum of Malaysia GDM2000 (2009) Peninsular Malaysia Geodetic Scientific Network 1994 (PMGSN94) 1 Geocentric Datum of Malaysia (GDM2000) East Malaysia Geodetic Scientific Network 1997 (EMGSN97) Malayan Revised Triangulation 1968 (MRT68) Borneo Triangulation 1968 (BT68) (Sabah) Borneo Triangulation 1968 (BT68) (Sarawak) 2 Peninsular Malaysia Geodetic Scientific Network 1994 (PMGSN94) East Malaysia Geodetic Scientific Network 1997 (EMGSN97) Malayan Revised Triangulation 1968 (MRT68) Borneo Triangulation 1968 (BT68) (Sabah) Borneo Triangulation 1968 (BT68) (Sarawak) 3 9 .1 Datum transformation is a computational process of converting a position given in one coordinate reference system into the corresponding position in another coordinate reference system. 2 Bursa-Wolf Datum Transformation Formulae 3. The transformation involves three geocentric datum shift parameters ( X . three rotation parameters and a scale. three rotation elements ( R X .2. R Z ) and a scale factor ( 1 L ). translation and scale. 1974).3. 3.2 The model in its matrix-vector form could be written as ( Burford 1985): RY X MRT RZ X WGS 84 X 1 L YWGS 84 Y R Z 1 L R X YMRT Z WGS 84 Z RY R X 1 L Z MRT 10 . This transformation model is more suitable for satellite datums on a global scale (Krakwisky and Thomson. A complete datum conversion is usually based on seven parameter transformations.2. They are generated through least square analysis using various models accepted by the global geodetic community.1.1. The parameters are computed based on coordinates of control stations which are common to different datums. 3. RY . which include three translation parameters. A simple three parameter conversion can be accomplished by conversion through Earth-Centred Earth Fixed (ECEF) cartesian coordinates from one reference datum to another by three origin offsets that approximate differences in rotation.2 The transformation parameter values associated with the transformation can be determined empirically from a measurement or a calculation process. 3. Y .3 Datum transformation can be accomplished by many different methods.1 Bursa-Wolf formulae is a seven-parameter model for transforming three-dimensional cartesian coordinates between two datums (see Figure 3). Z ). Z MRT : are the local datum (MRT) cartesian coordinates. In order to convert the cartesian coordinates of XYZ to the geographical coordinates of h. YWGS 84 . X MRT . YMRT .Figure 5: Bursa-Wolf 3-D model transformation where. X WGS 84 . Z WGS 84 : are the global datum (WGS84) cartesian coordinates. ellipsoid properties for the respective datum as listed in Table 2 below are used: 11 . 8017 4 6377298.e.3 Multiple Regression Model (i) Displacement Computation The computation and modelling of differences in the coordinate between the two systems.8017 3. a (m) 6378137. i.e.”GDM2000 (2009)) x 30 Height (U) = (hGDM2000 . and h). GDM2000 and GDM2000 (2009) were carried out using their respective geographical coordinates in the format of (. 1 Ellipsoid Geodetic Reference System 1980 (GRS80) World Geodetic System 1984 (WGS84) Modified Everest (Peninsular Malaysia) Modified Everest Malaysia) (East semi-major axis.000 flattening.063 300.2572236 3 6377304.000 298.Table 2: Ellipsoid Properties No. The differences in the three components are computed separately by using the following formulae: North (N) = (”GDM2000 . The conversion from geographical system to local geodetic system used the following factor. f (m) 298.2572221 2 6378137.”GDM2000 (2009)) x 30 East (E) = (”GDM2000 . i.hGDM2000 (2009)) 12 . These differences are then converted to the local geodetic horizon to avoid mathematical errors for some very small values. 1”= 30 meter.556 300. The coordinate differences are then gridded to derive the Regression Coefficient.(ii) Displacement Modelling The differences in coordinate of every GPS station are similarly computed between the two systems. The gridding method used is the polynomial regression with the power to the second order.e. East. i.N.e. GDM2000 and GDM2000 (2009) using their respective coordinates in the format of (E and N). The surface definition used the bi-linear saddle regression coefficient with the following formulae: Z(E.U) = A00 + A01 N + A10 E + A11 EN where. Z = Value of regression coefficient or the value of displacement correction for each component (i. North and Up) E = East coordinate in decimal degree N = North coordinate in decimal degree (iii) Basic Formulae To convert the coordinate in GDM2000 to GDM2000 (2009). the following formulae shall be used: GDM2000 (2009) = GDM2000 + correction ”GDM2000 (2009) ”GDM2000 (2009) hGDM2000 (2009) = = = ”GDM2000 + (ZN / 30) ”GDM2000 + (ZE / 30) hGDM2000 + ZU 13 . 14 .N. U) = A00 + A01 N + A10 E + A11 EN and A00. A10 and A11 are the coefficients of the multiple regression model.where. ” ” h” ZN ZE ZU = = = = = = Latitude in second of arc Longitude in second of arc Ellipsoidal height in metre Displacement correction in northing Displacement correction in easting Displacement correction in height where. A01. Z(E. 75616" Ellipsoidal Height (h) = 7.118 m 15 .04635 " Longitude = E 102 56 ' 31.04635" Longitude = E 102 56' 31.3.08807 " Longitude = E 102 56 ' 31.724 m BOX 7 Multiple Regression GDM2000 (2009) Coordinate Latitude = N 01 51' 27.724 m BOX 2 Bursa -Wolf 7Parameter Transformation PMGSN 94 Coordinate Latitude = N 01 51 ' 27.72178 " Ellipsoidal Height = 7 .04737" Longitude = E 102 56' 31.4 Test Examples (i) GDM2000 to GDM2000 (2009) GDM2000 Coordinate Latitude = N 01 51' 27.75616 " Ellipsoidal Height (h) = 7 .757 m (ii) GDM2000 to PMGSN94 GDM 2000 Coordinate Set Latitude = N 01 51 ' 27 .75374" Ellipsoidal Height= 7. 38567 " Longitude = E 102 56' 37.52660 " Ellipsoidal Height (h) = 2.75616 " Ellipsoidal Height = 7.724 m BOX 2 Bursa -Wolf 7-Parameter Transformation MRT Coordinate Latitude = N 01 51 ' 27.04635 " Longitude = E 102 56' 31.338 m 16 .(iii) GDM2000 / GDM2000 (2009) to EMGSN97 (iv) GDM2000 to MRT68 GDM 2000 Coordinate Latitude = N 01 51 ' 27 . 44225” 17 .58835" Ellipsoidal Height (h) = 52.04669" Longitude = E 116 50' 37.60565" Ellipsoidal Height (h) = 1.51240” ‘ 49.693 m (vi) GDM2000 / GDM2000 (2009) to BT68 (Sarawak) 54’ 41.620 m BOX 2 Bursa-Wolf 7-Parameter Transformation BT68 (Sabah) Coordinate Latitude = N 06 52' 46.(v) GDM2000 / GDM2000 (2009) to BT68 (Sabah) GDM2000/GSM2000(2009) Coordinate Latitude = N 06 52' 42.45022" Longitude = E 116 50' 47. 38567 " Longitude = E 102 56' 37.338 m (viii) EMGSN97 to BT68 (Sabah) 18 .08807 " Longitude = E 102 56' 31.52660 " Ellipsoidal Height (h) = 2.(vii) PMGSN94 to MRT68 PMGSN94 Coordinate Latitude = N 01 51 ' 27.72178 " Ellipsoidal Height = 7.118 m BOX 2 Bursa-Wolf 6-Parameter Transformation MRT Coordinate Latitude = N 01 51 ' 27. 000 m BOX 2 Bursa-Wolf 7-Parameter Transformation BT68 Coordinate (Sarawak) Latitude = N 01 51' 41.(ix) EMGSN97 to BT68 (Sarawak) EMGSN94 Coordinate Latitude = N 01 54' 39.51179" Longitude = E 112 51' 49.30200" Longitude = E 112 52' 01.25200" Ellipsoidal Height (h) = 977.44453" Ellipsoidal Height (h) = 933.784 m 19 . 1 The rectified skew orthomorphic (RSO) map projection (Figure 4) is an oblique Mercator projection developed by Hotine in 1947 (Snyder.4.1. All meridians and parallel are complex curves. Coordinate Reference System 1 Geocentric Datum of Malaysia GDM2000 (2009) 1 Geocentric Datum of Malaysia (GDM2000) Geocentric Datum of Malaysia GDM2000 (2009) 2 Geocentric Datum of Malaysia (GDM2000) Geocentric Datum of Malaysia GDM2000 (2009) 3 Geocentric Datum of Malaysia (GDM2000) 4 5 6 Malayan Revised Triangulation 1968 (MRT68) Rectified Skew Orthomorhic (Peninsular Malaysia) Borneo Triangulation 1968 (BT68) Map Projection System 1 Geocentric Cassini-Soldner Geocentric Malayan Rectified Skew Orthomorhic (Peninsular Malaysia) Geocentric Borneo Rectified Skew Orthomorhic (Sabah and Sarawak) Malayan Rectified Skew Orthomorhic (Peninsular Malaysia) Cassini-SoldnerOld Borneo Rectified Skew Orthomorhic (Sabah and Sarawak) 4.2 Rectified Skew Orthomorphic (RSO) Map Projection 4. Scale is approximately 20 .2. MAP PROJECTION 4. 1984). there are many different methods for projecting longitude and latitude in coordinate reference system 1 onto a flat map in the same system: No.1 Typically.1 Introduction 4. This projection is orthomorphic (conformal) and cylindrical. It is thus a suitable projection for an area like Switzerland. Oblique Mercator (Source: EPSG) 21 .2. 1947 (Snyder.2 The RSO provides an optimum solution in the sense of minimizing distortion whilst remaining conformal for Malaysia. New Zealand. Madagascar and Malaysia as well. Figure 4: Hotine. Italy. 1984). Table 3 tabulates the new geocentric RSO parameters for Peninsular Malaysia and East Malaysia. 4.true along a chosen central line (exactly true along a great circle in its spherical form). 1/f Longitude of Origin. o Azimuth of Central Line.000 Meters 298. c = longitude of center of the projection.2572221 Defined Parameters. GRS 80 6378137. a Flattening. c Scale factor. Rectified to Skew Grid. kc = scale factor at the center of the projection.2572221 East Malaysia BRSO Semi Major axis.2. c = azimuth (true) of the center line passing through the center of the projection.000 Meters 298.3 The notation adopted for use in this section is as follows: c = latitude of center of the projection.Table 3: The New Geocentric RSO Projection Parameters Peninsular RSO Ellipsoid Parameters Ellipsoid GRS 80 6378137. c = rectified bearing of the center line. o . = geographical latitude = geographical longitude to = isometric latitude for = 4 1 e sin c k to = log tan c log 4 2 2 1 e sin c or D ca efin n ed be p ob r oj ta ec t i n io ed n fro par m am J U et PE ers M 22 Latitude of Origin. k False Origin (Easting) False Origin (Northing) 4. p cos R =rectified convergence of meridians R 3652'11.t = isometric latitude o = basic longitude a = semi major axis of ellipsoid b = semi minor axis of ellipsoid f = flattening of ellipsoid f a b / a e = eccentricity of ellipsoid e2=(a2 – b2)/a2 e12=(a2 – b2)/b2 = radius of curvature in the meridian a 1 e 2 / 1 e 2 . FN = False Northing at the natural origin.6314" u = skew coordinate parallel to initial line v = skew coordinate at right angles to initial line N= Northing map coordinate East map coordinate E = FE = False Easting at the natural origin.Sin 2 3 / 2 = radius of curvature in the prime vertical 2 1 e1 .Cos 2 m = scale factor mo = scale factor at the origin = skew convergence at meridians p = distance from polar axis. 23 . 6 Or Basic Longitude: 24 .4 The Constants of the Projection (Table 4) are given as follows: 2 B 1 e1 . make D2 = 1.4. sin o 0. if D<1. cos 4 c 2 1 or A' B o o 2 C cosh 1 A Bt o p o 1 To avoid problems with computation of F.2. ): For the Hotine Oblique Mercator (where the FE and FN values have been specified with respect to the origin of the (u. o East Malaysia rs ete M ram E pa JUP n m tio ec d fro roj e d p btain e o fin De n be ca 4. v) axes): 25 .2. N) from a given (.5 (a) The projection formulae are as follows: Conversion of Geographical to Rectangular and vice versa Forward Case: To compute (E.Table 4: Constant of Projection Constant of Projection Peninsular Malaysia Parameter A Parameter A' Parameter B Parameter C Basic Longitude. The rectified skew coordinates are then derived from: Reverse Case: Compute (. 26 . ) from a given (E. N): For the Hotine Oblique Mercator: Then the other parameters can be calculated. (b) Convergence of Map Meridians The convergence of the map meridians is defined as the angle measured clockwise from True North to the Rectified Grid North.6314" where tan o sin Bo sinh Bt C cos Bo cosh Bt C tan sin Bv Bu tan o sinh A' mo A' m Bv Bu cos cosh A' mo A' mo 27 . and is denoted R (Figure 5): RGN TN R Figure 5: Convergence of Map Meridians R sin 1 0.6 3652'11. coordinate of the points: 28 . The scale factor for a line also may be evaluated from the following formula: A' mo B2 2 2 m u1 u1u 2 u 2 1 2 v m cos m cosh B m C 6 A' 2 mo where m and m are evaluated for the mid-latitude of the line u1 and u 2 are the u .(c) Scale Factor at any Point The formulas giving the scale factor “ m ” at any point in terms of isometric latitude and longitude and coordinate (v.u) are: Bu A' mo A' mo m p cosh Bt C cosh Bv A' mo A' mo p cos Bo cos It can be shown that the initial line of the projection has a scale factor that is nearly constant throughout its length. (d) Scale Factor for a Line The Scale factor for a line can be computed from the formula: m 1 m1 4m3 m2 6 where m1 . m2 are the scale factors at the ends of the line and m3 the scale factor at its mid-point. 3 1 21 2 and 1 2 is measured in seconds. then R R where is given in seconds by the formula B v2 v1 tanh B 2u1 u 2 2 A' mo sin 1" 3 2 A' mo 2 k1 o sin o sin 3 sin o 2 1 2 o where. it can therefore be safely neglected.007”. the maximum value of the second term of the formula is 0.(e) Arc-to-Chord Correction RGN TN R R Figure 8: Arc-to-Chord In Figure 6. 3 For a line not exceeding 113 km (70 miles) in length. if is the true azimuth of a line. R is the rectified grid bearing. 29 . 3.3.2 The mapping equations are as given in Richardus and Adler.(8 –T + 8C)T * A5/120] N = FN + M – Mo + v*tan[A2/2 + (5 – T + 6C)A4/24] where. N. = = = = = = = = = = Computed Cassini coordinate Cassini State origin coordinate ( . 4.4. E FE.o)Cos o T C V Longitude of computation point Longitude of state origin Latitude of computation point Tan2 e2 (1 e ) 2 Cos 2 Radius of curvature in prime vertical a (1 e Sin 2 )1/ 2 2 30 .3 Cassini-Soldner Map Projection 4. 1974 and the formulas to derive projected Easting and Northing coordinate are as follows: (a) Forward Computation E = FE + v[A – T(A3/6) . FN A where.1 There are nine state origins used in the coordinate projection in the cadastral system of Peninsular Malaysia. The Cassini-Soldner map projection has been used for over one hundred years and shall continue to be used for cadastral surveys in the new geodetic frame. Mo = the value of M calculated for the latitude of the chosen origin.(3e2/8 + 3e4/32 + 45e6/1024 + …)sin2 + (15e4/256 + 45e6/1024 +…)sin4 .) .(35e6/3072 + …)sin6 .)Sin21 +(21e12/16 55e14/32 + ..)Sin41 + (151e13/96 + …)Sin61 + (1097e14/512 ..M = = Meridianal arc distance a[1 – e2/4 – 3e4/64 – 5e6/256-..…)Sin81 + … a (1 e 2 ) (1 e 2 Sin 21) 3 / 2 1 v1 = = a (1 e Sin 21)1 / 2 2 where. 1 = 1 + (3e1/2 – 27e13/32+ ..] with is in radians. e1 = 1 (1 e 2 )1 / 2 1 (1 e 2 )1 / 2 1 = M1 a(1 e 2 / 4 3e 4 / 64 5e 6 / 256 .. (b) Reverse Computation 1 v1Tan 1 D 2 D4 (1 3T 1) ) 24 1 2 D3 D5 (1 3T 1)T 1 0 D T 1 / Cos 1 3 15 where.) 31 .. )" = .((Sino.Sin1"))E2 = = Cassini Coordinate Arc-to-Chord where.Sin1") Na.Coso)/(6R2. Ea. AB = (t – T)" = ((Nb – Na)(Eb + 2Ea))/(6R2. Eb (t – T)" Bearing correction:( . Nb.M1 = = Mo + (N – FN) Mo is the value of M calculated for the latitude of the origin tan21 (E – FE)/v1 T1 D = = (c) Scale and Arc-to-Chord Correction for Cassini Projection D = D' B' B s' o = T =t AB s A = A' Figure 7: Scale and Arc-to-Chord Refer to Figure 7. 32 . where.RSO Coordinate of Origin (North) ∆E(MRSO) : RSO Coordinate .(R2+XB1+YB2+XYB3+X2B4+Y2B5) where N(MRSO) : MRSO Coordinate in North Component E(MRSO) : MRSO Coordinate in East Component N(CAS) : Cassini Coordinate in North Component E(CAS) : Cassini Coordinate in East Component ∆N(CAS) : Cassini Coordinate .4 Polynomial Function The relationship between the MRSO coordinate and the Cassini SoldnerOld is defined by a series of polynomial function which make use of the coordinate of the origin of both projections for each state in Peninsular Malaysia.11678249 as the multiplying factor to convert from metres) 33 . The following are the formulae used: N(MRSO)= ∆N(CAS) + N(OMRSO) + (R1+XA1+YA2+XYA3+X2A4+Y2A5) E(MRSO)= ∆E(CAS) + E(OMRSO) + (R2+XB1+YB2+XYB3+X2B4+Y2B5) N(CAS)= ∆N(MRSO) + N(OCAS) .(R1+XA1+YA2+XYA3+X2A4+Y2A5) E(CAS)= ∆E(MRSO) + E(OCAS) . E2 s' = (E2a + EaEb + E2b) Linear correction:= s + [((Cos2o)/(6R2))E2]s Projected Coordinate on Cassini:Eb Nb = Ea + s'.Sin = Na + s'.Cassini Coordinate of Origin (North) ∆E(CAS) : Cassini Coordinate .Cassini Coordinate of Origin (East) ∆N(MRSO) : RSO Coordinate .RSO Coordinate of Origin (East) X : ∆N(CAS)/10000 or ∆N(MRSO)/10000 Y : ∆E(CAS)/10000 or ∆E(MRSO)/10000 All values of the coordinates must be in unit chains (use 0.Cos 4. 64488” 100 48’ 55.93860” 4 51’ 32.69308” 102 26’ 04.000 Easting 0. JUPEM had successfully carried out GPS observations at all the origins in August and October of 2005 and the results are given in Table 5.000 0.000 0.47038” 5 53’ 37.000 0.48581” 4 56’ 44.75188” 5 25’ 15. In this regard.92969” Cassini-Soldner Northing 0.00068” 4 56’ 44.000 34 .000 0.000 Gunung Sinyum Bukit Asa Gunung Gajah Trom Fort Cornwallis 3 42’ 38.64361” 100 48’ 55.97144” 102 53’ 37.92628” 2 42’ 43.93028” 5 57’ 52.24529” 0.20196” 103 33’ 39.000 0.000 0.000 0.000 0. the re-definition of the nine (9) state cadastral origins need to be carried out.000 0.97184” 102 53’ 37.00496” 5 25’ 15.07908” 102 10’ 32.20433” 100 20’ 40.000 0.5 Re-definition of State Origins in GDM2000 and GDM2000 (2009) In order to prepare the geodetic infrastructure fully for the implementation of e-Kadaster.46334” 4 51’ 32. Table 5: Re-definition of State Cadastral Origins in GDM2000 and GDM2000 (2009) Coordinates State Station Location Gunung Belumut GDM2000 (2009) GDM2000 Latitude (North) Longitude (East) 2 02’ 33.76024” 5 57’ 52.83730” 2 42’ 43.63412” 101 56’ 22.000 Johor Negeri Sembilan Melaka Pahang Selangor Terengganu Pulau Pinang Seberang Perai Kedah Perlis Perak Kelantan Gun Hill 0.000 0.000 0.82155” 100 38’ 10.60772” 3 40’ 48.69263” 102 26’ 04.83599” 2 02’ 33.4.20279” 103 33’ 39.63383” 101 56’ 22.07975” 102 10’ 32.000 Gunung Perak Gunung Hijau Larut Bukit Panau (Baru) 0.24004” 5 53’ 37.48130” 3 40’ 48.81981” 100 38’ 10.20204” 100 20’ 40.37778” 101 30’ 24.000 0.37751” 101 30’ 24.60447” 3 42’ 38. 57684" BOX 6 Cassini-Soldner Projection Geocentric Cassini Coordinate (Kedah/Perlis) Northing = 19364.4141 m (ii) GDM2000 to Geocentric Cassini GDM2000 Coordinate Latitude = N 06 08' 22.4.98892" Longitude = E 100 23' 6.98682 " Longitude = E 100 23' 6.4063 m 35 .183 m BOX 6 Cassini-Soldner Projection Geocentric Cassini (2009) Coordinate (Kedah/Perlis) Northing = 19364.56827" Ellipsoidal Height = -10 .3084 m Easting = -27805.6 Test Examples (i) GDM2000 (2009) to Geocentric Cassini (2009) GDM2000 (2009) Coordinate Latitude = N 06 08' 22.3195 m Easting = -27805. 98892" Longitude = E 100 23' 6.850 m = 266843 .(iii) GDM2000 (2009) to Geocentric MRSO (2009) GDM2000 (2009) Coordinate Latitude = N 06 8' 22.57684" BOX 6 Oblique Mercator Projection Geocentric RSO Coordinate Northing Easting = 679690.183 m BOX 6 Oblique Mercator Projection Geocentric RSO (2009) Coordinate Northing Easting = 679690 .914 m = 266843.56827" Ellipsoidal Height = -10.634 m (iv) GDM2000 to Geocentric MRSO GDM2000 Coordinate Latitude = N 06 8' 22.98682 " Longitude = E 100 23' 6.898 m 36 . 511 m 37 .20955" BOX 6 Oblique Mercator Projection MRSO Coordinate Northing Easting = 679686.047 m = 793704.631 m (vi) MRT68 to MRSO MRT Coordinate Latitude = N 06 8' 24.58835" BOX 6 Oblique Mercator Projection Geocentric RSO Coordinate Northing Easting = 762081.405 m = 267038.45022" Longitude = E 116 50' 47.(v) GDM2000 / GDM2000 (2009) to BRSO GDM2000 Coordinate Latitude = N 06 52' 42.50917" Longitude = E 100 23' 11. 60565" BOX 6 Oblique Mercator Projection BRSO Coordinate Northing Easting = 762132.405 m = 267038.(vii) MRSO to Cassini Soldner MRSO Coordinate Northing Easting = 679686.511 m BOX 4 Polynomial Transformation Coefficient Cassini-Soldner Coordinate (Kedah/Perlis) Northing = 19364.04669" Longitude = E 116 50' 37.884 m Easting = -27807.589 m (viii) BT68 to BRSO BT68 Coordinate Latitude = N 06 52' 46.383 m 38 .514 m = 793326. Bursa-Wolf transformation tormulae and multiple regression model. 5. JUPEM has produced various sets of datum transformation and map projection parameters to relate the different types of coordinate. 39 . JUPEM will continue to monitor the geodetic infrastructures for any significant displacement and produce up-dated parameters to relate the various coordinates accurately and timely. an in-depth knowledge of coordinates and datums was generally confined to users with specialised knowledge in surveying. Until recently.3 The parameter values relating to different coordinate reference systems are derived from standard coordinate conversion formulae. However.2 For this reason and with the availability of various systems in Malaysia. This technical guide is designed for those dealing with coordinates as a practical solution to the problems that may be encountered with datums and map projections.1 The determination of a position requires the choice of a coordinate reference system. In addressing the effects of plate tectonic motion as well as natural disasters particularly earthquakes. 5. A situation now exists where it is common for a user acquiring data to be using a set of coordinates that is completely different to the one in which the data will be ultimately required. the development of techniques such as GPS has made available entirely new method of acquiring accurate coordinates.5 CONCLUSION 5. Physical Geodesy. (1984). K. Geological Survey Bulletin 1532. 40 . Burford. Defence Mapping Agency (DMA). 6. Map Projections for Geodesists.S. Bulletin Geodesique. Peter. pp165-169. 209-214. Geometric connection and re-orientation of three-dimensional triangulation nets. Geological Survey. A further examination of datum transformation parameters in Australia The Australian Surveyor. (1962). (1985). Washington. The accuracy of geodetic latitude and height equations.. R. pp. Snyder. Studia Geophysica et Geodetica. Richardus.J.32 no.REFERENCES Bowring. 8350. (1963). and Adler. H. San Francisco. pp. no. No. W. USA.2.R. M. B. Geological Survey.P. Bursa. (1974). Map projections used by the U. H Freeman and Company. and the planes of the initial astronomic and geodetic meridians from observations of artificial Earth Satellites. Wolf. Cartographers and Geographers: Amsterdam. Co. Defence Mapping Agency. (1987). 536-558. Technical report no. Department of defence World Geodetic System 1984: its definition and relationship with local geodetic systems (second edition). vol. The theory of the determination of the non-parallelism of the minor axis of the reference ellipsoid and the inertial polar axis of the Earth. Survey Review 28(218): 202-206. J.S. U. B. 68. 7. (1985). North-Holland Pub. Heiskanen and Moritz (1967).