Question:

I get two coordinate pairs in the form `90°0′0″N 0°0′0″E`

as string and want to calculate the distance between those points on a sphere with radius R=6371km.

I found two formulas on the internet <a href="http://www.movable-type.co.uk/scripts/latlong.html" rel="nofollow">here</a>, the "haversine" and the "spherical law of cosines", but they don't seem to work. For a 90° angle which should return `2*pi*R / 4`

, the haversine operates correct but the cosines fail and return 0. A different point with more random coordinates returns false values with both algorithms: the haversine is too high and the cosines are too low.

Is my implementation wrong or did I chose an incorrect algorithm?

How should I make these calculations (coordinate pairs to distance on globe surface) instead?

(And yes, I know that I'm not checking for N/S and E/W yet, but the tested coordinates are all in the north-eastern hemisphere.)

Here's my Python 3 code:

<pre class="lang-py prettyprint-override">```
import math, re
R = 6371
PAT = r'(\d+)°(\d+)′(\d+)″([NSEW])'
def distance(first, second):
def coords_to_rads(s):
return [math.radians(int(d) +int(m)/60 +int(s)/3600) \
for d, m, s, nswe in re.findall(PAT, s)]
y1, x1 = coords_to_rads(first)
y2, x2 = coords_to_rads(second)
dx = x1 - x2
dy = y1 - y2
print("coord string:", first, "|", second)
print("coord radians:", y1, x1, "|", y2, x2)
print("x/y-distances:", dy, dx)
a = math.sin(dx/2)**2 + math.cos(x1) * math.cos(x2) * math.sin(dy/2)**2
c = 2 * math.atan2(math.sqrt(a), math.sqrt(1-a))
haversine = R * c
law_of_cosines = math.acos( math.sin(x1) * math.sin(x2) + \
math.cos(x1) * math.cos(x2) ) * R
print("HS:", round(haversine, 2), "LOC:", round(law_of_cosines, 2))
return haversine
#return law_of_cosines
if __name__ == '__main__':
def test(result, correct):
print("result: ", result)
print("correct:", correct)
test(distance("90°0′0″N 0°0′0″E", "0°0′0″N, 0°0′0″E"), 10007.5)
test(distance("51°28′48″N 0°0′0″E", "46°12′0″N, 6°9′0″E"), 739.2)
test(distance("90°0′0″N 0°0′0″E", "90°0′0″S, 0°0′0″W"), 20015.1)
test(distance("33°51′31″S, 151°12′51″E", "40°46′22″N 73°59′3″W"), 15990.2)
```

Here is some output:

```
coord string: 90°0′0″N 0°0′0″E | 0°0′0″N, 0°0′0″E
coord radians: 1.5707963267948966 0.0 | 0.0 0.0
x/y-distances: 1.5707963267948966 0.0
HS: 10007.54 LOC: 0.0
result: 10007.543398010286
correct: 10007.5
coord string: 51°28′48″N 0°0′0″E | 46°12′0″N, 6°9′0″E
coord radians: 0.8984954989266809 0.0 | 0.8063421144213803 0.10733774899765128
x/y-distances: 0.09215338450530064 -0.10733774899765128
HS: 900.57 LOC: 683.85
result: 900.5669567853056
correct: 739.2
```

Answer1:It looks like you mixed up `x`

and `y`

in your calculation of `a`

. You're supposed to take the cosine of latitude (`y`

), not longitude (`x`

).

I discovered this by changing your `distance`

to `angular_distance`

(i.e. don't multiply by `R`

) and adding some additional tests:

```
test(angular_distance("90°0′0″N 0°0′0″E", "89°0′0″N, 0°0′0″E"), math.radians(1))
test(angular_distance("90°0′0″N 0°0′0″E", "80°0′0″N, 0°0′0″E"), math.radians(10))
test(angular_distance("90°0′0″N 0°0′0″E", "50°0′0″N, 0°0′0″E"), math.radians(40))
test(angular_distance("90°0′0″N 0°0′0″E", "50°0′0″N, 20°0′0″E"), math.radians(40))
```