2  Coordinates

Exercise 2.1.

List three geographic measures that do not have a natural zero origin

• longitude: the zero meridian is arbitrary, 100 years ago there were many other zero meridians fashionable
• latitude: the equator may feel like a natural zero, but one could equally use the North Pole as zero, or choose entirely different origins and orientation for longitude and latitude.
• altitude (measured w.r.t. mean sea level, geoid, or ellispoid)

Exercises 2.2

(thanks to Jonas Hurst)

Convert the (x, y) point s (10, 2), (-10, -1), (10, -2) and (0, 10) to polar cooridnates

cart2polar = function(x, y){
  r = sqrt(x*x + y*y)  # compute r (distance from origin)
  phi = atan2(y, x)  # compute phi (angle between point and positive x axis in rad)
  phi_deg = phi * 180 / pi  #  compute angle in deg
  result = c(r, phi_deg)
  return(result)
}

cart2polar(10, 2)
# [1] 10.19804 11.30993
cart2polar(-10, -1)
# [1]   10.04988 -174.28941
cart2polar(10, -2)
# [1]  10.19804 -11.30993
cart2polar(0, 10)
# [1] 10 90

Exercise 2.3

Convert the polar (r, phi) points (10, 45°), (0, 100°) and (5, 259°) to Cartesian coordinates

deg2rad = function(angle_degree) {
  angle_degree * pi / 180
}

polar2cart = function(r, phi_deg){
  # phi must be in degrees
  phi_rad = deg2rad(phi_deg)  # convert phi in degrees to radians
  x = r * cos(phi_rad)
  y = r * sin(phi_rad)
  c(x, y) # return value
}

polar2cart(10, 45)
# [1] 7.071068 7.071068
polar2cart(0, 100)
# [1] 0 0
polar2cart(5, 259)
# [1] -0.954045 -4.908136

Exercise 2.4

assuming the Earth is a sphere with a radius of 6371 km, compute for (lambda, phi) points the great circle distance between (10, 10) and (11, 10), between (10, 80) >and (11, 80), between (10, 10) and (10, 11) and between (10, 80) and (10, 81).

distOnSphere = function(l1, phi1, l2, phi2, radius) {
  l1_rad = deg2rad(l1)
  l2_rad = deg2rad(l2)
  phi1_rad = deg2rad(phi1)
  phi2_rad = deg2rad(phi2)

  theta = acos(
    sin(phi1_rad) * sin(phi2_rad) +
    cos(phi1_rad) * cos(phi2_rad) * cos(abs(l1_rad - l2_rad))
  )
  radius * theta # return value
}

radius = 3671
distOnSphere(10, 10, 11, 10, radius)
# [1] 63.09763
distOnSphere(10, 80, 11, 80, radius)
# [1] 11.12568
distOnSphere(10, 10, 10, 11, radius)
# [1] 64.07104
distOnSphere(10, 80, 10, 81, radius)
# [1] 64.07104

Unit of all results are kilometers.