EDITION 1 September 1989 Defense Mapping Agency Fairfax, VA DMATM 8358.2 DMA TECHNICAL MANUAL THE UNIVERSAL GRIDS: Universal Transverse Mercator (UTM) and Universal Polar Stereographic (UPS) Approved for Public Release: Distribution Unlimited DMA Stock No. DMATM83582 TTTjrT.AKSTTTTFn_ SECURITY CLASSIFICATION OF THI5 PAGE REPORT DOCUMENTATION PAGE Form Approved OMB No. 0704-0138 la. REPORT SECURITY CLASSIFICATION PNr.T.AfifiTriED lb. RESTRICTIVE MARKINGS 2a. SECURITY CLASSIFICATION AUTHORITY 2b. DECLASSIFICATION /DOWNGRADING SCHEDULE 3. DISTRIBUTION / AVAILABILITY OF REPORT Distribution Unlimited 4 PERFORMING'ORGANIZATION REPORT NUMBER(S) PHft TM 8358■ Ž 5. MONITORING ORGANIZATION REPORT NUMBER(S) DMA TM 8 358.2 6a. NAME OF PERFORMING ORGANIZATION Defense Mapping Agency Hydrographie/Topographic 6b. OFFICE SYMBOL (If applicable) SDAG 7a. NAME OF MONITORING ORGANIZATION Defense Mapping Agency Plans and Requirements Directorate 6c ADDRESS (City. State, and ZIP Code) 6500 Brookes Lane Washington, D.C., 20315-0030 7b. ADDRESS (City, State, and ZIP Code) 8613 Lee Highway Fairfax, VA 22031-2137 8a. NAME OF FUNDING/SPONSORING ORGANIZATION DMAHTC 8b. OFFICE SYMBOL (If applicable) PP 9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER 8c ADDRESS (Crty, State, and ZIP Code) 6500 Brookes Lane Washington, D.C., 20315-0300 10 SOURCE OF FUNDING NUMBERS PROGRAM ELEMENT NO. PROJECT NO. TASK NO WORK UNIT ACCESSION NO. 11. TITLE (Include Security Classification) The Universal Grids: Universal Transverse Mercator (UTM) and Universal Polar Stereographic (UPS) (UNCLASSIFIED)_ 12. PERSONAL AUTHOR(S) ?er, John W. 13a. TYPE OF REPORT 13b. TIME COVERED 14. DATE OF REPORT (Year, Month, Day) 15. PAGE COUNT Final FROM TO 16. SUPPLEMENTARY NOTATION This manual replaces DA TM 5-241-8, title: Universal Transverse Mercator Grid and 17. COSATI CODES 18. SUBJECT TERMS (Continue on reverse if necestary and identify by block number) FIELD GROUP SUB-GROUP UTM projection parameters, accuracy of equations, coordinate conversions, convergence, scale factors, UPS projection parameters, grid tables. 19. ABSTRACT (Continue on reverse if necessary and identify by block number) This manual describes geographic to grid and grid to geographic conversions for the Universal Transverse Mercator (UTM) and the Universal Polar Stereographic (UPS) grids. It also discusses computations for convergence from geographic positions and from grid coordinates for the UTM grid, and convergence in the Polar Stereographic projection. Both mathematical and tabular methods are illustrated for the above items. A discussion of scale corrections is included. Transformations between the two grids, the Military Grid Reference System and the World Geographic Reference System are discussed. Finally, datum transformation methods, formulas and definitions are illustrated. Diagrams, textual information and software are provided. 20 DISTRIBUTION/AVAILABILITY OF ABSTRACT El UNCLASSIFIED/UNLIMITED □ SAME AS RPT. □ DTIC USERS 21. ABSTRACT SECURITY CLASSIFICATION irařTT.AKKTFT1ín 22a. NAME OF RESPONSIBLE INDIVIDUAL John W. Hager 22b. TELEPHONE (Include Area Code) «QLJ_227-22Ifi .... . 22c. OFFICE SYMBOL DD Form 1473, JUN 86 Previous ed/t/onj are otuo/ete. SECURITY CLASSIFICATION OF THIS PAGE UNCLASSIFIED DMA TM 8358.2 DEFENSE MAPPING AGENCY MISSION; To enhance national security and support the Office of the Secretary of Defense, the Joint Chiefs of Staff, Unified and Specified Commands, Military Departments, and other users, by producing and distributing timely and tailored mapping, charting, and geodetic products, services, and training, and advising on such matters. To provide nautical charts and marine navigational data to worldwide merchant marine and private vessel operators. To maintain liaison with civil agencies and other national and international scientific and other mapping, charting, and geodetic activities. iii DEFENSE MAPPING AGENCY DMA TM 8358.2 OFFENSE MAPPING AGENCYTFCHNICAL MANUAL B358.2 1 8 September 1989 THE UNIVERSAL GRIDS: UNIVERSAL TRANSVERSE MERCATOR (UTM) AND UNIVERSAL POLAR STEREOGRAPHIC fUPS) FOREWORD 1. This manual describes geographic to grid and grid to geographic conversions for the Universal Transverse Mercator (UTM) and the Universal Polar Stereographic (UPS) grids. It also discusses computations lor convergence from geographic positions and from grid coordinates for the UTM grid, and convergence in the Polar Stereographic projection. Both mathematical and tabular methods are illustrated for the above items. A discussion of scale corrections is included. Transformations between the two grids, the Military Grid Reference System and the World Geographic Reference System are discussed. Finally, datum transformation methods, formulas and definitions are illustrated. Diagrams, textual information and software are provided. All software for grid calculations, including those from manual DMA TM 8358.1, are included as ANNEXES A and B. 2. This publication contains no copyrighted material, and has been approved for public release. Distribution is unlimited. Copies may be ordered from the Defense Mapping Agency Combat Support Center, ATTN: DDCP, Washington, D.C. 20315-0020. STANLEY O. SMITH Brigadier General. USAF Chief of Staff RECORD OF CHANGES AND AMENDMENTS AMENDMENT OR CHANGE NO. DATE IMPLEMENTATION DATE REMARKS DMA TM 8358.2 CONTENTS PAGE FOREWORD............................................................................................................................... v CHAPTER 1 GENERAL 1-1 Authority......................................................................................................1-1 1-2 Cancellation................................................................................................1-1 1-3 Purpose......................................................................................................1-1 1-4 Scope..........................................................................................................1-1 1-5 Utilization....................................................................................................1-1 1-6 Definitions...................................................................................................1-2 1 -7 Cross Reference to Other Volumes.............................................................1 -3 1- 8 Technical Publication Information...............................................................1-3 CHAPTER 2 UNIVERSAL TRANSVERSE MERCATOR GRID (UTM) 2- 1 Description of UTM......................................................................................2-1 2-2 Summary of Notation...................................................................................2-1 2-2.1 Ellipsoid Parameters..............................................................................2-1 2-2.2 Universal Transverse Mercator Projection Parameters.........................2-2 2-2.3 Terms Used to Calculate General Equations........................................2-3 2-3 Specifications of the UTM............................................................................2-6 2-4 Accuracy of Equations.................................................................................2-6 2-5 Conversion of Geographic Coordinates to Grid Coordinates......................2-6 2-6 Conversion of Grid Coordinates to Geographic Coordinates......................2-6 2-7 Meridian Convergence from Geographic Coordinates................................2-6 2-8 Meridian Convergence from Grid Coordinates............................................2-7 2-9 Scale Factor for Geographic Coordinates...................................................2-7 2-10 Scale Factor for Grid Coordinates...............................................................2-7 2-11 Sample Output for the Preceding Computations.........................................2-7 ix DMA TM 8358.2 CHAPTER 3 UNIVERSAL POLAR STEREOGRAPHIC GRID (UPS) 3-1 Description of UPS......................................................................................3-1 3-2 Summary of Notation...................................................................................3-1 3-2.1 Ellipsoid Parameters..............................................................................3-1 3-2.2 Universal Polar Stereographic Projection Parameters..........................3-1 3-2.3 Formulas................................................................................................3-2 3-2.4 Specifications of the UPS......................................................................3-4 3-3 Conversion of Geographic Coordinates to Grid Coordinates......................3-4 3-4 Conversion of Grid Coordinates to Geographic Coordinates......................3-4 3-5 Convergence..............................................................................................3-6 3-6 Scale Factor for Geographic Coordinates...................................................3-6 3- 7 Sample Output for the Preceding Computations.........................................3-7 CHAPTER 4 SAMPLE GRID TABLES FOR THE UNIVERSAL TRANSVERSE MERCATOR (UTM) AND UNIVERSAL POLAR STEREOGRAPHIC (UPS) GRIDS 4- 1 General.......................................................................................................4-1 4-2 Design and Preparation of Universal Transverse Mercator Grid Tables.....4-1 4-3 Computation of UTM Grid Coordinates from Geographic Coordinates.......4-3 4-4 Computation of Geographic Coordinates from UTM Grid Coordinates.......4-3 4-5 Computation of the Convergence for the UTM............................................4-3 4-6 Scale Correction for the UTM......................................................................4-3 4-7 Sample Computation for the UTM...............................................................4-4 4-8 Design and Preparation of the Universal Polar Stereographic Grid............4-5 4-9 Formulas Needed to Use the UPS Tables...................................................4-5 4-10 Sample Calculation......................................................................................4-6 4-11 Sample Tables for the UTM and UPS Grids................................................4-7 x DMA TM 8358.2 CHAPTER 5 OTHER COORDINATE CONVERSIONS AND TRANSFORMATIONS 5-1 Grid Conversion Between Zones Within the UTM System and Grid Conversion Between the UTM and UPS Systems.......................................5-1 5-2 Military Grid Reference System...................................................................5-1 5-3 World Geographic Reference System.........................................................5-1 5-4 Datum to Datum Coordinate Transformations.............................................5-1 5-4.1 Molodenskiy Coordinate Transformation Formulas...............................5-1 5-4.1.1 Definition of Terms...........................................................................5-1 5-4.1.2 Standard Molodenskiy Formulas.....................................................5-2 5-4.1.3 Abridged Molodenskiy Formulas.....................................................5-2 5-4.2 Multiple Regression Equations..............................................................5-2 5-4.3 Accuracies.............................................................................................5-3 xi DMA TM 8358.2 ANNEX A SOFTWARE SUPPORT FOR UNIVERSAL GRIDS Section A-1 Software and Documentation for Geographic Coordinate Transformations to/from Grid Coordinates A-1.1 UTMGrid A-1.2 UPS Grid A-2 Software and Documentation for Datum to Datum Coordinate Transformations A-2.1 Molodenskiy A-2.2 Abridged Molodenskiy A-3 Software and Documentation for the UTM and the UPS Grid Coordinate Transformation to/from the MGRS A-4 Software and Documentation for Geographic Coordinate Transformation to/from GEOREF Coordinates A-5 Software and Documentation for Grid Coefficients and Latitude Functions ANNEX B SOFTWARE SUPPORT FOR NON-UNIVERSAL GRIDS Section B-1 Software and Documentation for the Transverse Mercator (TM) Grid B-2 Software and Documentation for the Mercator Grid B-3 Software and Documentation for the Lambert Conical Orthomorphic Grid B-4 Software and Documentation for the Madagascar Gauss Laborde Grid B-5 Software and Documentation for the Rectified Skewed Orthomorphic Grid B-6 Software and Documentation for the New Zealand Map Grid B-7 Software and Documentation for the Guam Azimuthal Equidistant Grid xii DMA TM 8358.2 CHAPTER 1 GENERAL 1-1 AUTHORITY, This document is issued under the authority delegated by DoD Directive 5105.40, subject: Defense Mapping Agency (DMA). 1-2 CANCELLATION. DMA TM 8358.2 replaces Department of the Army Technical Manuals 5-241-8 and 5-241-9. 1-3 PURPOSE, 1-3.1 This manual provides DoD Mapping, Charting and Geodetic (MC&G) production elements, product users and system developers with information necessary for the use of the UTM and UPS grids in surveying and mapping operations. 1-3.2 It also updates the methods and equations used to perform coordinate conversions, convergence and scale adjustments to efficiently program high-speed computers used in these operations. 1-4 SCOPE, 1 -4.1 This manual contains the recommended procedures and tables necessary to convert between grid and geographic coordinates within the Universal Transverse Mercator and Universal Polar Stereographic grid systems. Computations to determine convergence in the two systems and corrections for scale variations are also discussed and illustrated. One chapter is dedicated to the methods used to perform datum transformations and grid/geographic coordinate transformations. 1-4.2 Annex A refers to software and documentation available forthe two universal grids. Annex B refers to software and documentation available for selected non-universal grids. 1-5 UTILIZATION. 1-5.1 DMA TM 8358.2 is to be used by DoD MC&G production elements, product users and DoD system developers in the application of datums, ellipsoids, grids, grid/geographic coordinate conversions and grid reference systems. 1-5.2 Users are cautioned that the information contained herein applies to current and future MC&G production and does not necessarily apply to products that are currently available through the DoD supply system. 1-6 DEFINITIONS. 1 -6.1 Datum. As used in this manual, datum refers to the geodetic or horizontal datum. The classical datum is defined by five elements giving the position of the origin (two elements), the orientation of the network (one element) and the parameters of a reference ellipsoid (two elements). More recent definitions express the position and orientation as a function of the deviations in the meridian and in the prime vertical, the geoid-ellipsoid separations, and the parameters of a reference ellipsoid. The World Geodetic System datum gives positions on a specified ellipsoid with respect to the center of mass of the earth. 1-6.2 Easting. Eastward (that is left to right) reading of grid values on a map. 1-1 DMA TM 8358.2 1-6.3 Ellipsoid. A three-dimensional figure generated by the revolution of an ellipse about one of its axes. The ellipsoid that approximates the geoid is an ellipsoid rotated about its minor axis, or an oblate spheroid. 1-6.4 fifiojd- The equipotential surface in the Earth's gravity field approximates the undisturbed mean sea level extended continuously through the continents. The direction of gravity is perpendicular to the geoid at every point. The geoid is the surface of reference for astronomic observations and for geodetic leveling. 1 -6.5 Graticule. A network of lines representing parallels of latitude and meridians of longitude forming a map projection. 1 -6.6 Gjkj. Two sets of parallel lines intersecting at right angles and forming squares; a rectangular Cartesian coordinate system that is superimposed on maps, charts and other similar representations of the Earth's surface in an accurate and consistant manner to permit identification of ground locations with respect to other locations and the computation of direction and distance to other points. 1-6.7 Map Projection. An orderly system of lines on a plane representing a corresponding system of imaginary lines on an adopted terrestrial datum surface. A map projection may be derived by geometrical construction or by mathematical analysis. 1-6.8 Military Grid Reference System )3/2 v = radius of curvature in the prime vertical; also defined as the normal to the ellipsoid terminating at the minor axis p(1 + e'2cos2) (1 - e2sin V2 S = meridional arc, the true meridional distance on the ellipsoid from the equator = A' - B'sin2<(> + C'sirvty - D'sin&t) + E'sin8<|> 2-1 DMA TM 8358.2 where: A' 1-n+ |(n2-n3) + fl(n4-n5) + * -fa n-n2+ -£(n3-n4) + gnV C «^a 16 n2_n3+ 4(n4-n5) + 4 n3_n4+ J-!n5 + 16 E' 315 512 n4-n5 + (E' - 0.03mm) 2-2.2 Universal Transverse Mercator Projection Parameters. $ = latitude X = longitude 4' «= latitude of the foot of the perpendicular from the point to the central meridian Xo = longitude of the origin (the central meridian) of the projection AX. « k - Xo = difference of longitude from the central meridian (for general formula use, value is sign dependent; for use in tables, value always considered positive) ko «= central scale factor, an arbitrary reduction applied to all geodetic lengths to reduce the maximum scale distortion of the projection. FortheU.T.M.,ko = 0.9996 k = scale factor at the working point on the projection. FN » False Northing (0 for the Northern Hemisphere; 10,000,000 for the Southern Hemisphere) FE - False Easting (500,000) AE ■ E - FE (for general formula use, value is sign dependent; for use in tables, value always considered positive) E * grid easting N ' grid northing C * convergence of the meridians (i.e. the angle between true north and grid north) 2-2 DMA TM 8358.2 2-2.3 Terms Used to Calculate General Equations The following terms are used to calculate the general equations which follow in this chapter. With some modification, they are also used to produce the tables in chapter four. These terms are derived from the Functions found in the U.S. Department of Commerce, Special Publication No. 251. They differ slightly from the Functions used in DA TM 5-241-8 in that they are the fully expanded terms. T1 = Sko usirvt>cos$ko T2 T10 usin4>cos3<|>ko T3 * -(5-tan2<|» + 9e'2cos2«t» + 4e'4cos4 - 330tan2Ae'2cos2(|> + 445e'4cos4<)» + 324e'6cos6<|>- BSOtanV^cos4^ + ßSe^cos8* - eootan^^cos6^ - 192tan2$e'8cos8cos7<|>ko T5 = -(1385 - 3111 tan2 + 543tan4<> - tan6<») 40320 T6 = \XX>scos34>ko T7 = -(1 - tan2<() + e'2cos24i) 6 \x»s54>ko T8 = -(5 - 18tan2<{> + tan4<)> +14e'2cos2(|) - 58tan2 + 1 Se^cos4^ 120 + 4e'6cos6 - 64tan2e'4cos4<)) - 24tan2<|>e'6cos6) \xx>s7Qko T9 = -(61 - 479tan2<|> + 179tan4<|> - lan% 5040 tan*' 2puko2 tarvj)' T11 - -(5 + 3tanV+ e'^cosV - 4e'*cos V - 9tan^'e":cos':0') 24pv3k04 2-3 8.2 tarty' 720p\)5ko6 (61 + 90tanfy + 46e'2cos2f+ 45tanV - 252tanVe'2cos2' - Se^cosV + 10Oe^cosV - eĎtanfye^cosV - 90tanVe'2cosV + SSe^cosV + 225tanVe'4cosV + 84tanfye'6cosy - l92tanVe'8cos8(|»') tai>)>' —— (1385 + 3633tanV + 4095tanV + 1 575tan6<>') 7|y 8 40320p\/ko' 1 VCOS^ťko 6v3cos<)>'ko3 1 (1 + 2tanV + e'2cosV) ^O^cosir'ko5 --— (5 + 6e'2cosV+ 28tan2(|)'- Se^cosV + 8tanVe'2cosV + 24tan4<|)' - 4e'6cosy + 4tanVe'4cosV + 24tanVe'6cos6') -^-- (61 + 662tan2f + 1320tanV + 720tan6') 5040u cos^lío sin sirtycos24> -(1 + 3e'2cos2<(> + 2e'4cos4<| sirtycos44> ---(2 - tanz<|> + 15e'zcosz<|> + SSe^cos4^ - 15tan2<(>e'2cosz<|> + SSe^cos6^ 15 - SOtan^^cosV +11 e^cos6^ - eotan^^cos6«}» sirtycos6<|> 315 tarty' - 24tan2<|>e'8cos8' + 3tanV + tan2d/e'2cos2' + ge^cosV 15u5ko5 + 20e'6cosV - 7tan2<(>'e'4cosV - 27tanfye'6cosV +11 e^cosV - 24tanfye'8cos8') tan<(>' T25 = -(17 + 77tan V + 105tanV + 45tanV) 315u7ko7 cos2o> T26 = -(1 + e'2cos2) cos 4» T27 = -(5 - 4tan2<)> + 14e'2cos2<|> + 1 Se^cos4«) - 28tan2e'2cos2 + 4e'6cos6 24 -^etan^^cos4^ - 24tan2) cos6 T28 = -(61 - 148tan2<|> + 1 etan4«» 720 T29 - —-(1 + e'2cos2<)>') 2u2ko2 T30 = ---(1 + 6e'2cos2f + Se^cos V + 4e'6cos V - 24tan2'e'4cos V 24v4ko4 - 24tan2fe/6cos$') T31 720\)6ko6 2-5 DMA TM 8358.2 2-3 SPECIFICATIONS OF THE [JTM. Projection: Transverse Mercator (Gauss-Kruger type), in zones 6° wide. Unit of measurement: Meter Zone numbering: Starting with 1 for the zone from 180°W to 174°W, and increasing eastward to 60 for the zone from 174°E to 180°E. Latitude limits: North: 84°N South: 80°S Zone limits and overlap: The zones are bounded by meridians whose longitudes are multiples of 6° west or east of Greenwich. On large-scale maps, an overlap of approximately 40km on either side of the junction is provided for engineer surveyors and for artillery survey and firing. This overlap is never used in giving a grid reference. Polar region overlap: The U.T.M. overlaps 30' onto the Universal Polar Stereographic Grid, which extends from the poles to 83°30'N or 79°30'S respectively. 2-4 ACCURACY OF THE EQUATIONS. The computations in this chapter, using geodetic latitude, are accurate to the nearest .001 arc second for geographic coordinates and to the nearest .01 meter for grid coordinates. More accurate formulas are available. These formulas contain more terms and utilize isometric latitude. If desired, FORTRAN subroutines and programs utilizing these formulas may be obtained from HQ DMA, ATTN: PR. 2-5 CONVERSION OF GEOGRAPHIC COORDINATES TO GRID COORDINATES. The general formulas for the computation of N and E are: N m FN + (T1 + AX212 + AX4T3 + A?„6T4 + AX8T5) E = FE + (AXT6 + AX^n + AX^8 + AA7T9) 2-6 CONVERSION OF GRID COORDINATES TO GEOGRAPHIC COORDINATES. The general formulas for the computation of <|> and X are: = geodetic latitude X = isometric latitude = an auxiliary latitude used in the conformal mapping of the spheroid on a sphere. By transforming geographic latitudes on the spheroid into isometric latitudes on a sphere, a conformal map projection (the Mercator) may be calculated, using spherical formulas, for the plotting of geographic data. X - longitude z = isometric colatitude = 90° minus isometric latitude. 3-1 DMA TM 8358.2 ko = scale factor at the pole, an arbitrary reduction applied to all geodetic lengths to reduce the maximum scale distortion of the projection. k = scale factor at the working point on the projection. - R = radius of the parallel of latitude from the pole. N = Northing E = Easting FN = False Northing FE = False Easting AN = N - FN AE = E - FE 3-2.3 Formulas. r -e/2 2a ri-e Co = constant = — 2vi* |_1+e. _e/2 z r 1 + e sin<|> tan— = -- 2 L 1 - e sin. R = radius = koCotan — 2 tan ,---- L4 2. Constants for computing isometric to geodetic latitude: e2 5e4 e6 13e8 A = — + - + — + - 2 24 12 360 7e4 29e6 81 le8 B = C = D = 48 240 11520 7e6 81e8 120 1120 4279e8 161280 The preceding terms are used to calculate the general equations which follow in this chapter and the tables in chapter four. 3-2 DMA TM8358.2 180° FIGURE 1(a). North Zone o° 180« FIGURE 1 (b). South Zone 3-3 DMA TM 8358.2 3-2.4 Bonifications of the UPS. Unit of Measurement: Meter Ellipsoid: WGS 84 False Northing: 2,000,000 meters False Easting: 2,000,000 meters Scale Factor at the Origin: 0.994 Orientation of the grid systems: In both zones, the 2,000,000 meter easting line coincides with the 0° and the 180° meridians and grid north is equal to true north on the 0° meridian (See figure 1). Limits of system: North zone: The north polar area 84° - 90° South zone: The south polar area 80° - 90° Overlap with the UTM: The U.P.S. grid will be extended to 83°30'N and 79°30'S to provide a 30' overlap with the Universal Transverse Mercator Grid System. 3-3 CONVFRSION OF GEOGRAPHIC COORDINATES TO GRID COORDINATES. The general formulas for the computation of N and E are: N = FN - RcosX. for the North Zone N = FN + RcosX for the South Zone E = FE + RsinX for Both Zones 3-4 CONVERSION OF GRID COORDINATES TO GEOGRAPHIC COORDINATES. 3-4.1 From the notation in Section 3-2.3, it is given: AN = N - FN AE = E - FE To compute the longitude, let: L = arctan for the North Zone where N * 0 -AN L = arctan —- for the South Zone where N * o AN A single argument arctan routine returns an angle 'L' in the range of n/2 (90°) to -n/2 (-90°). The longitude of the point is equal to 'L' if the denominator of the above equations is positive. If the denominator is negative, then the longitude is obtained as follows: 3-4 DMA TM 8358.2 X = 7c + L when AE is positive and angle 'L' is negative X = -7i + L when AE is negative and angle 'L' is positive A two argument arctan routine returns the value of X directly (range of n to -n) except in the case where both arguments are equal to zero. If AN = 0, then X ■ 90° E or W, depending on the sign of AE (see figure 1). If AN = AE = 0, then the point is at the pole and X is undefined. When computing longitude by hand, use the above arctan functions when AE is numerically less than AN. When AE is numerically greater than AN, use the following arccot functions for greater accuracy: arccot -AN AE for the North Zone arccot AN AE for the South Zone NOTE: The programmed equations utilize only the arctan function since this routine is standard in most systems. Normally, using only the arctan function will not result in any loss of accuracy on the computer as the machine defaults to the arccot routine where appropriate (i.e. above 45°). 3-4.2 The computation of latitude begins by defining R: R = AE sin X where AE * 0 if AE - AN = 0, then $ = ~ = 90 = pole if AN = 0, if AE « 0, then R = I AE | then R = I AN I When computing R by hand, use the previous equation when AN is numerically less than AE. When AN is numerically greater than AE, use the following equation for greater accuracy: R = AN cos X where AN = 0 From R and X, determine the isometric colatitude (z) from the following: R tan |-2 3-5 DMA TM 8358.2 Using the isometric latitude, determine <> from the following: = X + Asin2X + Bsin4X + Csin6X + Dsin8Z where: NOTE: Latitude, in the Polar Stereographic System, is always treated as positive, regardless of hemisphere. 3-5 CONVERGENCE. Convergence in the Polar Stereographic Projection is equal to the longitude, k, in numeric value. In the north polar area, it has the same sign as k and in the south polar area it has the opposite sign. Unlike most grid systems, where convergence is a small angle, the polar system may have convergence angles up to 180° east or west. 3-6 SCALE FACTOR FOR GEOGRAPHIC COORDINATES. The scale factor is the factor by which true distances over short lines must be multiplied to obtain grid distances. The scale factor is constant for any given latitude. The general formula for the scale factor is: NOTE: The computation of both convergence and scale factor has generally been replaced by geodetic inverse and forward computer programs to determine geodetic azimuths and distances. Formulas such as those by Vincenty and Sodano are available from HQ DMA, ATTN: PR. 3-6 DMA TM 8358.2 3-7 SAMPLE OUTPUT FOR THE PRECEDING COMPUTATIONS. COORDINATE CONVERSIONS ON THE UNIVERSAL POLAR STEREOGRAPHIC (UPS) GRID ELLIPSOID DATA A 1/F NAME 6378137.000 298.257223563 WGS-34 B E"2 EB~2 6356911.94613 .006694379990 .006739406742 UNITS METERS LATITUDE AND LONGITUDE -TO- NORTHING AND EASTING ID LATITUDE LONGITUDE NORTHING EASTING CONVERGENCE 1 84 17 14.042N 132 14 52.761W 2426773.60 1530125.78 132 14 52.76 W 2 73 0 .000N 44 0 .O00E 632668.43 3320416.75 44 0 .00 E 3 87 17 14.400S 132 14 52.303E 1797474.90 2222S79.47 132 14 52.30 W NORTHING AND EASTING -TO- LATITUDE AND LONGITUDE ID LATITUDE LONGITUDE NORTHING EASTING CONVERGENCE 4 84 17 14.042N 132 14 52.762W 2426773.60 1530125.78 132 14 52.76 W 73 0 .0O0N 44 0 .000E 632668.43 3320416.75 44 0 .00 E 83 38 14.343S 135 0 .000E 1500000 00 2500000.00 135 0 .00 E SCALE FACTOR .99647445 1.01619505 .99455723 SCALE FACTOR .09647445 1.01619505 .99707070 3-7 DMA TM 8358.2 CHAPTER 4 SAMPLE GRID TABLES FOR THE UNIVERSAL TRANSVERSE MERCATQR (UTM) AND UNIVERSAL POI AR STFREOGRAPHIC (UPS\ GRIDS 4- 1 GENERAL. Tables were traditionally used to solve complex, repetitive calculations. With the advent and ready access to computers and hand-held programmable calculators, tables are now rarely used. Their discussion has been included in this manual, however, to maintain continuity with manuals DA TM 5- 241-8 and DA TM 5-241-9, which this manual replaces. The equations have been adapted from those in the previous manuals to produce the tables in this manual. A representative example of the function tables for the International Ellipsoid is included at the end of this chapter. 4-2 DESIGN AND PREPARATION OF UNIVERSAL TRANSVERSE MERCATQR GRID TABLES. 4-2.1 The tables in this chapter were computed using the International Ellipsoid where: a = 6,378,388 meters 1/f = 297 b - 6,356,911.946 meters e2 = 0.006 722 670 022 Tables for other ellipsoids should be constructed using the constants provided in DMA TM 8358.1. 4-2.2 Other Notation: ko = -9996 p = .0001AX" q = .000001AE where AE = IE - FEI (FE = false easting) S = A> - B'sin2<|> + C'sin4<(> - D'sin6 + E'sin8<|> where (forthe International Ellipsoid): A' = 6,367,654.500 058 meters B' = 16,107.034 678 meters C - 16.976 211 meters D' - 0.022 266 meters E' - 0.000 032 meters NOTE: The values A' through E' were derived from the formulas listed in Section 2-2.1. 4-2.3 The functions represented by Roman Numerals (I) to (XIX) in DA TM 5-241-8, and functions (XX) and (XXI) are calculated from the T terms of Section 2-2.3 herein as follows: 4-1 DMA TM 8358.2 (1) T1 = Sko (II) (T2sin21") x108 (III) = (T3sin41")x 1016 (IV) = (T6sin1")x 104 (V) = (T7sin31'0x 1012 (VII) = (T10/sin1")x 1012 (VIII) = (T11/sin1")x 1024 (IX) = (T14/sin1*)x106 (X) = (T15/sin1")x 1018 (XII) = T18x 104 (XIII) = (T19sin21")x 1012 (XV) = (T22/sin 1") x 10s (XVI) = (T23/sin1")x 1018 (XVIII) = T29x 1012 (XIX) - T30x1024 (XX) = (T26sin21") x 108 (XXI) = (T27sin41")x 1016 4-2.4 The terms represented graphically in DA TM 5-241-8 are represented as values in the new tables. They are calculated from the following: Ae = (T4sin61")x 1024 B5 = (T8sin51")x*>n20 C5 - (T20sin41")x 1020 De = (Tia/sinl^xlO36 Es - (Tie/sinnxlO30 Fs -= ^/sinnxlO30 NOTE: Unlike DA TM 5-241-8, the terms p and q are not incorporated into the above terms, but rather, are included in the equations used to perform grid/geographic conversions and convergence computations when using the tables. 4-2 DMA TM 8358.2 4-2.5 Two correction terms, A2(IV) and A2(IX), which were shown graphically in the old tables were eliminated in the construction of the new tables. They are difficult to interpolate with precision and therefore, for accuracy requirements greater than that provided by the table, it is recommended that calculations be done using the computer programs listed in the appropriate Annexes. NOTE: The tables presented in this chapter are constructed at 1' intervals, however, they can be requested from HQ DMA, ATTN: PR at any desired interval. 4-3 COMPUTATION OF UTM GRID COORDINATES FROM GEOGRAPHIC COORDINATES. The formulas for the computation of E' and N are: N = (I) + (ll)p2+ (lll)p4 +Asp6 E' = (IV)p + (V)p3+ B5p5 NOTE: E' is added to or subtracted from 500,000, depending on whether the point is east or west of the central meridian. This E' should not be confused with the E' in Section 4-2.2. South of the equator: N = 10,000,000 - [(I) + (ll)p2 + (lll)p4 + Aep6] 4-4 COMPUTATION OF GFOGRAPHIC COORDINATES FROM UTM GRID COORDINATFS 4-4.1 The formulas for computing geographic coordinates from UTM grid coordinates are: $ = f - (Vll)q2 + (Vlll)q4 - Dsq6 AX = (IX)q-(X)q3 + E5q5 X = Xo ± AX where Xo is the longitude of origin of the projection (the central meridian). 4-4.2 The footpoint latitude (<(»') is obtained by entering the table through Function I with N as the argument in the northern hemisphere or 10,000,000 - N as the argument in the southern hemisphere, and making an inverse interpolation. 4-5 COMPUTATION OF THE CONVERGENCE FOR THE UTM. The formula for the computation of convergence from geographic coordinates is: C = (Xll)p + (Xlll)p3 + Csp5 and from grid coordinates is: C = (XV)q - (XVI)q3 + F5q5 4-6 SCALE CORRECTION FOR THE UTM. The formula for scale correction lor geographic coordinates is: k = k0[1 + (XX)p2 + (XXI)p4] 4-3 DMA TM 8358.2 and for grid coordinates is: k = kop +(XVIII)q2 + (XIX)q4] NOTE: Functions (XVIII) and (XIX) can be determined from the tables by either N orfootpoint latitude. 4-7 SAMPLE COMPUTATION FOR THE IJTM. A sample computation is provided for computing UTM grid coordinates from geographic coordinates as follows: Given: Latitude = 34° 15'34".742 N Longitude - 96° 02' 43*.158 E Central Meridian - 99° 00' OCT E From the proceeding, it follows: AX - 2° 57'16".842 - 10.636.842" p = 1.0636842 To determine the Northing, derive the following functions using the tables: Function I: Even minutes of = 3789935.119 Interpolation for seconds of 0 « 1070.106 I - 3791005.225 Function II: Even minutes of 4» «= 3489.536 Interpolation for seconds of $ = 0.465 II - 3490.001 Function III: III - 2.138 Function A6: Afi - 0.0009 The above functions are then multiplied by the appropriate powers of p as indicated in the formula and then summed to give: I - 3791005.225 lip2« 3948.671 lllp4« 2.737 Asp6- 0.001 3794956.630 - NORTHING 4-4 DMA TM 8358.2 To determine the Easting, derive the following functions using the tables: Function IV: Even minutes of 0 = 255779.038 Interpolation for seconds of $ ■ -29.205 IV - 255749.833 Function V: Even minutes of ■ 37.040 Interpolation for seconds of <|> = -0.036 V - 37.004 Function B5: Bs - -0.0175 The proceeding functions are then multiplied by the appropriate powers of p as indicated in the formula and then summed to give: IVp = 272037.057 Vp3 = 44.534 B5p5 = -0.024 plus False Easting* 500000.000 772081.570 - EASTING •NOTE: False Easting is either added or subtracted dependent on the location of the point relative to the central meridian. All other computations are performed in a similar fashion and therefore need not be discussed in this manual. 4-8 DESIGN AND PREPARATION OF THE UNIVERSAL POLAR STEREOGRAPHIC GRID. 4-8.1 The table of radii is produced from the basic formula for the Polar Stereographic Projection as follows: R = ko[(2a/(1 - e2)1/2l[(l - e)/(l + ej^an^) z/2 « the isometric semi-colatitude 4-8.2 The tables of scale factors were computed from the following formula: k = R/(ucos<(») v « the radius of curvature in the prime vertical. 4-9 FORMULAS NEEDED TO USE THE UPS TABLES. The formulas to utilize the tables for the Universal Polar Stereographic Grid System are identical to those presented in chapter 3 of this manual. 4-5 DMA TM 8358.2 4-10 SAM PI F CALCULATION. A sample computation is provided for computing UPS grid coordinates from geographic coordinates as follows: Given: From the table: 4> . 84° 17'14*042 N X - 132° 14' 52".761 W R (even minutes of ) Interpolation for seconds of (j> R Therefore: And finally: cosX sinX Northing Easting COS(-132°14'52"\761) -0.67234083 sin(-132° 14' 52".761) -0.74024172 2,000,000 - RcosX 2,000,000 + RsinX. 635191.905 ^434.088 634757.817 -C0S(47° 45' Or.239) -Sin(47° 45' Or.239) 2426773.60 1530125.78 4-6 DMA TM 8358.2 4-11 SAMPLF TABLES FOR THE UTM AND UPS GRIDS. For the UTM grid: ELLIPSOID DATA A 1/F NAME UNITS 6378388.000 297.000000000 INTERNATIONAL METERS B E"2 EB"2 6356911.94613 .006722670022 .006768170197 LATITIDE N DIFF. V DIFF. r 34 34 34 34 34 6385102.774 6385108.576 6385114.378 6385120.183 6385125.988 .09669 .09671 .09674 .09676 .09678 6355538.173 6355555.496 6355572.824 6355590.157 6355607.493 .28873 .28880 .28887 .28894 .28901 34 34 34 34 34 5 0 6 0 7 0 8 0 9 0 6385131.795 6385137.604 6385143.413 6385149.224 6385155.037 .09681 .09683 .09685 .09687 .09690 6355624.833 6355642.178 6355659.526 6355676.879 6355694.236 .28907 .28914 .28921 .28928 .28935 3410 34 11 34 12 34 13 34 14 6385160.851 6385166.666 6385172.482 6385178.300 6385184.119 .09692 .09694 .09696 .09699 .09701 6355711.597 6355726.962 6355746.331 6355763.704 6355781.081 .28942 .28948 .28955 .28962 .28969 34 15 34 16 34 17 34 18 34 19 6385189.940 6385195.762 6385201.585 6385207.410 6385213.236 .09703 .09705 .09708 .09710 .09712 6355798.462 6355815.848 6355833.237 6355850.631 6355868.028 .28976 .28982 .28989 .28996 .29003 34 20 34 21 34 22 34 23 34 24 6385219.063 6385224.891 6385230.721 6385236.553 6385242.385 .09714 .09717 .09719 .09721 .09723 6355885.430 6355902.835 6355920.245 6355937.658 6355955.076 .29009 .29016 .29023 .29029 .29036 34 25 34 26 34 27 34 28 34 29 6385248.219 6385254.055 6385259.891 6385265.729 6385271.569 .09726 .09728 .09730 .09732 .09734 6355972.498 6355989.923 6356007.353 6356024.787 6356042.224 .29043 .29049 .29056 .29063 .29069 34 30 0 6385277.409 .09737 6356059.666 .29076 4-7 DMA TM 8358.2 METERS PER SECOND R*SIN1" N*SIN1"*COS(LAT) LATITUDE MERIDIONAL ARC DIFF.1" LATITUDE LONGITUDE 34 0 0 3763719.865 30.81256 30.81252 25.66356 34 1 0 3765568.619 30.81264 30.81260 25.65855 34 2 0 3767417.378 30.81273 30.81269 25.65354 34 3 0 . 3769256.141 30.81281 30.81277 25.64852 34 4 0 3771114.910 30.81290 30.81285 25.64350 34 5 0 3772963.684 30.81298 30.81294 25.63848 34 6 0 3774812.463 30.81306 30.81302 25.63345 34 7 0 3776661.247 30.81315 30.81311 25.62843 34 8 0 3778510.036 30.81323 30.81319 25.62340 34 9 0 3780358.830 30.81332 30.81328 25.61837 34 10 0 3782207.629 30.81340 30.81336 25.61334 34 11 0 3784056.433 30.81349 30.81344 25.60830 34 12 0 3785905.242 30.81357 30.81353 25.60326 34 13 0 3787754.056 30.81365 30.81361 25.59822 34 14 0 3789602.875 30.81374 30.81370 25.59318 34 15 0 3791451.700 30.81382 30.81378 25.58814 34 16 0 3793300.529 30.81391 30.81386 25.58309 34 17 0 3795149.363 30.81399 30.81395 25.57805 34 18 0 3796998.203 30.81408 30.81403 25.57300 34 19 0 3798847.048 30.81416 30.81412 25.56794 34 20 0 3800695.897 30.81424 30.81420 25.56289 34 21 0 3802544.752 30.81433 30.81429 25.55783 34 22 0 3804393.611 30.61441 30.81437 25.55277 34 23 0 3806242.476 30.81450 30.81446 25.54771 34 24 0 3805091.346 30.81458 30.81454 25.54265 34 25 0 3809940.221 30.81467 30.81462 25.53758 34 26 0 3811789.101 30.81475 30.81471 25.53252 34 27 0 3813637.986 30.81484 30.81479 25.52745 34 28 0 3815486.876 30.81492 30.81488 25.52238 34 29 0 3817335.771 30.81500 30.81496 25.51730 34 30 0 3819184.672 30.81509 30.81505 25.51223 4-8 DMA TM 8358.2 LATITUDE (I) DIFF. 1" (II) DIFF.1" (III) (A6) 34 0 0 3762214.378 30.80024 3477.361 .01367 2.147 .0009 34 1 0 3764062.392 30.80032 3478.181 .01365 2.147 .0009 34 2 0 3765910.411 30.80040 3478.999 .01363 2.146 .0009 34 3 0 3767758.435 30.80049 3479.817 .01361 2.145 .0009 34 4 0 3769606.464 30.80057 3480.633 .01359 2.145 .0009 34 5 0 3771454.499 30.80066 3481.449 .01357 2.144 .0009 34 6 0 3773302.538 30.80074 3482.263 .01355 2.144 .0009 34 7 0 3775150.582 30.80082 3483.076 .01353 2.143 .0009 34 8 0 3776998.632 30.80091 3483.887 .01351 2.142 .0009 34 9 0 3778846.686 30.80099 3484.698 .01349 2.142 .0009 34 10 0 3780694.746 30.80108 3485.507 .01347 2.141 .0009 34 11 0 3782542.810 30.30116 3486.315 .01345 2.140 .0009 34 12 0 3784390.880 30.30124 3487.122 .01343 2.140 .0009 34 13 0 3786238.955 30.80133 3487.928 .01341 2.139 .0009 34 14 0 3788087.034 30.80141 3488.733 .01339 2.139 .0009 34 15 0 3789935.119 30.80150 3489.536 .01337 2.138 .0009 34 16 0 3791783.209 30.80158 3490.339 .01335 2.137 .0009 34 17 0 3793631.304 30.80167 3491.140 .01333 2.137 .0009 34 18 0 3795479.404 30.80175 3491.940 .01331 2.136 .0009 34 19 0 3797327.509 30.80183 3492.738 .01329 2.135 .0009 34 20 0 3799175.619 30.80192 3493.536 .01327 2.135 .0009 34 21 0 3801023.734 30.80200 3494.332 .01325 2.134 .0009 34 22 0 3802871.854 30.80209 3495.127 .01323 2.134 .0009 34 23 0 3804719.979 30.80217 3495.921 .01321 2.133 .0009 34 24 0 3806568.110 30.80226 3496.714 .01319 2.132 .0009 34 25 0 3808416.245 30.80234 3497.506 .01317 2.132 .0009 34 26 0 3810264.335 30.80243 3498.296 .01315 2.131 .0009 34 27 0 3812112.531 30.80251 3499.036 .01314 2.130 .0009 34 28 0 3813960.681 30.80259 3499.874 .01312 2.130 .0009 34 29 0 3815808.837 30.80268 3500.661 .01310 2.129 .0009 34 30 0 3817656.998 30.30276 3501.447 .01308 2.128 .0009 NORTHING - (I) + (ll)P"2 + (lll)P"4 + A(6)P"6 NORTHERN HEMISPHERE SUBTRACT FROM 10 MILLIONS FOR SOUTHERN HEMISPHERE P - .0001 DELTA(LONGITUDE) IN SECONDS FROM CENTRAL MERIDIAN 4-9 DMA TM 8358.2 LATITUDE (IV) DIFF. 1" (V) DIFF. 1" (B5) 34 0 0 256532.988 -.83519 37.976 -.00103 -.0170 34 1 0 256482.877 -.83555 37.905 -.00103 -.0170 34 2 0 256432.744 -.83591 37.843 -.00103 -.0170 34 3 0 256382.589 -.83627 37.782 -.00103 -.0171 34 4 0 256332.413 -.83664 37.720 -.00103 -.0171 34 5 0 256282.214 -.83700 37.658 -.00103 -.0172 34 6 0 256231.994 -.83736 37.596 -.00103 -.0172 34 7 0 256181.753 -.83772 37.534 -.00103 -.0172 34 8 0 256131.489 -.83808 37.472 -.00103 -.0173 34 9 0 256081.204 -.83845 37.410 -.00103 -.0173 34 10 0 256030.898 -.83881 37.349 -.00103 -.0173 34 11 0 255980.569 -.83917 37.287 -.00103 -.0174 34 12 0 255930.219 -.83953 37.225 -.00103 -.0174 34 13 0 255879.847 -.83989 37.163 -.00103 -.0175 34 14 0 255829.454 -.84025 37.101 -.00103 -.0175 34 15 0 255779.038 -.84062 37.040 -.00103 -.0175 34 16 0 255728.601 -.84098 36.978 -.00103 -.0176 34 17 0 255678.143 -.84134 36.916 -.00103 -.0176 34 18 0 255627.662 -.84170 36.854 -.00103 -.0176 34 19 0 255577.161 -.84206 36.793 -.00103 -.0177 34 20 0 255526.637 -.84242 36.731 -.00103 -.0177 34 21 0 255476.092 -.84278 36.669 -.00103 -.0178 34 22 0 255425.525 -.84314 36.607 -.00103 -.0178 34 23 0 255374.936 -.84353 36.546 -.00103 -.0178 34 24 0 255324.326 £4386 36.484 -.00103 -.0179 34 25 0 255273.694 -.84423 36.442 -.00103 -.0179 34 26 0 255223.041 -.84459 36.360 -.00103 -.0179 34 27 0 255172.365 -.84495 36.299 -.00103 -.0180 34 28 0 255121.669 -.84531 36.237 -.00103 -.0180 34 29 0 255070.950 -.84567 36.175 -.00103 -.0181 34 30 0 251020.210 -.84603 36.113 -.00103 -.0181 DELTA(EASTING) - (IV)P + (V)P"3 + (B5)P"5 ADD OR SUBTRACT FROM FALSE EASTING (500,000) P - .0001 DELTA(LONGITUOE) IN SECONDS FROM CENTRAL MERIDIAN 4-10 DMA TM 8358.2 LATITUDE (I) DIFF. 1" (VII) DIFF. 1" (VIII) (D6) 34 0 0 3762214.378 30.80024 1715.576 .01784 22.286 .32 34 1 0 3764062.392 30.80032 1716.646 .01785 22.306 .32 34 2 0 3765910.411 30.80040 1717.717 .01785 22.326 .32 34 3 0 3767758.435 30.80049 1718.788 .01786 22.346 .32 34 4 0 3769606.464 30.80057 1719.860 .01787 22.366 .32 34 5 0 3771454.499 30.80066 1720.932 .01788 22.386 .32 34 6 0 3773302.538 30.80074 1722.004 .01788 22.406 .32 34 7 0 3775150.582 30.80082 1723.077 .01789 22.426 .32 34 8 0 3776998.632 30.80091 1724.151 .01790 22.446 .32 34 9 0 3778846.686 30.80099 1725.224 .01790 22.466 .32 34 10 0 3780694.746 30.80108 1726.299 .01791 22.486 .32 3411 0 3782542.810 30.80116 1727.373 .01792 22.506 .32 34 12 0 3784390.830 30.80124 1728.448 .01792 22.526 .32 34 13 0 3786238.955 30.80133 1729.524 .01793 22.546 .32 34 14 0 3788087.034 30.80141 1730.600 .01794 22.566 .32 34 15 0 3789935.119 30.80153 1731.676 .01794 22.586 .32 34 16 0 3791783.209 30.80158 1732.752 .01795 22.606 .32 34 17 0 3793631.304 30.80167 1733.830 .01796 22.626 .33 34 18 0 3795479.404 30.80175 1734.907 .01797 22.646 .33 34 19 0 3797327.509 30.80183 1735.935. .01797 22.666 .33 34 20 0 3799175.619 30.80192 1737.063 .01798 22.686 .33 34 21 0 3801023.734 30.80200 1738.142 .01799 22.707 .33 34 22 0 3802871.854 30.80209 1739.221 .01799 22.727 .33 34 23 0 3804719.979 30.80217 1740.301 .01800 22.747 .33 34 24 0 3806568.110 30.80226 1741.381 .01801 22.767 .33 34 25 0 3808416.245 30.80234 1742.462 .01802 22.788 .33 34 26 0 3810264.335 30.80243 1743.542 .01802 22.808 .33 34 27 0 3812112.531 30.80251 1744.624 .01803 22.828 .33 34 28 0 3813960.681 30.80259 1745.706 .01804 22.849 .33 34 29 0 3815808.837 30.80268 1746.738 .01804 22.869 .33 34 30 0 3817656.998 30.80276 1747.870 .01805 22.889 .33 DELTA(LATITUDE) « (VII)Q"2 - (VIII)Q**4 + (D6)Q"6 SUBTRACT FROM FOOTPOINT LATITUDE IN SECONDS Q - .000001 DELTA(EASTING) - DIFFERENCE IN EASTING FROM FALSE EASTING (500,000) TABLE FOR USE WITH FOOTPOINT LATITUDE 4-11 DMA TM 8358.2 LATITUDE (IX) DIFF. 1" (X) DIFF. 1" (E5) 34 0 0 38981.341 .12694 305.344 .00402 4.454 34 1 0 38988.958 .12704 305.585 .00402 4.460 34 2 0 38996.580 .12714 305.827 .00403 4.466 34 3 0 39004.209 .12725 306.063 .00403 4.473 34 4 0 39011.844 .12735 306.310 .00404 4.479 34 5 0 39019.485 .12746 306.553 .00404 4.486 34 6 0 39027.133 .12756 306.795 .00405 4.492 34 7 0 39034.786 .12767 307.038 .00405 4.499 34 8 0 39042.447 .12778 307.281 .00406 4.505 34 9 0 39050.113 .12788 307.525 .00406 4.512 34 10 0 39057.786 .12799 307.747 .00407 4.518 34 11 0 39065.465 .12809 308.013 .00407 4.525 34 12 0 39073.151 .12820 308.257 .00408 4.532 34 13 0 39080.842 .12830 308.502 .00408 4.538 34 14 0 39088-541 t .12841 308.747 .00409 4.545 34 15 0 39096.245 .12851 308.992 .00409 4.551 34 16 0 39103.956 .12862 309.233 .00410 4.558 34 17 0 39111.673 .12873 309.484 .00411 4.564 3418 0 39119.397 .12883 309.730 .00411 4.571 34 19 0 39127.127 .12894 309.977 .00412 4.578 34 20 0 39134.863 .12905 310.224 .00412 4.584 34 21 0 39142.606 .12915 310.471 .00413 4.591 34 22 0 39150.355 .12926 310.719 .00413 4.598 34 23 0 39158.111 .12936 310.966 .00414 4.604 34 24 0 39165.873 .12947 311.215 .00414 4.611 34 25 0 39173.641 .12958 311.463 .00415 4.618 34 26 0 39181.416 .12969 311.712 .00415 4.625 34 27 0 39189.197 .12979 311.961 .00416 4.631 34 28 0 39196.984 .12990 312.211 .00416 4.638 34 29 0 39204.778 .13001 312.460 .00417 4.645 34 30 0 39212.578 .13011 312.710 .00417 4.652 DELTA(LONGITUDE) - (IX)Q - (X)Q"3 + (E5)Q"5 ADD OR SUBTRACT FROM CENTRAL MERIDIAN IN SECONDS Q- .000001 DELTA(EASTING) - DIFFERENCE IN EASTING FROM FALSE EASTING (500,000) TABLE FOR USE WITH FOOTPOINT LATITUDE 4-12 LATITUDE (XII) DIFF.1" (XIII) (C5) 34 0 0 5591.929 .04019 3.053 .00154 34 1 0 5594.340 .04018 3.053 .00154 34 2 0 5596.751 .04017 3.054 .00154 34 3 0 5599.162 .04017 3.054 .00154 34 4 0 5601.572 .04016 3.054 .00154 34 5 0 5603.981 .04015 3.054 .00153 34 6 0 5606.390 .04014 3.054 .00153 34 7 0 5608.798 .04013 3.054 .00153 34 8 0 5611.206 .04013 3.054 .00153 34 9 0 5613.614 .04012 3.054 .00153 3410 0 5616.021 .04011 3.054 .00153 3411 0 5618.428 .04010 3.054 .00153 34 12 0 5620.834 .04009 3.054 .00153 3413 0 5623.239 .04009 3.055 .00153 34 14 0 5625.645 .04008 3.055 .00152 34 15 0 5628.049 .04007 3.055 .00152 3416 0 5230.453 .04006 3.055 .00152 34 17 0 5632.857 .04005 3.055 .00152 34 18 0 5635.260 .04005 3.055 .00152 3419 0 5637.663 .04004 3.055 .00152 34 20 0 5640.066 .04003 3.055 .00152 34 21 0 5642.467 .04002 3.055 .00152 34 22 0 5644.869 .04001 3.055 .00151 34 23 0 5647.270 .04001 3.055 .00151 34 24 0 5649.670 .04000 3.055 .00151 34 25 0 5652.070 .03999 3.055 .00151 34 26 0 5654.469 .03998 3.055 .00151 34 27 0 5656.868 .03997 3.056 .00151 34 28 0 5659.267 .03997 3.056 .00151 34 29 0 5661.665 .03996 3.056 .00151 34 30 0 5664.062 .03995 3.056 .00151 CONVERGENCE IN SECONDS - (XII)P + (Xlll)P"3 + (C5)P"5 P - .0001 DELTA(LONGITUDE) IN SECONDS FROM CENTRAL MERIDIAN DMA TM 8358.2 LATITUDE (XV) DIFF.1" (XVI) (F5) 34 0 0 21798.090 .2277 258.7 4.297 34 1 0 21811.750 .2278 258.9 4.304 34 2 0 21825.416 .2279 259.2 4.310 34 3 0 21839.087 .2279 259.5 4.317 34 4 0 21852.763 .2280 259.7 4.324 34 5 0 21866.445 .2281 260.0 4.330 34 6 0 21880.132 .2282 260.3 4.337 34 7 0 21893.825 .2283 260.5 4.343 34 8 0 21907.523 .2284 260.8 4.350 34 9 0 21921.226 .2285 261.1 4.357 34 10 0 21934.935 .2286 261.3 4.363 34 11 0 21948.649 .2287 261.6 4.370 34 12 0 21962.368 .2287 261.9 4.377 34 13 0 21976.093 .2288 262.1 4.383 34 14 0 21989.824 .2289 262.4 4.390 34 15 0 22003.559 .2290 262.7 4.397 34 16 0 22017.301 .2291 262.9 4.403 34 17 0 22031.047 .2292 263.2 4.410 34 18 0 22044.799 .2293 263.5 4.417 34 19 0 22058.557 .2294 263.7 4.424 34 20 0 22072.320 .2295 264.0 4.430 34 21 0 22086.088 .2296 264.3 4.437 34 22 0 22099.862 .2297 264.6 4.444 34 23 0 22113.641 .2297 264.8 4.451 34 24 0 22127.426 .2298 265.1 4.458 34 25 0 22141.216 .2299 265.4 4.464 34 26 0 22155.011 .2300 265.6 4.471 34 27 0 22168.813 .2301 265.9 4.478 34 28 0 22182.619 .2302 266.2 4.485 34 29 0 22196.431 .2303 266.5 4.492 34 30 0 22210.249 .2304 266.7 4.499 CONVERGENCE IN SECONDS - (XV)Q + (XVI)Q"3 + (F5)Q"5 Q « .000001 DELTA(EASTING)« DIFFERENCE IN EASTING FROM FALSE EASTING (500,000) TABLE FOR USE WITH FOOTPOINT LATITUDE 4-14 LATITUDE (XX) (XXI) 34 0 0 .00081149 .00000035 34 1 0 .00081117 .00000035 34 2 0 .00081085 .00000035 34 3 0 .00081053 .00000034 34 4 0 .00081021 .00000034 34 5 0 .00080989 .00000034 34 6 0 .00080957 .00000034 34 7 0 .00080925 .00000034 34 8 0 .00080893 .00000034 34 9 0 .00080861 .00000034 34 10 0 .00080829 .00000034 34 11 0 .00080797 .00000034 34 12 0 .00080765 .00000034 34 13 0 .00080733 .00000034 34 14 0 .00080701 .00000034 34 15 0 .00080668 .00000034 34 16 0 .00080636 .00000034 34 17 0 .00080604 .00000034 34 18 0 .00080572 .00000034 34 19 0 .00380540 .00000034 34 20 0 .00080508 .00000034 34 21 0 .00080476 .00000034 34 22 0 .00080444 .00000033 34 23 0 .00080411 .00000033 34 24 0 .00080379 .00000033 34 25 0 .00080347 .00000033 34 26 0 .00080315 .00000033 34 27 0 .00080283 .00000033 34 28 0 .00080250 .00000033 34 29 0 .00380218 .00000033 34 30 0 .00380186 .00000033 SCALE FACTOR - .9996( 1.0 + (XX)P"2 + (XXI)P"4) P - .0001 DELTA(LONGITUDE) IN SECONDS FROM CENTRAL MERIDIAN DMA TM 8358.2 LATITUDE (I) (XVIII) (XIX) 34 0 0 3762214.378 .0123310 .000026 34 1 0 3764062.392 .0123309 .000026 34 2 0 3765910.411 .0123309 .000026 34 3 0 3767758.435 .0123308 .000026 34 4 0 3769606.464 .0123308 .000026 34 5 0 3771454.499 .0123307 .000026 34 6 0 3773302.538 .0123307 .000026 34 7 0 3775150.582 .0123307 .000026 34 8 0 3776998.632 .0123306 .000026 34 9 0 3778846.686 .0123306 .000026 34 10 0 3780694.746 .0123305 .000026 34 11 0 3782542.810 .0123305 .000026 34 12 0 3784390.880 .0123304 .000026 34 13 0 3786238.955 .0123304 .000026 34 14 0 3788087.034 .0123303 .000026 34 15 0 3789935.119 .0123303 .000026 34 16 0 3781783.209 .0123303 .000026 34 17 0 3793631.304 .0123302 .000026 34 18 0 3795479.404 .0123302 .000026 34 19 0 3797327.509 .0123301 .000026 34 20 0 3799175.619 .0123301 .000026 34 21 0 3801023.734 .0123300 .000026 34 22 0 3802871.854 .0123300 .000026 34 23 0 3804719.979 .0123299 .000026 34 24 0 3806568.110 .0123299 .000026 34 25 0 3808416.245 .0123298 .000026 34 26 0 3810264.385 .0123298 .000026 34 27 0 3812112.531 .0123298 .000026 34 28 0 3813960.681 .0123297 .000026 34 29 0 3815808.837 .0123297 .000026 34 30 0 3817656.998 .0123296 .000026 SCALE FACTOR - .9996( 1.0 + (XVIII)Q"2 + (XIV)Q"4 ) Q « .000001 DELTA(EASTING) « DIFFERENCE IN EASTING FROM FALSE EASTING (500,000) TABLE FOR USE WITH FOOTPOINT LATITUDE 4-16 DMA TM 8358.2 For the UPS Grid: ELLIPSOID A 1/F NAME UNITS 6378137.00000 298.257223563 WGS-84 METERS LATITUDE R DIFF. 1" SCALE FACTOR DIFF 84 0 0 666727.704 -30.92101 .9967300 -2.53 84 1 0 664872.443 -30.92056 .9967148 -2.52 64 2 0 663017.210 -30.92011 .9966997 -2.51 84 3 0 661162.003 -30.91966 .9966846 -2.51 64 4 0 659306.824 -30.91921 .9966696 -2.50 84 5 0 657451.671 -30.91877 .9966546 -2.49 84 6 0 655596.545 -30.91832 .9966396 -2.49 84 7 0 653741.446 -30.91788 .9966247 -2.48 84 8 0 651886.373 -30.91743 .9966098 -2.47 84 9 0 650031.327 -30.91699 .9965950 -2.47 84 10 0 648176.308 -30.91655 .9965802 -2.46 84 11 0 646321.315 -30.91612 .9965654 -2.45 84 12 0 644466.348 -30.91568 .9965507 -2.44 84 13 0 642611.407 -30.91524 .9965361 -2.44 84 14 0 640756.492 -30.91481 .9965214 -2.43 84 15 0 638901.604 -30.91437 .9965069 -2.42 84 16 0 637046.742 -30.91394 .9964923 -2.42 84 17 0 635191.905 -30.91351 .9964778 -2.41 8418 0 633337.094 -30.91308 .9964634 -2.40 84 19 0 631482.310 -30.91265 .9964490 -2.39 84 20 0 629627.550 -30.91223 .9964346 -2.39 84 21 0 627772.817 -30.91180 .9964203 -2.38 84 22 0 625918.109 -30.91137 .9964060 -2.37 84 23 0 624063.427 -30.91095 .9963918 -2.37 84 24 0 622208.769 -30.91053 .9963776 -2.36 84 25 0 620354.138 -30.91011 .9963634 -2.35 84 26 0 618499.531 -30.90969 .9963493 -2.34 84 27 0 616644.950 -30.90927 .9963352 -2.34 84 28 0 614790.394 -30.90885 .9963212 -2.33 84 29 0 612935.862 -30.90844 .9963072 -2.32 84 30 0 611081.356 -30.90802 .9962933 -2.32 84 31 0 609226.875 -30.90761 .9962794 -2.31 84 32 0 607372.418 -30.90720 .9962655 -2.30 84 33 0 605517.986 -30.90679 .9962517 -2.30 84 34 0 603663.579 -30.90638 .9962380 -2.29 84 35 0 601809.196 -30.90597 .9962242 -2.28 84 36 0 599954.838 -30.90556 .9962105 -2.27 84 37 0 598100.504 -30.90516 .9961969 -2.27 8438 0 596246.195 -30.90475 .9961833 -2.26 84 39 0 594391.910 -30.90435 .9961697 -2.25 4-17 DMA TM 8358.2 84 40 0 592537.649 -30.90395 .9961562 -2.25 84 41 0 590683.412 -30.90355 .9961427 -2.24 84 42 0 588829.199 -30.90315 .9961293 -2.23 84 43 0 586975.010 -30.90275 .9961159 -2.22 84 44 0 585120.845 -30.90235 .9961026 -2.22 84 45 0 583266.704 -30.90196 .9960893 -2.21 84 46 0 581412.586 -30.90156 .9960760 -2.20 84 47 0 579558.493 -30.90117 .9960628 -2.20 84 48 0 577704.422 -30.90078 .9960496 -2.19 84 49 0 575850.376 -30.90039 .9960365 -2.18 84 50 0 573996.352 -30.90000 .9960234 -2.18 84 51 0 572142.352 -30.89961 .9960103 -2.17 84 52 0 570288.376 -30.89922 .9959973 -2.16 84 53 0 568434.422 -30.89884 .9959844 -2.15 84 54 0 566580.492 -30.89845 .9959714 -2.15 84 55 0 564726.585 -30.89807 .9959586 -2.14 84 56 0 562872.700 -30.89769 .9959457 -2.13 84 57 0 561018.839 -30.89731 .9959329 -2.13 84 58 0 559165.000 -30.89693 .9959202 -2.12 84 59 0 557311.185 -30.89655 .9959075 -2.11 85 0 0 555457.391 -30.89618 .9958948 -2.10 NORTHING - 2,000,000 - (R)COS(LONGITUDE) NORTH ZONE NORTHING - 2.000.000 + (R)COS(LONGITUDE) SOUTH ZONE EASTING -= 2,000,000 + (R)SIN(LONGITUDE) BOTH ZONES 4-18 DMA TM 8358.2 CHAPTFR5 OTHER COORDINATE CONVERSIONS AND TRANSFORMATIONS 5-1 GRID CONVERSION BETWEEN ZONES WITHIN THE UTM SYSTEM AND GRID CONVERSION BETWEEN THE UTM AND UPS SYSTEMS. 5-1.1 To accurately convert grid coordinates between zones or systems, the grid coordinates are first converted to geographic coordinates in the known zone or system. Once the grid coordinates have been converted to geographic coordinates, they are then converted to grid coordinates in the new zone or system. These conversions utilize the formulas found in Chapters 2 and 3 of this manual. 5-1.2 Although direct grid to grid conversions exist, which can be performed with the use of tables, this method is not as accurate as the conversion method discussed above. Other apparent direct grid to grid conversions exist, however they involve the intermediate conversion to geographic coordinates. 5-2 MILITARY GRID REFERENCE SYSTEM. 5-2.1 The U.S. Military Grid Reference System (MGRS) is designed for use with the UTM and UPS grids. An MGRS position location is an alpha-numeric version of a numerical UTM or UPS grid coordinate. 5-2.2 Chapter 3 and Appendix B of DMA TM 8358.1 describes and shows the method for finding the 100,000-meter square identifications. Software to convert between UTM or UPS coordinates and MGRS positions is listed as Annex A of this manual and can be obtained from DMA HTC/PRT. 5-3 WORLD GEOGRAPHIC REFERENCE SYSTEM. 5-3.1 The World Geographic Reference System (GEOREF) is a system used for position reporting. It is not a military grid, but rather an area-designation method. Positions are expressed in a form suitable for reporting and plotting on any maporchart graduated in latitude and longitude (with Greenwich as prime meridian) regardless of map projection. 5-3.2 Section 5-4 of DMA TM 8358.1 describes and illustrates the World Geographic Reference System. Software to convert between UTM or UPS coordinates and GEOREF positions is provided as Annex A-4 of this manual. 5-4 DATUM TO DATUM COORDINATE TRANSFORMATIONS. 5-4.1 Molodenskiy Coordinate Transformation Formulas. 5-4.1.1 Definition Of Terms. Terms previously defined in Chapters 2 and 3 are not redefined below. , X. = geodetic coordinates (input ellipsoid) H = N + h = the distance of a point from the ellipsoid center measured along the ellipsoidal normal through the point 5-1 DMA TM 8358.2 N ■= geoid-ellipsokj separation - the distance of the geoid above (+ N) or below (- N) the ellipsoid h = distance of a point from the geoid = the elevation above or below mean sea level A<$>, AX, AH = corrections to transform the geodetic coordinates from the input datum to the output datum (output minus input) AX, AY, AZ = shifts between ellipsoid centers of the input datum and output datum (output minus input) Aa, Af = differences between the parameters of the input ellipsoid and the output ellipsoid (output minus input) p - radius of curvature in the meridian v - radius of curvature in the prime meridian 5-4.1.2 Standard Molnrtenskiy Formulas The Standard Molodenskiy formulas are as follows: A$" = {-AXsirn)>cosX • AYsirxfisinX + AZcos + Aa(FtNe2sin<|>cos)/a + Af[RM(a/b) + RN(b/a)] sir*t>cos} / [(Rm + H)sin 1*] AX" - [-AXsinX + AYcosX] / [(Rn + H)cossin 1"] AH - AXcos4>cosX + AYcosj>sinX + AZsirty - Aa(a/RN) + Af (b/a) RNSin24> 5-4.1.3 Abridged Molodenskiy Formulas. The Abridged Molodenskiy formulas are as follows: A4>" > [-AXsirvt^sX - AYsir^sinX + AZcos+(aAf+ fAa)sin24>] / [RMSin 1*] AX" « [-AXsinX + AYcosX] / [RNCos<|>sin 1"] AH « AXcosfrcosX + AYcos^sinX + AZsirvf) + (aAf + f Aa)sin24» - Aa 5-4.2 MuHiple Rftgrassion Fqiiationfi. Multiple regression equations are developed from polynomial equations to the nth degree to fit predetermined accuracy requirements. The terms that are included in the multiple regression equations are a function of the desired accuracy of the solution. In developing the solution, the terms of first degree to nth degree are tested and in the process of deriving the solution to fit the accuracy requirements, certain terms drop out. Because of this, there is no generalized equation to present as a gukfline. However, as an example, a datum shift from European Datum to WGS1972 is 5-2 DMA TM 8358.2 presented to illustrate multiple regressions. This shift is not to be construed as the most up-to-date transformation for the area under consideration. For a more in-depth discussion of The Multiple Regression Equations, see DMA TR 8350.2, Department of Defense World Geodetic System 1984, Its Definition and Relationships with Local Geodetic Systems. A" = - 2.5830 + 2.0782X + 0.6631 Y + 0.6144X2 + 1.0456XY - 0.7752Y2 + 0.8414X3 + 0.1058XY2 - 14.4049X3Y + 4.4291 XY3 + 0.0166Y4 + 59.9408X5Y - 4.2792X3Y3 - 3.9642XY5 + 0.7818Y6 - 93.5475X7Y - 4.5053X7Y2 + 48.8445X9Y + 11.0197X5Y7 + 4.5980X9Y4 - 5.4256X5Y9 AX"= - 4.8255 - 1.8094X + 1.8479Y - 2.0174X2 + 1.1912XY - 0.3288X3 + 3.3287X4 - 4.1036X3Y + 1.6161XY3 + 0.6259Y4 + 1.5379XY4 - 2.5285XY5 -1.1917X8 + 4.8445X7Y - 10.0979X6Y2 - 0.7021 Y9 + 10.6185XV + 6.2369*^ - 9.1252X6Y8 AHm= 36.604 - 28.206X -18.351 Y + 9.525X2 + 2.107Y2 + 11.094X3 - 0.684X2Y + 25.268Y3 + 86.856X2Y2 + 6.040XY3 - 34.469Y5 - 22.520X5Y - 77.583X4Y2 - 125.318XV - 18.485XY6 + 57.003X5Y3 + 72.140X2Y6 - 24.950X8Y + 16.896Y9 + 37.821 X*Y2 + 41.265X4Y9 Where: X = 3$ - 2.7969898 Y = 3k - 0.5248325 , X = ED 50 latitude and longitude (in radians) AHm = ED 50 height (in meters) 5-4.3 ACCURACIES 5-4.3.1 The Multiple Regression Equations will provide a more accurate fit than either the Standard or Abridged Molodenskiy Transformations within its specific area of application. Outside of the specific area of application, the accuracies deteriorate rapidly. 5-3 DMA TM 8358.2 5-4.3.2 Within a small area, such as a degree square, locally determined Molodenskiy Transformation constants will generally provide accuracies commensurate with the multiple regression. The simplicity and versatility of the Molodenskiy Transformation are also advantageous. 5-4.3.3 As long as the accuracy of the positions determined by satellite point positioning is in the 2-meter range, the Standard Molodenskiy Transformation will not give significantly more accurate results than the Abridged Molodenskiy Transformation. 5-4.3.4 When available, the Multiple Regression Equations will be the preferred method of datum transformation. This is followed by the Standard Molodenskiy Transformation and then the Abridged Molodenskiy Transformation. 5-4 DMA TM 8358.2 ANNEXES A and B These annexes list the software and documentation available for the transformation of geographic coordinates to/from grid coordinates. These products can be obtained by contacting DMA HTC(PRT). ANNEX A SOFTWARE SUPPORT FOR UNIVERSAL GRIDS Section A-1 Software and Documentation for Geographic Coordinate Transformations to/from Grid Coordinates A-1.1 UTMGrid A-1.2 UPS Grid A-2 Software and Documentation for Datum to Datum Coordinate Transformations A-2.1 Molodenskiy A-2.2 Abridged Molodenskiy A-3 Software and Documentation for the UTM and the UPS Grid Coordinate Transformation to/from the MGRS A-4 Software and Documentation for Geographic Coordinate Transformation to/from GEOREF Coordinates A-5 Software and Documentation for Grid Coefficients and Latitude Functions ANNEX B SOFTWARE SUPPORT FOR NON-UNIVERSAL GRIDS Section B-1 Software and Documentation for the Transverse Mercator (TM) Grid B-2 Software and Documentation for the Mercator Grid B-3 Software and Documentation for the Lambert Conical Orthomorphic Grid B-4 Software and Documentation for the Madagascar Gauss Laborde Grid B-5 Software and Documentation for the Rectified Skewed Orthomorphic Grid B-6 Software and Documentation for the New Zealand Map Grid B-7 Software and Documentation for the Guam Azimuthal Equidistant Grid A-1 B-1