The Education Master 1994 (4th Edition)

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1 GEOD USER'S MANUAL version 1.1 (c) Copyright 1992-1993 H. S. Aurand All rights reserved 4 June 1993 GETTING STARTED If you are viewing this document on the screen of a personal computer, you have already decompressed GEODLHA.EXE to obtain the GEOD.DOC which contains this manual. You probably have done this by executing the installation file, INSTGEOD.EXE. In any event you should have the following files in a single directory. INSTGEOD.EXE Installation file GEODLHA.EXE Self-extracting compressed file containing: GEOD.EXE GEOD.MSG GEOD.MIX GEOD.DOC User's Manual (this file) README.DOC Shareware information VENDOR.DOC Vendor Information FILE_ID.DIZ BBS Information If these files are not present in a desired directory, you should install them in such a directory as follows: Place the GEOD distribution disk containing INSTGEOD.EXE and GEODLHA.EXE in a floppy drive. If the letter for that drive is n, type n:instgeod <Enter>. Answer the questions (there may be only one), and you will be at the initial GEOD screen. Please read the information there. Then press any key, and you will be at a screen that permits input and execution as described in this document. At any menu the different help screens and the background and explanation screens that make up this document can be viewed by pressing F1 or F8 respectively. If want to save disk space you may delete all but GEOD.EXE, GEOD.MSG and GEOD.MIX, but be sure to have INSTGEOD.EXE and GEODLHA.EXE saved on a backup floppy disk. 2 GEOD - BACKGROUND AND EXPLANATION GEOD is a program to solve shortest-distance (geodetic) lines on an ellipsoidal earth. It also compares these results with three different spherical earth solutions. Standard ellipsoids stored and used in the program are the Bessel- Encke(1852), International or Hayforth (1910) and Clarke (1866). Other ellipsoids may be entered and used. Also several different ways are provided to calculate the earth's mean radius for these ellipsoids. Different units may be specified for length and are changed automatically throughout the program. Those available are meters, kilometers, feet, statute miles and nautical miles. Nautical miles have no standard definition. A value of 10800 feet is used in this program. There are seven variables that are defined by a line between two points on the earth's surface - the latitude and longitude of one point, the latitude and longitude of the other, the distance between them and the azimuths of the two points from each other. Four of these variables must be specified in order to solve for the other three. Of the several cases that ensue, the two most interesting, and the ones solved by this program, are the vector, wherein a point and the distance and azimuth to the other are specified, and the inverse, wherein the two points are specified and the azimuths and distance between them determined. In addition to single solutions for each of these two cases, there is another mode in which multiple solutions are provided for the vector case only. In this mode the distance or the azimuth can be varied between two values in specified increments. Output to a printer can also be provided. Comparisons with three different spherical solutions are made in the single solution case. In the first, the geodetic latitudes are used as they are specified. This is called the geographical solution. In the other two, the specified geodetic latitudes are altered before calculation. After calculation the inverse alteration is performed on a calculated latitude. These two cases are called the geocentric and reduced respectively. In the geocentric spherical solution a latitude is changed to one specified by a line running from the center of the meridional ellipse to the point. In the reduced spherical solution a latitude is altered by reducing it in accordance with a geometrical construction used in calculating the meridional ellipse. 3 It is beyond the scope of this brief note to explain and justify these methods of calculation, but the user can run the program and see that by altering the latitudes in the geocentric and reduced spherical calculations a closer approximation to the geodetic line can often be achieved. To understand this and other aspects of geodetic calculations, the user is referred to a general text on geodesy such as "Geodesy", G. Bomford, Oxford, 1971. The author is not schooled in geodesy, but is an engineer. On occasion I have had the need to calculate geodetic lines. In reviewing the literature, I found that the several formulas were predominately special purpose approximations that minimized computational effort, but could be used only for limited ranges of distance. Before the advent of computers this is understandable. However, I also noticed references to Helmert's method. These references stated that his solutions could be made as exact as desired for any distance, albeit at the expense of computational effort. I found this to be so in his book, "Die Mathematischen und Physikalischen Theorieen der Höheren Geodäsie", F. R. Helmert, B. G. Teubner Verlagsgesellschaft, Leipzig, 1880, republished by Minerva GMBH Frankfurt/Main, 1962. This book contains an elegant solution for the vector case which relies on strongly convergent series and is ideally suited for use on small computers. Helmert's solution for the inverse case not only has this feature, but uses a strongly convergent iteration. The algorithms looks like they were developed with computer solutions in mind! Helmert's methods are used in this program. Helmert also addresses the case where the two points are antipodal or near antipodal. The results are counter-intuitive. In the vector case the locus of points having a constant geodetic distance is not a continuous curve around the antipodal point, but has four well-defined cusps. This can be clearly seen by using the multiple solution case varying the azimuth from 0° to 90° with a value of eta of .5. In the inverse case there are two solutions for the geodetic line for the near antipodal case. In this program only the shorter is calculated and displayed. There is another counter-intuitive aspect of the geodetic line on an ellipsoid that should be mentioned. In general the geodetic line is not contained in a plane containing the vertical at one end point and the other end point. Such a plane does not even contain the vertical at the other end point (as does a great circle on a sphere). In fact the geodetic line is not contained in a plane at all. Beware, there are other counter-intuitive aspects of geodetic lines. For further explanation consult a geodesy text such as Bomford's. In this program there are two levels of accuracy that can be selected. The first, labeled Helmert, follows Helmert's book closely except that fifteen significant figures instead of seven are used. The second, labeled Helmert extended, in addition to this greater number of significant figures, adds to the number of terms in his series, 4 decreases the epsilon at which iteration terminates in the inverse case and adds an additional iteration to this case in which Newton's Method is used to adjust for greater accuracy. This program has been tested for accuracy in two ways. First it has been checked against the examples given in Helmert's book. Second it has been checked by running the vector and inverse cases sequentially and observing the change from the initial values. In this particular attention has been paid to singular cases such as polar and antipodal ones. Nevertheless, bugs and mistakes are bound to slip through. This version, 1.1, is a considerable improvement in this regard to version 1.0 which saw only limited release. The first registered user who reports a non-trivial bug or error will be provided with a free upgrade. This program is written in Microsoft C, version 6.0. Double precision is used for all calculations. These have been encapsulated in a few functions separate from the user interface. Copies of the source code can be licensed along with a mathematical explanation of the methods used to extend the accuracy of the calculations beyond that given by Helmert. For information please contact the author at the address listed below. Your comments, critique and suggestions would be appreciated. Henry S.Aurand 734 Camino Catalina Solana Beach, CA 92075 Telephone: 619-481-0781 5 EXPLANATION OF MENUS MAIN MENU Provides access to all other menus. Use the highlighted and capitalized letter key or move the pointer using the up and down arrow keys to the selection of your choice, then press ENTER. The selections are: Quit - Exits the program Spheroid - Permits the selection of ellipsoids Radius of sphere - Permits choice of several means for determining the radius of an approximating sphere used in calculating spherical solutions Units - Permits the selection of the units used for the ellipsoid axes, distance and spherical radius Accuracy - Permits the selection of Helmert or Helmert extended accuracy along with selection of the epsilon for error termination and maximum number of iterations sIngle execute - Goes to Single/Input Output Menu where data can be entered and a solution obtained for either vector or inverse cases muLtiple execute - Goes to Multiple Input/Output Menu where data can be entered and solutions obtained for the vector case. Azimuth or distance can be varied in steps. Output is to the screen and may be sent to a printer Zero variables - Zeros all seven input/output variables. Does not affect ellipsoid parameters View descriptive information - Displays several pages of background 6 Keys which operate at all menus are as follows: F1 and H - Help F2 - Go to single execution input/output menu F3 - Go to multiple execution input/output menu F4 - Go to spheroid menu F5 - Go to radius of approximating sphere menu F6 - Go to units menu F7 and Z - Zeros variables. Zeros all seven input/output variables. Does not affect ellipsoid parameters. Z is only effective in the two Input/Output menus. F8 - View descriptive information. Displays five pages of background F9 - Go to Accuracy menu 7 SPHEROID MENU Permits the selection of different spheroids. Use the highlighted and capitalized letter key or move the pointer using the up and down arrow keys to the selection of your choice, then press ENTER. Quit - Quits program Main - Return to Main Menu Radius of sphere - Goes to Radius of Sphere Menu Units Menu - Goes to Units Menu Single Execute - Goes to Single/Input Output Menu where data can be entered and a solution obtained for either vector or inverse cases. muLtiple execute - Goes to Multiple Input/Output Menu where data can be entered and solutions obtained for the vector case. Azimuth or distance can be varied in steps. Output is to the screen and may be sent to a printer Bessel - Changes to the Bessel standard ellipsoid haYforth - Changes to the Hayforth standard ellipsoid Clarke - Changes the Clarke standard ellipsoid Other - Permits direct entry of semimajor and semiminor axes of any ellipsoid 8 RADIUS OF SPHERE MENU Permits the selection of the radius for the approximating sphere. Use the highlighted and capitalized letter key or move the pointer using the up and down arrow keys to the selection of your choice, then press ENTER. Quit - Quits program Main menu - Return to Main Menu Spheroid menu - Goes to Spheroid Menu Units Menu - Goes to Units Menu Single execute - Goes to Single/Input Output Menu where data can be entered and a solution obtained for either vector or inverse cases muLtiple execute - Goes to the Multiple Input/Output Menu where data can be entered and solutions obtained for the vector case. Azimuth or distance can be varied in steps. Output is to the screen and may be sent to a printer Geometric mean - Sets the radius of the approximating sphere equal to the geometric mean of the axes of the ellipsoid used Equal volume - Sets the radius of the approximating sphere equal to that of a sphere having the same volume as the ellipsoid used Equal area - Sets the radius of the approximating sphere equal to that of a sphere having the same surface area as the ellipsoid used Other proportional - Sets the radius of the approximating sphere equal to c * a0 ^ rho * b0 ^ (1-rho). c and rho must be entered. a0 and b0 are the semimajor and semiminor axes respectively of the ellipsoid used Enter - The radius of the approximating sphere is entered by the user 9 UNITS MENU Permits the selection of units for all distance variables and parameters. Use the highlighted and capitalized letter key or move the pointer using the up and down arrow keys to the selection of your choice, then press ENTER. Quit - Quits program Main menu - Return to Main Menu sPheroid - Goes to Spheroid Menu Radius of sphere - Goes to Radius of Sphere Menu Single Execute - Goes to Single/Input Output Menu where data can be entered and a solution obtained for either vector or inverse cases. muLtiple execute - Goes to Multiple Input/Output Menu where data can be entered and solutions obtained for the vector case. Azimuth or distance can be varied in steps. Output is to the screen and may be sent to a printer When one of the following is selected, the units for all variables and parameters are changed to the value selected. Kilometers Nautical miles Statute miles mEters Feet 10 SINGLE INPUT/OUTPUT MENU Permits the entry of input and displays output for a single solution. The display indicates whether the calculation is vector or inverse as well as the spheroid, the units, the radius approximation and the accuracy used. In the inverse case the number of iterations required is also shown and if extended accuracy is used the number of iterations for this purpose is also shown. Use the highlighted and capitalized letter key or move the pointer using the up and down arrow keys to the selection of your choice, then press ENTER. When you make a selection requiring input of an angle, you must press ENTER after each entry to the "deg?", "min?" and "sec?" prompts. For latitude an "n" or "s" and for longitude a "w" or "s" are required to complete the entry. If out of range entries are made, the display so indicates. To return to inputting press any key. Quit - Quit the program Main menu - Return to Main Menu mUltiple execute - Goes to MULTIPLE INPUT/OUTPUT MENU Type of calculation toggle - Toggles between vector and inverse calculations Go - Performs calculation after input data has been entered. The results are displayed alongside the selections and at the bottom of the screen. The computed results of the three spherical approximations are compared with the ellipsoidal results. See the Main Menu selection, "View descriptive information" or press F8, to see an explanation of these approximations. In the following you are permitted to enter only the variables needed. 1 Enter Lat1 - Enter the latitude of the first point 2 Enter Long1 - Enter the longitude of the first point 3 Enter Lat2 - Enter the latitude of the second point 4 Enter Long2 - Enter the longitude of the second point 5 Enter Az1,2 - Enter the azimuth, clockwise from north, of the second point from the first 6 Az2,1 - Azimuth, clockwise from north, of the first point from the second. Never can be entered, results of calculation only are displayed. 7 Enter distance, s - Enter the distance between two points in the units shown 11 MULTIPLE INPUT/OUTPUT MENU Permits the entry of input and displays output for a range of solutions. Use the highlighted and capitalized letter key or move the pointer using the up and down arrow keys to the selection of your choice, then press ENTER. When you make a selection requiring input of an angle, you must press ENTER after each entry to the "deg?", "min?" and "sec?" prompt. For latitude an "n" or "s" and for longitude a "w" or "s" only are required to complete the entry. If out of range entries are made, the display so indicates. To return to inputting press any key. Quit - Quit the program Main menu - Return to Main Menu Units - Goes to Units Menu Single execute - Goes to SINGLE INPUT/OUTPUT MENU Variable toggle - Successively selects Az1,2, distance and distance parameter, eta, to be varied Go - Calculates vector case in accordance with beginning, ending and difference variables entered. Outputs the results below these and scrolls the display up as required. Outputting can be stopped for viewing by pressing any key except ESC. Pressing any key except ESC will resume outputting. Pressing ESC will return to the menu 12 In the following you are permitted to enter only the variables needed. If selected, 1 Enter Lat1 - Enter the latitude of the first point 2 Enter Long1 - Enter the longitude of the first point 3 Lat2 - Latitude of the second point 4 Long2 - Longitude of the second point 5 Enter Az1,2 - Enter the azimuth, clockwise from north, of the second point from the first. No entry permitted if you are varying this azimuth 6 Az2,1 - Azimuth, clockwise from north, of the first point from the second. Cannot be entered, results of calculation only are displayed 7 Enter distance, s - Enter the distance between two points in the units shown. s and eta cannot be entered if you are varying either. 8 Enter eta - Enter this dimensionless parameter, called offset ratio, that sets distance. Used primarily for antipodal or near antipodal cases. When eta is 0, the distance, s, is set at pi times the semiminor axis, b0. When 1, s is set at pi times the semimajor axis, a0. When .5, s is set to the average of these two values. This is just slightly smaller than the meridional distance from equator to equator. The correct value for this distance can be determined using the inverse case of the single input/output case 9 Print toggle - When "on" outputs to standard printer in the same format as the output to the screen. In the following, input is made to a variable determined by the Variable toggle set above. The three types are: Az1,2, distance and offset ratio. A Enter starting (variable name) B Enter ending (variable name) C Enter delta (variable name) Pressing F7 or 'Z' zeros the seven variables, but not the ellipsoid parameters 13 ACCURACY MENU Permits the selection of variables that affect the accuracy of calculations. Use the highlighted and capitalized letter key or move the pointer using the up and down arrow keys to the selection of your choice, then press ENTER. Quit - Quits program Main menu - Return to Main Menu sPheroid - Goes to Spheroid Menu Radius of sphere - Goes to Radius of Sphere Menu Units menu - Goes to Units Menu sIngle Execute - Goes to Single/Input Output Menu where data can be entered and a solution obtained for either vector or inverse cases. muLtiple execute - Goes to Multiple Input/Output Menu where data can be entered and solutions obtained for the vector case. Azimuth or distance can be varied in steps. Output is to the screen and may be sent to a printer Accuracy toggle - Toggles between Helmert's method and extended accuracy achieved by extending his series and adding a Newton's Method approximation in the inverse case. Resets the epsilon used in terminating iterations in the inverse case. Uses 1.0e-9 for Helmert and 1.0e-15 for Helmert extended accuracies. enter Epsilon - Permits the user to select the epsilon used in terminating iterations. Note that this value is overwritten by the Accuracy toggle. enter max inVerse iterations - Permits the user to select the maximum number of iterations used in the inverse case. Default value is 100 enter max inverse iterations for Newton's method - Permits the user to select the maximum number of iterations used in the inverse case when Helmert extended accuracy is used. Default value is 100.