Global Navigation: Prototype Moving-Map Navigation System

A collection of separate navigation functions written in C, which were used as the basis for the moving map navigator implemented as the Java applet on this website [see contents list on the left]. [PDF]

This Moving Map was built as a small demonstrator for the EBS Nomad Project. The function MapBox() creates a square map window centred on the aircraft. Within this map window ShowNode() displays solid circles. These represent way points such as geographical features or cities. The scale of the moving map can be varied over a wide range. When the scale covers a very wide area, the function NodeTilt() is called to transform the solid circles into ellipses. This mimics the effect of their angl­ing due to the curvature of the Earth.

The way-points move on the map as the aircraft travels. The names and coordin­ates of all the way points are kept in a file called WayPts.DAT. After calling MapIni() to initialise the map display, main() cycles endlessly through WayPts.DAT calling RandB() to compute (using full spherical geometry) the distance and bearing of each way point in turn from the aircraft. These are then converted by RBtoXY() to logical screen co-ordinates. Its circle (or ellipse) is then re-displayed in the map win­dow. It also calls ShowData() which displays the aircraft's current co-ordinates, heading, height and speed. To do this it calls RadKm() and RadDeg() to convert the Earth radians used in the computations to kilometres and degrees for display. De­tails about the next way-point are also shown.

This test-bed program flies the aircraft at 1000km/hr from Stansted to Ringway. Its motion is effected by GndTrk(). This contains a radial intercept function which brings the aircraft on to Ringway's approach radial from its take-off heading at Stansted.

From this test-bed project, I developed a waypoint radial capture function to be used in navigation applications.

The MapWin.C File 07 JUNE 1991

#include <graph.h>
#include <math.h>
#include <malloc.h>

/* Define ERESCOLOR raster's true aspect ratio on an IBM 8513 
VGA screen: */ #define ASPECTRATIO .6834532737

/* Define the number of kilometres equivalent to one Great 
Circle radian: */ #define KMPERRADIAN 6366.197723857773

/* Define the number of radians/second equivalent to one 
kilometre/hour */ #define KMH_RADS .4363323129861111E-7

/* Define the number of degrees/minute equivalent to one 
radian/second: */ #define RADS_DEGM 3437.74677

#define X_DIAM 100 //Diameter of scale square on map in Xpixels
#define X_SQUR 200 //Diameter of scale square on map in Xpixels

extern char *RadDeg(double c);


extern double           //All measured in radians
  ObsLat, ObsLng,       //Observer's lat/long
  DstLat, DstLng,       //Destination lat/long
  ObsHdg, ObsVel,       //Observer's direction and velocity

  Range,     //Distance of feature (eg city) from Observer
  Bearing,   //Bearing of the feature as viewed by Observer
  ObsRot,    //Observer's rate of rotation (radians/sec)
  ReqHdg,    //Commanded heading to bring vehicle on to Radial
  Radial;    //Destination radial on which observer must travel
Set up an area of memory to hold:

If compiling with CL, a static array char far Graph[2156] can be used. However, QC does not support far static data, so we must use malloc() to allocate an area of memory dynamically.

  far *GraphBuf,  //Pointer to `city blob' image
  far *Letters;   //Pointer to letter images

The MapBox() Function

Display the basic map screen on the currently-active page.
MapBox() {
  _settextcolor(6); _settextposition(1,1);
  _moveto(-X_DIAM, Y_SQUR + 17);
  _lineto( X_DIAM, Y_SQUR + 17);
  _settextposition(24, 53); _outtext("100km");
  _settextposition(4,1); _outtext("NEXT WAY POINT");
  _settextposition(5,1); _outtext("LATITUDE");
  _settextposition(6,1); _outtext("LONGITUDE");
  _settextposition(7,1); _outtext("DISTANCE TO GO       km");
  _settextposition(8,1); _outtext("INBOUND RADIAL");
  _settextposition(10,1); _outtext("VEHICLE");
  _settextposition(11,1); _outtext("LATITUDE");
  _settextposition(12,1); _outtext("LONGITUDE");
  _settextposition(13,1); _outtext("HEADING");
  _settextposition(14,1); _outtext("SPEED                km/hr");
  _settextposition(16,1); _outtext("REQ'D HEADING");
  _settextposition(17,1); _outtext("RATE OF TURN         ø/min");

The MapIni() Function

Set up the graphics images necessary for displaying annotated map features (such as cities) within the map window:

MapIni() {
  int a, ox, oy;              //Logical origin of Map Window
  char far *Letter;
  struct videoconfig config;  //Defines CGA, EGA, VGA etc.
  _setvideomode(_ERESCOLOR);  //Mode to 16-colour 640 x 350
  _setactivepage(1);          //Set Page 1 active while
  _setvisualpage(0);          //displaying Page 0

  /* Allocate 2156 bytes of buffer and put its start address 
     in the far char pointer 'GraphBuf'. 76 bytes are for the 
     city blob, the rest is for the captured text characters 
     at 40 bytes per character. */

  GraphBuf = (char far *)malloc((unsigned int) 2156);
  Letters = GraphBuf + 76;

  /* Form the blob to represent a city and store its image in 
     the graphics image buffer pointed to by `graphbuf' this 
     image occupies first 76 bytes of the buffer */ 

  _moveto(4,0); _lineto( 8,0);
  _moveto(2,1); _lineto(10,1);
  _moveto(1,2); _lineto(11,2);
  _moveto(0,3); _lineto(12,3);
  _moveto(0,4); _lineto(12,4);
  _moveto(0,5); _lineto(12,5);
  _moveto(1,6); _lineto(11,6);
  _moveto(2,7); _lineto(10,7);
  _moveto(4,8); _lineto( 8,8);

  /* Display the alphabet on the screen, then capture each 
     letter as a graphics image and store it in the graphics 
     buffer from byte 76 onwards. The image of each letter 
     occupies 40 bytes of buffer */

  for(a = 0, Letter = Letters; a < 416; a += 8, Letter += 40)
    _getimage(0 + a, 2, 7 + a, 10, Letter);

  /* Set the logical co-ordinates origin to the centre of the 
     map display window. (Effective for both video pages.) */ 

  ox = config.numxpixels / 2 + 115;
  oy = config.numypixels / 2;
  _setlogorg(ox, oy);

  // Set up the main screen format on both video pages 0 and 1
  MapBox(); _setactivepage(0); _setvisualpage(1); MapBox();

  /* Limit graphics area to window. Effective on BOTH video 
     pages. */
    ox - X_SQUR, oy - Y_SQUR + 1, ox + X_SQUR, oy + Y_SQUR

The RBtoXY() Function

Compute X,Y screen co-ordinates from the range & bearing in radians. Assumes logical co-ordinates are at centre of 640 by 350 pixel VGA screen. If scale <= 1 nautical mile per pixel, curvature of screen ÷ that of the Earth so the actual range on the Earth's surface is used to compute the x,y screen co-ordinates. If scale > 1 nautical mile per pixel the surface range is projected onto the Earth's tangential plane at the observer point (centre-screen) before x,y are computed. Returns zero co-ordinates if the 'observed' is out of screen range.

int *RBtoXY(float s) {
  static int a[2];              //array for screen co-ordinates
  double x, y, z,
    d = Range * KMPERRADIAN,    //in km (20,000/pi km/radian)
    b = Bearing - ObsHdg;       //bearing in radians + heading

  // If scale > 2km/pixel, project range onto tangential plane.
  if(s > 2) d *= sin(Range);

  if(s > 0) d /= (z = s);           //Re-scale from km to pixels
  x = d * sin(b);                   //screen x-pixel
  y = d * cos(b);                   //screen y-pixel
  if(x > 0) x += .5; else x -= .5;  //Do appropriate negative
  if(y > 0) y += .5; else y -= .5;  //or positive rounding.
  if(x > 185 || x < -185 ||         //safe screen x-range limit
    y > 185 || y < -185) {          //safe screen y-range limit
    x = 1000; y = 1000;             //set zero co-ordinates
  }                                 //if target is off screen

  /* Store the x-displacement as is, and the y-displacement
     scaled to the MRES screen aspect ratio. Then return the 
     pointer to the co-ordinates array. */

  *a = x; *(a + 1) = -y * ASPECTRATIO; return(a);

The ShowNode() Function

Display a city or way point on the Earth's surface as a small annotated circle or ellipse.

ShowNode (
  int *q,     //Pointer to city's SCREEN co-ordinates array.
  char *n     //Pointer to city's name string.
) {
  int x = *q, y = *(q + 1), //x-y co-ordinates of city
  i,                        //Letter Nº within City Name
  l,                        //Nº of characters in annotation
  m;                        //repeat of l

  _putimage(x-6, y-4, GraphBuf, _GPSET);  //Display the city
  for(l = 0; *n != ' '; l++, n++);  //Find length of its name
  n -= l;

  m = l;
  switch(*(n + 15)) {
    case 'L': x -= (8 + (m <<= 3)); break;    //annotate left
    case 'R': x += 10; break;                 //annotate right
    case 'A': x -= (m <<= 2); y -= 13; break; //annotate above
    case 'B': x -= (m <<= 2); y += 13;        //annotate below

  /* Display the name of the city by displaying in sequence at 
     the city's screen co-ordinates the letters of its name as 
     graphics images of the normal text letters . */

  for(i = 0; i < l; i++, x += 8)  //For each letter of the name
  _putimage(x, y - 4, Letters + (*(n + i) - 39) * 40, _GPSET);

Moving map project: annotated explanation of the _putimage() command.

The RadKm() Function

Converts a 'double' in Earth-radians or Earth-radians/second to a 5-byte string showing km or km/hr respectively:

char *RadKm(double d, int e) {
  int a, b, c, latch = 0, x = ' ';
  char *s = "      ", *t = s, *o = "OVFLOW";
  if(e == 0) d *= KMPERRADIAN;
  else if (e == 1) d /= KMH_RADS;
  else if (e == 2) d *= RADS_DEGM;
  if(d > 32767 || d < -32767) s = o;
  else {
    b = d;
    if(b < 0) { b = -b; x = '-'; }
    for(a = 10000; a > 1; a /= 10, t++) {
      for(c = 0; b >= a; b -= a, c++);
      if(!latch) {          //If latch not yet set,
        if(c == 0)          //if digit value is zero
          *t = ' ';         //store a leading space
        else {              //else if first non-zero digit
          latch = 1;        //set the latch
          *t++ = x;         //store the sign
          *t = c + 48;      //store character, including zeros.
      } else *t = c + 48;   //store character, including zeros.
    if(!latch) *t++ = x;    //store the sign
    *t = b + 48;

The ShowData() Function

Updates the text-based navigation data on the screen:

ShowData() {
  static int pg;
  _setcolor(7);              //Set to display in white 
  _moveto(-X_DIAM,-Y_DIAM);  //Re-display the inner square
  _lineto(X_DIAM,-Y_DIAM);  _lineto(X_DIAM,Y_DIAM); 
  _lineto(-X_DIAM,Y_DIAM);  _lineto(-X_DIAM,-Y_DIAM);
  //Re-display the cross-hairs at the centre of the map window
  _moveto(-14,  0); _lineto( 14,  0); 
  _moveto(  0, 10); _lineto(  0,-10);
  _settextposition( 5,16); _outtext(RadDeg(DstLat));
  _settextposition( 6,16); _outtext(RadDeg(DstLng));
  _settextposition( 7,16); _outtext(RadKm(Range,0));
  _settextposition( 8,16); _outtext(RadDeg(Radial));
  _settextposition(11,16); _outtext(RadDeg(ObsLat));
  _settextposition(12,16); _outtext(RadDeg(ObsLng));
  _settextposition(13,16); _outtext(RadDeg(ObsHdg));
  _settextposition(14,16); _outtext(RadKm(ObsVel,1));
  _settextposition(16,16); _outtext(RadDeg(ReqHdg));
  _settextposition(17,16); _outtext(RadKm(ObsRot,2));
  _setvisualpage(pg);    //Display completed graphics page
  pg ^= 1;               //Flip video page: 0 -> 1; 1 -> 0
  _setactivepage(pg);    //Set old display page for updating

The NodeTilt() Function

Displays a feature on the Earth's surface as a small filled circle which is tilted if necessary so that it appears as an ellipse whose major to minor axis ratio shows the effect of the Earth's curvature according to the feature's displacement from the point of observation at the centre of the screen.

ShowNode (
  double *p,             //Pointers to feature's EARTH &
  int *q,                //SCREEN co-ordinates arrays.
  char *n,               //Pointer to annotation name.
  int c                  //Display colour (can be black)
) {
    tilt = *(p + 2),     //tilt of feature on the Earth
    CosPsi = *(p + 3),   //Cosine of bearing
    TanPsi = *(p + 4),   //Bearing of feature from centre
    t,                   //Angle in generation of ellipse
    z, w;                //Working variables

    x = *(q + 1),        //x-position of observed object
    y = *(q + 2),        //y-position of observed object
    i = 1,               //`first time through' switch
    r = 7,               //major axis radius of ellipse
    u,                   //Ellipse-gen X-displacement
    v,                   //Ellipse-gen Y-displacement
    l = 0,               //Nº of characters in annotation
    c = *(n + 14) - 48;  //Display colour

  If(CosPsi != 0) {      //To prevent division by zero
    if(s) c = 0;         //set colour black if re-display

  //Take t round in a full circle in 0.1 radian steps

    for(t = 6.5; t > 0; t -= .1) {
      w = tilt * r * cos(t);
      z = CosPsi * (r * sin(t) - w * TanPsi);
      v = w / CosPsi + z * TanPsi;
      u = z;
      if(i == 0)                 //If not first pass of loop,
        _lineto( x + u, y + v ); //draw line to next valid point
      else {
        i = 0;                   //else clear `first pass' flag
        _moveto( x + u, y + v ); //and set position to start of 
      }                          //ellipse locus.
    _floodfill(x, y, c);         //Fill in the ellipse
    while(*n != ' ') {l++; n++;} //Find length of city name
    n -= l;

    /* Annotate the ellipse with the name of the city or way 
       point it represents. Place the name at the left, right,
       above or below the ellipse as appropriate. */

    switch(*(n + 15)) {
      case 'L': x -= (7 + l * 7); break;            //left
      case 'R': x += 10; break;                     //right
      case 'A': x -= (l * 7 / 2); y -= 13; break;   //above
      case 'B': x -= (l * 7 / 2); y += 13;          //below
    } annotate( x, y, l, n );
Please note that this function is not used in the current demonstrator.

The Way Points Database

I had to find a way to represent the Earth in terms of numeric data. This was long before Geographic Information Systems and databases became generally available. The purpose at hand did not require a high degree of resolution or detail. The marking of coastlines was unnecessary. I decided to build my geographic data model of the Earth entirely of way points placed on a true sphere. Nothing more would be gained by representing the Earth as a distorted ellipsoid with land relief and oceanic bulges.

I conveniently found a world-wide listing of 11,000 cities and major geographical features together with their co-ordinates in the back of my daughter's school atlas. As I was going to use the data for experimental purposes only and not for commer­cial gain I decided that it would be permissible to use it. Entering this list was a long and boring job. I entered the names and co-ordinates 10 at a time in odd sessions over a long period. This is the Earth model used as the basis for this Moving Map project.

The main() Way-Point File Scanning Function

This file contains all the C code for my demo flight system which flies an ideal aircraft from Stansted (London) to Ringway (Manchester). It displays point features (or nodes) on the Earth's surface from their latitudes and longitudes. The screen represents a spheroidal section of the earth's surface rather than a `projected' map. Unlike a map projection, therefore, all the variables; range, bearing, shape and area are correct. Note: the RandB() function is in the file RandB.C.

#include <graph.h>
#include <math.h>
#include <stdio.h>
extern double DegRad(char *p, int l);  //in RandB.C
extern char *RadDeg(double c);         //in RandB.C
extern double RandB(double RefLat, double RefLng, int Swap);
extern GndTrk();                       //in file GndTrk.C
extern int *RBtoXY(float s);
extern int MapInit();
extern ShowNode(int *q, char *n);      //in file NodeTilt.C
char *RadKm(double d, int e);
#define Pi 3.1415926535
#define TwoPi 6.283185307
#define CITIES 24        //Nº of cities (sizeof city array)
#define CITYRL 81        //80 data bytes + \0 terminator
#define OBSERVER 5       //City Nº of observer
#define LATBYTE 17       //Byte Nº of lat within city record
#define LNGBYTE 28       //Byte Nº of lng within city record
#define SCALE .5         //Map scale in km per X-pixel
#define VEH_SPEED 1000   //Vehicle speed in Km/hr
#define WINDOW_WIDTH 200 //½-width of map window (pixels)
#define KMH_RADS .4363323129861111E-7  //km/hr to rad/sec
char city[CITYRL];       //Holds data relating to one city 

  ObsLat,  //Observer's (or vehicle's) current latitude
  ObsLng,  //Observer's (or vehicle's) current longitude
  RefLat,  //Reference-point's (or city's) latitude
  RefLng,  //Reference-point's (or city's) longitude
  MaxRng,  //Cosine of maximum range we are interested in
  ObsVel,  //Observer's current velocity (radians/sec)
  ObsHdg,  //Observer's current heading (radians)
  ObsRot,  //Observer's rate of turn (radians/sec)
  DstLat,  //Latitude of destination (or way point)
  DstLng,  //Longitude of destination (or way point)
  Range,   //Range of Reference object from observer
  Bearing, //Bearing of Reference object from observer
  CosRng,  //Cosine of the range
  CosBrg,  //Cosine of the bearing
  TanBrg,  //Tangent of the bearing
  ReqHdg,  //Required Heading to come smoothly on to radial
  Radial;  //Radial which vehicle must endeavour to follow

The following is the basis for the dLat test which determines the approximate in-window range before RandB() is used. The size of a latitude-radian is fixed at 40,000 / 2pi kilometres. The size of a Longitude-radian varies with latitude.

Moving map project: annotated explanation of the window height calculation.

double window_height = WINDOW_WIDTH * SCALE * Pi / 20000;
double window_width;

main() {
  int active = 1;            //loop active
  int trigger = 1;           //first-time-through trigger
  static char *p, 
    *q = city + CITYRL;      //Ptr to byte after last byte
  static int i;
  static long fp;            //file position pointer
  static FILE *fh;           //declare file handle pointer
  static double mrofi,       //Maximum range of interest =
    *d, *r;                  // ½(width of map window)
  double x, dlat, dlng;      /* lat/long difference between 
                                observer and observed */

  MapInit();                 //perform the map initialisation
  trigger = 0;               //set 'initialisation done'
  /* Set up observer's name & co-ordinates in same 
     format as the city data is presented in disk file. 
     Load them into elements [0]& [1] of coords array */ 

  p = "DESTINATION      53.28.100N 002:31.850W";
  DstLat = DegRad(p + LATBYTE,0);
  DstLng = DegRad(p + LNGBYTE,1);
  p = "OBSERVER         51:52.000N 000:11.000E";
  ObsLat = DegRad(p + LATBYTE,0);
  ObsLng = DegRad(p + LNGBYTE,1);

  /* This `leg' of the journey, ie the Destination Radial 
     on which vehicle must try to travel, is the bearing 
     of the origin from the destination's point of view: */ 

  MaxRng = -1;
  Radial = RandB(DstLat, DstLng, 1);
  ObsVel = VEH_SPEED * KMH_RADS;    /* Observer's speed 
                                         (radians/sec) */
  ObsHdg = 1.5;          //Observer's heading (radians) 

  /* Set max range at which a city is of interest. Root 2 
     times (window_height) is the radius of the smallest 
     circle within which the square window will fit. */

  MaxRng = (mrofi = cos(1.414213562 * window_height));

  /* Open disk file containing names and co-ordinates of 
     cities to be displayed as reference points for the 
     observer and allocate a 4k buffer to the file. */ 

  fh = fopen("WayPts.DAT", "r");
  setvbuf(fh, NULL, _IOFBF, 4096);
  /* Skip over the column titles at the beginning of the 
     file and set the file pointer to the start of the data 
     for the first city. */ 

  fseek(fh, fp = CITYRL, SEEK_SET);
  i = 0;          //Set index to the first city in the list.

  while(active) {            //while scan OK and no key hit
    if(kbhit()) active = 0;  //if key struck, kill program
    else { 

  / *Compute range & bearing of each city to be displayed 
    in the map window */

      if (i <= CITIES) {      //If not all cities done

        /* Advance to [next] city in list, get its Earth 
           co-ordinates and convert them from degrees and
           minutes to radians. */ 
        for(p = city; p <= q; p++) *p = getc(fh);
        RefLat = DegRad(city + LATBYTE,0);
        RefLng = DegRad(city + LNGBYTE,1);

        /* Provided the centre of the map window is not at one 
           of the poles, make the East-West aperture of map 
           window equal to its North-South aperture (expressed 
           in great-circle radians) times the cosine of the 
           latitude of the window's centre (ie of observer) */ 

        window_width = window_height;
        if((x = cos(ObsLat)) > 0) window_width /= x;

        /* Compute the latitude offset and longitude offset of 
           the city from the centre of the map window */ 

        dlat = fabs(RefLat - ObsLat);
        if (dlat > Pi) dlat = TwoPi - dlat;
        dlng = fabs(RefLng - ObsLng);
        if (dlng > Pi) dlng = TwoPi - dlng;

        /* Test if city is approximately within range of map 
           window first by seeing if its latitude offset is 
           within window's vertical (north-south) limits, if 
           so, see if its longitude offset is within the map 
           window's horizontal (east-west) limits */

        if ((dlat <= window_height) && (dlng <= window_width )) {

          /* if the city is known to be within the map window, 
             or at least close to the map window's boundary, 
             compute its range and bearing. If the RandB() 
             function found its range to be within 'maximum 
             range of interest' given in coords[4], then 
             compute its screen x, y co-ordinates and display 
             the city's position and name in the map window */ 

          RandB(RefLat, RefLng, 0);
          if(Range > 0) ShowNode(RBtoXY(SCALE), city);
      /* Display the graticule and swap the display and active 
         graphics pages etc. */
      else {  
        MaxRng = -1;         //Disable range limit for RandB()
        GndTrk();            //Advance observer's position
        MaxRng = mrofi;      //Reset Max Rng to window radius
        fseek(fh, fp, SEEK_SET); //Re-initialise file pointer
        i = 0;               //Set to re-start display cycle

The RandB() Function

This function computes distance and bearing of point latitude RefLat, longitude RefLng from the point latitude ObsLat, longitude ObsLng. Range & bearing in radi­ans is computed to approximately 52-bit precision in about 40 milliseconds on a PS/2 30-286.

Since in most cases, range & bearing are measured from the point of view of the observer, RandB() picks up the lat/long of the observer directly from the externals ObsLat, ObsLng. The lat/long of the point being observed is passed to RandB() by the calling function. The resulting range & bearing are placed in the externals Ran­ge and Bearing, and Bearing is also returned by the function. The 'Swap' switch when set to non-zero causes the Observer and the Observed to be interchanged so that the bearing of the Observer from the point of view of the Observed is returned.

#include <math.h>
#define Pi 3.1415926535  //Last modified 28 May 1991
extern double 
  ObsLat, ObsLng,        //Lat/long of observer
  MaxRng,                //Cosine of max range of interest
  Range, Bearing,        //Computed range & bearing
  CosRng, CosBrg;        //Cosines of the range & bearing

double RandB(double RefLat, double RefLng, int Swap) {
  temp, dlong, 
  SinObsLat, CosObsLat, 
  SinRefLat, CosRefLat, 
  if(Swap) {
    temp = ObsLat; ObsLat = RefLat; RefLat = temp;
    temp = ObsLng; ObsLng = RefLng; RefLng = temp;
  dlong     = RefLng - ObsLng, //Longitudinal difference
  SinObsLat = sin(ObsLat),     //Sine of latitude of observer
  CosObsLat = cos(ObsLat),     //Cosine latitude of observer
  SinRefLat = sin(RefLat),     //Sine of latitude of observed
  CosRefLat = cos(RefLat),     //Cosine latitude of observed
  SinRng    = 0;
  Range = 0; Bearing = 0;      //initialise externals
  CosRng = 1; CosBrg = 1;
  if(dlong >  Pi) dlong -= Pi;
  if(dlong < -Pi) dlong += Pi;
  CosRng = CosObsLat * CosRefLat * cos(dlong)
         + SinObsLat * SinRefLat;
  if(CosRng > +1) CosRng = +1;
  if(CosRng < -1) CosRng = -1;

  /* If computed range within specified maximum `range of 
     interest' to calling function (expressed in MaxRng as 
     cosine of maxi), then proceed with full range and 
     bearing computation, else leave range as zero and exit */ 

  if(CosRng > MaxRng) {
    Range = acos(CosRng);    //Compute the range in radians 


    /* Evaluate bearing only if Sin(Range) is non-zero 
       otherwise a division by zero situation will arise 
       in the following formula. If Sin(Range) is exactly 
       zero, then exit leaving Bearing as zero */ 

    if((SinRng = sin(Range)) > 0)
       /* Standard spherical geometry formula for bearing */
       CosBrg = (SinRefLat - SinObsLat * CosRng)
              / (CosObsLat * SinRng);

       /* Avoid possible acos domain errors resulting from 
          rounding errors in evaluating the above formula */ 

       if(CosBrg >=  1) Bearing = 0;
       if(CosBrg <= -1) Bearing = Pi;

       /* If within acos domain (-1 to +1), evaluate the 
          actual bearing */

       else {
         Bearing = acos(CosBrg);
         if(dlong < 0)            //If longitudinal difference
         Bearing = -Bearing;      //is -ve, the bearing is -ve
  //Swap observer's and observed's lat/longs back if necessary
  if(Swap) {
    temp = ObsLat; ObsLat = RefLat; RefLat = temp;
    temp = ObsLng; ObsLng = RefLng; RefLng = temp;

Convert Degrees and Minutes to Radians

Latitudes must be expressed in range 90:00.000S to 90:00.000N. Longitudes must be expressed in range 180:00.000W to 179:59.999E. Returns a double length quant­ity in radians. Input is passed as a pointer to a string containing degrees, minutes and seconds of arc in the format "000:00:00X". Switch = 0 to pick up a latitude and 1 to pick up a longitude.

double DegRad(char *p, int Switch) {
  int x, y;  //Save first character of field
  double c;
  //Convert the whole degrees to integral Nº of minutes of arc
  x = (*p++ - 48) * 10 + *p++ - 48;
  if(Switch == 1)
    x = x * 10 + *p++ - 48;
  x *= 60;                            //degrees to minutes
  p++;                                //jump over the colon
  x += 10 * (*p++ - 48) + *p++ - 48;  //+ tens, units of mins
  p++;                                //jump over decimal point
  y = (*p++ - 48) * 100               //100s of milliminutes
    + (*p++ - 48) * 10                //10s of milliminutes
    + *p++ - 48;                      //single milliminutes
  c = x;
  c *= 1000;
  c += y;                             //form total as a double
  c *= 2.908882086574074E-7;          //milliminutes to radians
  x = *p;                             //Get N S E W letter
  if ( x == 'W' || x == 'S' ) c = -c; //West & South are -ve

The RadDeg() Function

Converts 'double' radians to a degrees, minutes, seconds display string:
char *RadDeg(double c) {   //c = input in radians
  long s;                   //long interger seconds
  int h = 0, i, q;          //int degrees, minutes, seconds
  char *p = "   ø  '  ";    //output string
  if(c < 0) {               //If angle is negative, then
    q = '-';                //set up the minus sign
    c = -c;                 //and make it positive.
  } else q = '+';           //else set up the set minus sign
  c *= 57.2957795141996;    //convert radians to degrees
  while(c > 100) {          //find hundreds of degrees `h'
    c -= 100; h++;
  if(h == 0) *p = q;
  else *p = h + 48;
  for(i = 0; i < 3; i++) {
    p++;                    //Skip over `hundreds' of colon
    s = c;                  //interger of `c'
    *p++ = s / 10 + 48;     //Set tens
    *p++ = s % 10 + 48;     //Set units
    c -= s;                 //chop integral part of `c'
    c *= 60;                //multiply down to next unit
  return (p -= 9);          //return pointer to deg/min/sec

Please note that in the final implementation, trig frunctions are replaced by fun­ctions which use look-up and interpolation tables. These are designed to return values to an accuracy equal to the precision of the C-language's int type (normally 32 bits). And of course, they are considerably faster than the trig functions they replace.

The GndTrk() Function

Given the vehicle's current position, speed and heading this function updates the vehicle's position and heading based on the time that has elapsed since the last pass.

#include <sys\types.h>  //getting System Time each pass
#include <sys\timeb.h>  //Include library functions for
#include <math.h>       //for sin, cos etc.
#define Pi 3.1415926535
#define TwoPi 6.283185307
#define RotMax 0.03    //Max rate-of-turn (radians/sec)
#define RotDmp 0.02    //Rate-of-turn damping factor
#define RotFac 0.03    /*Converts heading deviation in radians 
                         to rate-of-turn in radians/second) */
extern double          //gets range and bearing of destination
  RandB(double Lat, double Lng, int Swap),
  ObsLat, ObsLng,      //Observer's (vehicle) lat/long
  RefLat, RefLng,      //Reference Object's lat/long
  DstLat, DstLng,      //Destination lat/long
  ObsHdg, ObsVel,      //Observer's heading and speed
  ObsRot,              //Observer's Rate of turn (radians/sec)
  Bearing,             //bearing of reference object
  ReqHdg,              //Heading Command
  MaxRng,              //Maximum range of interest
  Radial;              //Bearing of origin from destination

double BrgDif(double a, double b) {     //BEARING SUBTRACTOR
  double c = a - b;
  while(c < -Pi) c += TwoPi;
  while(c >  Pi) c -= TwoPi;

double RatAng(double a) {      //EXPRESS AN ANGLE AS A VALUE
  while(a < 0    ) a += TwoPi;
  while(a > TwoPi) a -= TwoPi;

GndTrk() {                //UPDATE HEADING AND GROUND TRACK
  static struct timeb T;  //System time at last pass
  static int ftt;         //first-time-through flag
  double  HdgDot,         //Hdg deviation/increment this pass
    d,                    //Dist moved along track this pass
    et,                   //Elapsed time (secs) since last pass
    t,                    //double accommodation for T.millitm
    lat,                  //Average lat during move this pass
    dlat;                 //Latitude component of vehicle

  et = -(t = T.millitm) / 1000 - T.time;
  ftime(&T);              //Get system time now 
  et += (t = T.millitm) / 1000 + T.time;

  /* If this is the first pass of the program since switch-on, 
     simply put the System Time in the `T' time data structure 
     to act as the reference from which the elapsed time will 
     be measured during the second pass. Then exit without 
     updating heading and ground track.  */ 

  if(ftt == 0) { ftt = 1; et = 0; }
  else {


       First compute the required heading ReqHdg the vehicle 
       must turn towards to bring it smoothly on to the 
       required radial of the destination waypoint. */

    ReqHdg = RatAng(
      RandB(DstLat, DstLng, 0) +    //Brg of dest from vehicle
      RandB(DstLat, DstLng, 1) -    //Brg of vehicle from dest
      Radial                        //Brg of origin from dest
    /* Compute a rate of turn HdgDot proportional to current 
    heading error:*/ HdgDot = RotFac * BrgDif(ReqHdg, ObsHdg);

    // Impose a maximum limit to this rate of turn:
    if(HdgDot >  RotMax) HdgDot =  RotMax;
    else if (HdgDot < -RotMax) HdgDot = -RotMax;

    /* Apply damping to rate of change of observer's rate of 
    turn: */  ObsRot += RotDmp * et * (HdgDot - ObsRot);

    // Update the observer's current heading:
    ObsHdg = RatAng(ObsHdg += ObsRot * et);

    /* Compute distance moved along ground track since last 
    program pass: */  d = ObsVel * et;


       Shift in latitude is simply distance along track x 
       cos(heading). Shift in longitude is distance along 
       track x sin(heading) x cos(average latitude during 
       the shift). */

    dlat = d * cos(ObsHdg);     //change in latitude this pass
    lat = ObsLat + dlat / 2;    //average lat during travel
    ObsLat += dlat;             //perform the latitude shift


       The distance in full `great-circle' radians moved in 
       the West - East Direction is d.sin(heading). But we 
       need to express this in terms of the (smaller) 
       `longitudinal radians' for the latitude concerned. 
       This (larger) figure is obtained by dividing 
       d.sin(heading) by cos(lat), where `lat' is the average
       latitude of the vehicle while it is travelling the 
       distance `d' during this pass of the program. */

    ObsLng += d * sin(ObsHdg) / cos(lat);

Waypoint Encounter: the geometry involved in navigating an aircraft or ship to­wards an en-route way point, including the implementation of a demonstration applet.

Barges follow the course of the rivers and canals on which they float. These natur­ally lead them to their destinations. Trains are guided by the tracks on which they travel. Cars and trucks follow roads. All they have to do to reach their destinations is to take the right turn at each junction along the way. But ships at sea and aero­planes have no permanent way to guide them. They rely on natural geographical features, lighthouses and other artificial navigational aids to guide them on their journeys.

For a ship, aeroplane or off-road vehicle a journey is made up of a sequence of way points. A way point is simply a point on the Earth's surface. It may be the location of a hill, mountain, lake, town or some other geographical feature. It may be the loca­tion of a lighthouse or radio navigation station. Or it may be simply a latitude and longitude recorded in the memory of a G.P.S. [global positioning system] receiver or a futuristic astral navigator which fixes positions using signals from pulsars.

The featureless separation between way points plays no part in guiding the traveller to his destination.

This project is concerned essentially with the navigation of aircraft. They are cap­able of flying over any part of the Earth's surface at high speed. This allows them to follow paths much closer to mathematical ideals than can other kinds of vehicle. There is a strong tendency to think the ideal path for an aircraft to follow to be a succession of hyperbolic encounters with its way points - a path rather like that of a small positively charged particle weaving between a sequence of large stationary particles which are also positively charged:

Moving map project: the hyperbolic intercept of waypoints.

However, aircraft are not ideal particles or planets moving in free space: they are vehicles battling through a rather dense complex dynamical atmosphere. For them, therefore, the ideal - the most economical - path is not a series of hyperbolae, but one of straight lines joined by small circular arcs whose radii are equal to the radius of the aircraft's minimum comfortable turning circle.

The aircraft's route is set by the radial on which the aircraft must leave each way point. Consequently the aircraft must fly over the way point in the direction of the next way point en-route. To achieve this, we construct a circle passing through the way point we are approaching such that the tangent to the circle at the way point is the direction of the next way point.

Moving map project: the full geometry of the waypoint intercept attractor.

All angular computations are then performed relative to the centre of the constr­ucted circle. The aircraft never passes near the centre of the circle. Nasty infinities therefore can never occur to cause trigonometrical confusion. If the aircraft drifts off course, a new circle is constructed, thus maintaining the integrity of the trigon­ometry.

As computer software, this could form the basis of an automatic flight control sys­tem of the future. Integrated with international flight plan processing, it could form the basis of a completely automated system of global air highways.

I have designed, programmed and exhaustively tested a software kernel to provide this navigational functionality. I have coded part of it in Java and installed it herewith as a demonstration applet to give some idea of my capabilities in this area. How­ever, please be aware that a vast amount of applied expertise is needed (hopefully some of mine) to transform the core functions demonstrated in the applet into a safe automatic flight control system.

Mathematical Philosophy

The spherical range and bearing computations described in this document's parent use the universally familiar sin and cos functions quite liberally. Many people see these pure mathematical functions as quite beautiful. They see them as a reflection of nature. Or even as part of the very fabric of the universe. This is undoubtedly because, when viewed against a fixed frame of reference in space and time, they trace out apparently smooth (and beautiful) periodic curves. The idea of replacing such things of beauty with something as crude and frumpy as a look-up table is con­siderd to be pure anathema.

This beauty is however like all beauty: in the eye of the beholder. Nature does not set out to make sine curves. They are merely one consequence of one microscopic event following another ad infinitum. They are the global outcome of a myriad iterations of a single fractal processes. They become visible to us only when we take our observer's licence of freezing time and rigidifying space with our artificial systems of co-ordinates and units of measurement. They are nothing more than human aids to comprehension.

The lowly look-up table, on the other hand, makes no pretence of being the fabric of the universe. Nevertheless, it does define the same set of waypoints which the nat­ural fractal process visits on its journey through time and space. And it finds them for you much more quickly than does the 'pure' mathematical function. The truth is that neither one is a fundamental component of the universe. Reality is relativistic and fractal. You cannot freeze time or rigidify space. Motion is key to existence.

The mathematician and the physicist with their beautiful functions therefore have no cause to look down upon the engineer and the programmer for using look-up tables.

© 1997 Robert John Morton