Someone sent me email regarding my online calculator for computing the distance between to locations given their longitude and latitude values. He wants to do sort of the opposite. Starting with the longitude and latitude of one location, he wants to find the longitude and latitude of locations moving north/south or east/west of that location. I like to answer reader questions when I can, so here goes. I’ll give a theoretical derivation followed by some Python code.

Longitude and latitude and are usually measured in degrees, but theoretical calculations are cleaner in radians. Someone using the Python code below could think in terms of degrees; radians will only be used inside function implementations. We’ll use the fact that on a circle of radius r, an arc of angle θ radians has length rθ. We’ll assume the earth is a perfect sphere. See this post for a discussion of how close the earth is to being a sphere.

### Moving North/South

I’ll start with moving north/south since that’s simpler. Let R be the radius of the earth. An arc of angle φ radians on the surface of the earth has length M = Rφ, so an arc M miles long corresponds to an angle of φ = M/R radians. Moving due north or due south does not change longitude.

### Moving East/West

Moving east/west is a little more complicated. At the equator, the calculation is just like the calculation above, except that longitude changes rather than latitude. But the distance corresponding to one degree of longitude changes with latitude. For example, one degree of longitude along the Arctic Circle doesn’t take you nearly as far as it does at the equator.

Suppose you’re at latitude φ degrees north of the equator. The circumference of a circle at constant latitude φ, a circle parallel to the equator, is cos φ times smaller than the circumference of the equator. So at latitude φ an angle of θ radians describes an arc of length M = R θ cos φ. A distance M miles east or west corresponds to a change in longitude of θ = M/(R cos φ). Moving due east or due west does not change latitude.

### Python code

The derivation above works with angles in radians. Python’s cosine function also works in radians. But longitude and latitude are usually expressed in degrees, so function inputs and outputs are in degrees.

import math
# Distances are measured in miles.
# Longitudes and latitudes are measured in degrees.
# Earth is assumed to be perfectly spherical.
earth_radius = 3960.0
degrees_to_radians = math.pi/180.0
radians_to_degrees = 180.0/math.pi
def change_in_latitude(miles):
"Given a distance north, return the change in latitude."
return (miles/earth_radius)*radians_to_degrees
def change_in_longitude(latitude, miles):
"Given a latitude and a distance west, return the change in longitude."
# Find the radius of a circle around the earth at given latitude.
r = earth_radius*math.cos(latitude*degrees_to_radians)
return (miles/r)*radians_to_degrees

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Spherical trigonometry

Finding distances using latitude and longitude

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