Exo Cruiser: Certain Direction from a Point

Suppose you want to know where you go if you start from a particular town in a particular direction and keep going straight (along a great circle). You can calculate the coordinates of points along the route as follows:

Fig. 2. Mercator and Hondius

["Jodocus Hondius (Latinized version of his Dutch name: Joost de Hondt) (14 October 1563 – 12 February 1612) was a Flemish engraver and cartographer. He is sometimes called Jodocus Hondius the Elder to distinguish him from his son Jodocus Hondius II. Hondius is best known for his early maps of the New World and Europe, for re-establishing the reputation of the work of Gerard Mercator, and for his portraits of Francis Drake. One of the notable representatives in the Golden Age of Dutch/ Netherlandish cartography, he helped establish Amsterdam as the center of cartography in Europe in the 17th century."]

Let's say you start from Amsterdam (52°22' north, 4°54' east) by going straight to the east, and you keep going straight. If you keep this up for 1000 km, then where are you? Most of calculation is similar to the part 1 of this article series.

1. Call the polar coordinates (longitude and latitude) of the city l1 and b1, and the direction in which you start gamma, measured from south to west (so south = 0°, west = 90°, north = 180°, east = 270°).

  l1 = 4.9°;
  b1 = 52.37°;
  gamma = 270°

2. Translate the polar coordinates of the city to the corresponding Cartesian coordinates x1, y1, z1 according to formulas 1-3 already shown in part 1.

  x1 = cos(l1)*cos(b1) (1)
  y1 = sin(l1)*cos(b1) (2)
  z1 = sin(b1) (3)

  x1 = 0.6083285;
  y1 = 0.05215215;
  z1 = 0.7919701

3. Calculate the Cartesian coordinates of the corresponding south point with

  lsouth = l1;
  bsouth = b1 - 90°

if b1 is positive (i.e., in the Northern hemisphere), and

  lsouth = l1 + 180°;
  bsouth = -90° - b1

if b1 is negative (i.e., in the southern hemisphere).

  xsouth = cos(lsouth)*cos(bsouth)
  ysouth = sin(lsouth)*cos(bsouth)
  zsouth = sin(bsouth)

  xsouth = 0.7890756;
  ysouth = 0.06764765;
  zsouth = -0.6105599

4. Calculate the cartesian coordinates of the corresponding west point with

  lwest = l1 - 90°;
  bwest = 0.

  xwest = cos(lwest)*cos(bwest)
  ywest = sin(lwest)*cos(bwest)
  zwest = sin(bwest)

  xwest = 0.08541692;
  ywest = -0.9963453;
  zwest = 0

5. Calculate the cartesian coordinates x3, y3, z3 of the great circle point at 90° from the city:

  x3 = xsouth*cos(gamma) + xwest*sin(gamma) (10)
  y3 = ysouth*cos(gamma) + ywest*sin(gamma)
  z3 = zsouth*cos(gamma) + zwest*sin(gamma)

  x3 = -0.08541692;
  y3 = 0.9963453 ;
  z3 = 0.0000000,

  (yielding P3:

  l3 = 94.90000000;
  b3 = 0.00000000)

so in this case each Cartesian coordinate of point 3 is the opposite of the corresponding coordinate of the west point, which was to be expected because we start out going straight to the east. And so P3 is the opposite point of Pwest.

The distance per degree u across the Earth sphere is equal to

  u = r*pi/180, where r = 6378 km
  u = 111.317 km/°

or 111.317 km per degree. Since dist = 1000 km corresponds to an angle phi of

  phi = dist/u
  phi = 8.98335°

we can now calculate the point in question.

6. Now use formulas 6, 7 and 8 (in part 1) to calculate the desired Cartesian and polar coordinates.

  x = x1*cos(phi) + x3*sin(phi) (6)
  y = y1*cos(phi) + y3*sin(phi)
  z = z1*cos(phi) + z3*sin(phi)

  x = 0.587529;
  y = 0.2070892;
  z = 0.7822556,

and to polar coordinates

  b = asin(z) (7)
  l = atan2(y,x) (8)

  b = 51.46756°;
  l = 19.41627°.

This is a location in the middle of Poland (see figure 1.)