02.qmd

# 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

```{r}
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)
cart2polar(-10, -1)
cart2polar(10, -2)
cart2polar(0, 10)
```

## Exercise 2.3 {-}

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

```{r}
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)
polar2cart(0, 100)
polar2cart(5, 259)
```

## 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).

```{r}
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)
distOnSphere(10, 80, 11, 80, radius)
distOnSphere(10, 10, 10, 11, radius)
distOnSphere(10, 80, 10, 81, radius)
```
Unit of all results are kilometers.