Question

find the distance between two latitudes and longitudes while also dealing with unit conversions

1. As pseudocode: showing a single correct solution path, expressed in natural language, ignoring programming syntax, but providing enough detail, hierarchy and order so that a peer can easily understand the steps.
   The pseudocode must include a desk-check, that is to say, it must be traced by hand using ‘real’ numbers, showing the values/calculations of all of the intermediate steps, and then indicating the ‘algorithm-calculated’ final result.

Answer 1

We first derive the relevant formula, then give the algorithmic explanation for said formula and then lastly do a desk-check of the values.

The formula for the haversine of an angle is given as:

.

Now, let the central angle between the radius of the earth and the distance between any two given points be d. Then,

Where: r is the radius of the Earth (6371 km), d is the physical distance between the two points under consideration, is latitude of the two points and is longitude of the two points respectively.

Now, we take h = haversine(d/r). Then:

(we take haversine = hsine for brevity in the derivation below)  

Then,

Which gives us:

Now, algorithmically (we assume that the internal sin() and cos() functions take radians in arguments):

1. Take lat_a, long_a, lat_b and long_b as inputs - here, lat_a and long_a are the latitude and longitude of a point "a" and lat_b and long_b are the latitude and longitude of another point "b".
2. Take pi = 3.141257.   
3. Calculate delta_lat and delta_long as shown:
   1. Calculate delta_lat = (lat_b - lat_a) * pi /180 (in radians)
   2. Calculate delta_long = (long_b - long_a) * pi /180 (in radians)
4. Convert the input latitudes to radians:
   1. lat_a = (lat_a) * pi / 180
   2. lat_b = (lat_b) * pi / 180
5. We calculate h = sqrt(sin(delta_lat/2)² + cos(lat_a) * cos(lat_b) * sin(delta_long/2)²)
6. We calculate d = 2 * 6371 * sin^-1(h).

Desk-check:

1. Given: London (51.5007° N, 0.1246° W) and New York (40.6892° N, 74.0445° W).
2. Then, lat_a = 51.5007, long_a = 0.1246, lat_b = 40.6892, long_b = 74.0445.
3. delta_lat = (40.6892 - 51.5007) * 3.141257 / 180 = -0.1886 radian
4. delta_long = (74.0445 - 0.1246) * 3.141257 / 180 = 1.29 radian.
5. lat_a = (51.5007 * 3.141257) / 180 = 0.899 radian
6. lat_b = (40.6892 * 3.141257) / 180 = 0.71 radian.
7. Then, h = = 0.4236
8. Then, d = 2 * 6371 * 0.4236 = 5397.5112 km (approximately)

The actual distance between the points is:

Thus our final result is very close to the actual distance between these two points, with some error induced due to rounding.

(If you liked this answer, feel free to give it a thumbs up. Thank you so much!)