I’ve got this map of a ficticious world losely based on earth. I know the distance between two cities on it, (approximately 90 miles east-west), and I know that they are apparently 3 degrees off of each other on an east-west axis, and that, straight-line, they’re about 400 miles appart. Is there anything I can do, mathematically speaking, to then determine other aspects of the world from this data? ie- size, area, anything like that? The big question about this ficticious world has always been ‘where is the equator’. Any way to figure that out either? I know it’s not much to go on, but I’m hoping with a little geometry and a little cartography to be able to shed at least -some- light on it all.
Sorry- to clarify- The two cities are 90 miles apart in an east-west direction, but do not lie directly east-west of each other. City ‘A’ is 400 miles north-west of city ‘B’, but the east-west distance is 90 miles (which, through geometry, I believe makes them approximately 390 miles apart north-south)
If you assume that the planet is spherical you can use the ratio that:
90 miles is to 3[sup]o[/sup] as the circumferance of a great circle (pi * diameter) is to 360[sup]o[/sup]. This will give you the diameter which turns out to be:
3*360/pi or about 340 miles. Which, of course makes the original assumption of a spherical planet questionable.
Anyway, if it is spherical you can take it from there.
East-west is not measured along a great circle except at the equator. What we know is that a latitude line between the two cities has a diameter of 340 miles.
Hold on guys. 3 degrees of arc is 1/120 of a circle. If you are keeping the angle in degrees rather than radians there’s no need to drag pi into the calculation. 3 degrees of arc = 90 miles indicates a latitude line with a circumference of 10,800 miles. Consistent with a roughly Earth-sized planet unless we are talking an extremely high latitude. At least 1/2 the diameter of the Earth. Longitude lines will be about 30 miles apart at about 60 degrees latitude on the Earth.
If you know that latitude of the measurement, divide by the cos() of the latitude to get the circumference at the equator.
The formulas for Great circle distances and azimuths are quite simple and I posted them just some months ago in another thread. I hate to do the coding again.
Ok, I succumbed to the temptation. Here are the formulas which you can easily implement in a spreadsheet:
The great circle distance in degrees between
Point1 (lat1, lon1) and Point2 (lat2, lon2)
is easily calculated:
cos(D)= Sin(lat1) * Sin(lat2) + cos(lat1) * cos(lat2) * cos(lon2 - lon1)
or what is the same:
D= arccos(Sin(lat1) * Sin(lat2) + cos(lat1) * cos(lat2) * cos(lon2 - lon1))
In this paper we assume the following signs N=+, S=-, W=+, E=- .
cos(D) will always fall in the range -1 < cos(D) < 1 and (D) will be
a positive angle: 0° < D < 180°. You can use a formula that will use
the arctan function but this will give a result in the -90° to +90°
range and you need to add 180° if the result is negative.
multiply D by 60 to obtain distance in nautical miles or
multiply D by 69 to obtain distance in statute miles or
multiply D by 111.12 to obtain distance in Km
The azimuth or initial course from point 1 (origin) to point 2
(destination) Zn can also be easily calculated.
Zn is always measured from North to East from 0° to 360°.
First we calculate Z.
If we have considered West longitudes as positive we use this formula:
Sin(lon2-lon1)
Tan(Z) = -----------------------------------------------------
sin(lat1) * cos(lon2-lon1) - cos(lat1) * tan(lat2)
(If we had considered East longitudes as being positive we
would need to change the sign on one side of this equation.)
The function ATAN() returns a value between -90° and +90°
so Z needs to be adjusted to the right quadrant in order
to obtain Zn: 0° < Zn < 360°
If sign(sin(Z)) = sign(sin(lon2-lon1)) then Zn = Z + 180
Else, If Z is negative then Zn = Z + 360
Else Zn = Z
The following formula adds 180° if needed to reduce Zn
to the correct quadrant:
Zn = Z + 90 * (1 + sign( sin(lon2-lon1) * sin(Z) ) )