Calculation example
This step-by-step example shows how to calculate the geographic midpoint (center of gravity) for three cities: New York, Chicago, and Atlanta. The three cities will be weighted by time.
Convert all coordinates into radians
All latitude and longitude data must be converted into radians. If the coordinates are in degrees.minutes.seconds format, they must first be converted into decimal format. Then convert each decimal latitude and longitude into radians by multiplying each one by PI/180 as in the table below.
| City | DMS coordinates | Decimal | Radians | |
|---|---|---|---|---|
| New York lat | 40° 42' 51.6708" N | 40.7143528 | 0.710599509 | |
| New York lon | 74° 0' 21.5022" W | -74.0059731 | -1.291647896 | |
| Chicago lat | 41° 52' 41.2098" N | 41.8781136 | 0.73091096 | |
| Chicago lon | 87° 37' 47.2722" W | -87.6297982 | -1.5294285 | |
| Atlanta lat | 33° 44' 56.382" N | 33.7489954 | 0.58903108 | |
| Atlanta lon | 84° 23' 16.7346" W | -84.3879824 | -1.47284814 |
weighting factors
The three cities will be weighted by time. For each city, the time is converted into days.
New York, 3 years
w1 = 365.25*3 = 1095.75
Chicago, 2 years
w2 = 365.25*2 = 730.5
Atlanta, 1 year
w3 = 365.25*1 = 365.25
Note that instead of time, a simple weighting factor such as population could also have been stored in w1, w2, and w3. If no weighting factor is desired, set w1=1, w2=1, and w3=1. Now sum the weights for all three cities.
totweight=w1 + w2 + w3
totweight=1095.75 + 730.5 + 365.25 = 2191.5
Convert lat/long to cartesian (x,y,z) coordinates
formulas:
X1 = cos(lat1) * cos(lon1)
Y1 = cos(lat1) * sin(lon1)
Z1 = sin(lat1)
New York:
X1 = cos(0.710599509) * cos(-1.291647896)
= 0.20884915
Y1 = cos(0.710599509) * sin(-1.291647896)
= -0.728630226
Z1 = sin(0.710599509)
= 0.65228829
Chicago:
X2 = cos(0.73091096) * cos(-1.5294285)
= 0.03079231
Y2 = cos(0.73091096) * sin(-1.5294285)
= -0.74392960
Z2 = sin(0.73091096)
= 0.66754818
Atlanta:
X3 = cos(0.58903108) * cos(-1.47284814)
= 0.08131173
Y3 = cos(0.58903108) * sin(-1.47284814)
= -0.82749399
Z3 = sin(0.58903108)
= 0.55555565
Compute combined weighted cartesian coordinate
X = (X1*w1 + X2*w2 + X3*w3)/totweight
= (0.20884915*1095.75 + 0.03079231*730.5 + 0.08131173*365.25)/2191.5
= 0.12824063
Y = (Y1*w1 + Y2*w2 + Y3*w3)/totweight
= ((-0.728630226)*1095.75 + (-0.74392960)*730.5 + (-0.82749399)*365.25)/2191.5
= -0.75020731
Z = (Z1*w1 + Z2*w2 + Z3*w3)/totweight
= (0.65228829*1095.75 + 0.66754818*730.5 + 0.55555565*365.25)/2191.5
= 0.64125282
Convert cartesian coordinate to latitude and longitude for the midpoint
Note that in Excel and possibly some other applications, the two parameters in the atan2 function must be reversed, for example: use atan2(X,Y) instead of atan2(Y,X).
Lon = atan2(y, x)
= (atan2(-0.75020731, 0.12824063)
= -1.40149245
Hyp = sqrt(x * x + y * y)
= sqrt(0.12824063*0.12824063 + (-0.75020731)*(-0.75020731))
= 0.76108913
Lat = atan2(z, hyp)
= atan2(0.64125282, 0.76108913)
= 0.70015084
Convert midpoint lat and lon from radians to degrees
lat= 0.70015084 * (180/PI)
= 40.11568861
lon= -1.40149245* (180/PI)
= -80.29960280
The weighted midpoint is located near Washington, Pennsylvania.