I've got the start and end gps location from an excursion across town and need to determine the travel distance
start: 43.059535, -71.013171
end: 43.083620, -70.892085
You could always use google maps, but that would just be cheating.
The goal for this formula is to calculate the shortest great circle distance between two points on the globe designated by latitude and longitudes. The added benefit of the Haversine equation is that it calculates the central angle as well where $s = r\theta$.
The Haversine formula is mainly based on calculation of the central angle, $\theta$, between two gps coordinates. Using the formula for arc length on a sphere
$$ s = r \theta $$where $r$ is the Earth's radius, and $\theta$ is the central angle calculated as
$$ \theta = 2 \arcsin\left( \sqrt{\sin^2 \left(\frac{\phi_2-\phi_1}{2}\right) + \cos(\phi_1)\cos(\phi_2)\sin^2 \left( \frac{\lambda_2-\lambda_1}{2} \right) } \right) $$with:
$$ \begin{align} \phi &= \text{latitude}\\ \lambda &= \text{longitude}\\ \end{align} $$
In [1]:
import numpy as np
import math
# Mean radius of the earth
EARTH_RADIUS = 6371.009
def haversine(lat1, lon1, lat2, lon2):
"""
Calculate the great circle distance between two points
on the earth (specified in decimal degrees)
Return (central angle, distance between points in km)
"""
# convert decimal degrees to radians
lat1, lon1, lat2, lon2 = [math.radians(x) for x in [lat1, lon1, lat2, lon2]]
# haversine formula
dlon = lon2 - lon1
dlat = lat2 - lat1
a = math.sin(dlat/2)**2 + math.cos(lat1) * math.cos(lat2) * math.sin(dlon/2)**2
central_angle = 2 * math.asin(math.sqrt(a))
# s = r * theta
km = EARTH_RADIUS * central_angle
return (central_angle, km)
In [2]:
start = (43.059535, -71.013171)
end = (43.083620, -70.892085)
central_angle, km = haversine(*(start + end))
print("Central Angle of %g radians" % central_angle)
print("Arc length distance of %g km" % km)