Great Circle Calculator

This is a collection of equations and formulas that I've been using across my many projects to compute various distances using great circle calculations. The formulas here were adapted into python from here and here.

Because I've been using these equations across several projects, I decided to upload this to PyPI for ease of keeping updated and distribution. Feel free to clone, fork, or modify the code as needed. I believe there are more complete packages out there. One example is geodesy.

Any questions, feel free to get in touch.

How to install

Clone/download the package to your project or use pip install great-circle-calculator

• PyPI
• GitHub

How to use

The convention of this package is for the spatial points to be represented as a tuple of length 2 with longitude being the first element and latitude being the second element, i.e. (lon, lat). One should be able to pass a Shapely Point object if the point is in the form of (lon, lat) as well.

NEW FOR 2.1 : functions that take a point or return a point now have the parameter lat_lon. Set lat_lon=True to pass points in the form (lat, lon).

Here is an outline of the functions available to you.

Library great_circle_calculator

Depending on my needs, I typically import this library as

import great_circle_calculator as gcc

The functions are

• distance_between_points
• bearing_at_p1
• bearing_at_p2
• midpoint
• intermediate_point
• point_given_start_and_bearing

Function distance_between_points()

Function distance_between_points(p1: Point | tuple, p2: Point | tuple, unit: str = "meters", haversine: bool = True, planet_radius: float | None = None, body: CelestialBody = EARTH) Computes the distance between two points, in the unit given by the 'unit' parameter. Uses the haversine formula unless haversine=False, in which case the (typically faster, less precise for short distances) law of cosines is used. p1: tuple point of (lon, lat). p2: tuple point of (lon, lat). You can also pass a Shapely Point instead of a tuple. unit: unit of measurement. See great_circle_calculator._constants.eligible_units(). You can also submit your own units. planet_radius: optional custom sphere radius, expressed in unit (Although unit has no effect if planet_radius is given). Parameter planet_radius Overrides body. Use this for a quick one-off body without constructing a CelestialBody. body: which CelestialBody's radius to use if 'planet_radius' isn't given. Defaults to EARTH; pass e.g. elestial_body.MARS to compute distances on Mars. This returns a float which is the distance between p1 and p2, in 'unit'.

Function bearing_at_p1()

Function bearing_at_p1(p1, p2) computes the bearing (i.e. course, compass direction) at p1 given a destination of p2. Use in conjunction with midpoint() and intermediate_point() to find the course along the route. Use bearing_at_p2() to find the bearing at the endpoint, p2. p1 and p2 are points (i.e. tuples, lists of length 2) in the form of (lon, lat).

Function bearing_at_p2

Function bearing_at_p2(p1, p2) computes the bearing (i.e. course) at p2 given a start of p1. Use in conjunction with midpoint() and intermediate_point() to find the course along the route. Use bearing_at_p1() to find the bearing at the starting point, p1. p1 and p2 are points (i.e. tuples, lists of length 2) in the form of (lon, lat) in decimal degrees.

Keep in mind that bearing_at_p2(p1, p2) does not show the direction of you standing at p2 and facing p1, but rather as if you're traveling from p1 to p2, what direction would you be facing at p2

Equivalently, you can also determine the direction you are enroute by

p1, p2 = (lon1, lat1), (lon2, lat2)
frac_along_route = 0.2
course_enroute = gcc.bearing_at_p2(p1, gcc.intermediate_point(p1, p2, frac_along_route))

Function midpoint()

Function midpoint(p1, p2) is the half-way point along a great circle path between the two points. p1 and p2 are points (i.e. tuples, lists of length 2) in the form of (lon, lat) in decimal degrees. For example, say p3 = midpoint(p1, p2), distance_between_points(p1, p3) == distance_between_points(p2, p3) (although considering floating point maths, these will be close if not equal)

Function intermediate_point()

Function intermediate_point(p1, p2, fraction=0.5) an intermediate point along the course laid out by p1 to p2 given the fractional distance. fraction is the fraction of the distance between p1 and p2, where 0 is p1, 0.5 is equivalent to midpoint(), and 1 is p2.

Function point_given_start_and_bearing()

Function point_given_start_and_bearing(p1, course, distance, unit='meters') is given a start point p1, initial bearing course, and distance distance, this will calculate the destination point bearing travelling along a (shortest distance) great circle arc. unit is a unit of measurement that can be accessed by great_circle_calculator.constants.eligible_units, default is 'meters'.

Custom units

As of 2.0.0, unit is no longer limited to a fixed list. great_circle_calculator.eligible_units() returns the currently-registered unit names (a decent default set is built in: meters, kilometers, centimeters, millimeters, miles, feet, inches, yards, nautical_miles, fathoms, furlongs). Add your own with register_unit:

import great_circle_calculator as gcc

gcc.register_unit('shufflin_step', 12)
p1 = (45.496547, -73.569841)
p2 = (40.750455, -73.993824)
gcc.distance_between_points(p1, p2) # 154703.2456...
gcc.distance_between_points(p1, p2, unit='shufflin_step') # 1856438.9473...

Other planetary bodies

distance_between_points and point_given_start_and_bearing accept a body parameter (default EARTH) so you can run the same great-circle math on other spheres. MOON, MARS, VENUS, and MERCURY are predefined, or build your own CelestialBody:

import great_circle_calculator as gcc

gcc.distance_between_points(p1, p2, unit='kilometers', body=gcc.MARS)

ceres = gcc.CelestialBody('Ceres', radius_meters=470_000)
gcc.distance_between_points(p1, p2, unit='kilometers', body=ceres)

For a quick one-off sphere without building a CelestialBody, pass radius= directly (interpreted in unit, and it takes priority over body if both are given):

gcc.distance_between_points(p1, p2, planet_radius=1000)

midpoint() and intermediate_point() are calculated without the need for knowing which planet you're on.

Library compass

This library was created to let me call, say Compass.east so I can get 90deg. I thought it helped with code readability at first, kept it because it might be useful...

It has two classes called CompassSimple and CompassComplex. CompassComplex is still in the todo list, but it contains more information about each compass point.

To see the eligible points, see here. Simply use the terms in "Compass point", use lower case and underscores where there are spaces or dashes. Alternatively you may use the "Abbreviation" with the appropriate case to call the same value.

To use CompassSimple:

from great_circle_calculator.compass import CompassSimple as compass

print(compass.east)  # prints 90
print(compass.north)  # prints 0
print(compass.northwest_by_north)  # prints 326.250
print(compass.SEbE)  # Southeast by east, prints 123.750
print(compass.SWbS == compass.southwest_by_south)  # prints True

Library _constants

This was created for two purposes:

1. To easily store the radius of the earth in various units

2. To have thin wrappers for the trig functions in particular for the arcsin and arctan which have domain limits of -1 to 1, and due to floating point errors, numbers can occationally be outside those domain limits and I didn't want the code crashing about this.

To see the available units, call _constants.eligible_units and a list of the units that are available will be given.

eligible_units

To see the available units, call _constants.eligible_units and a list of the units that are available will be given.

Libraries __conversion and __error_checking

Private libraries that convert (__conversion.py) values between radians and degrees as the default option for python's math package is radians.

The error checking library (__error_checking.py) is used to ensure that the points passed to through the code make sense in the project. That is, it checks two things. First, that the point passed to the function is a tuple or other Point of length 2. If this check fails, a TypeError is thrown.

The second is that the information contained within the tuple matches the expected range of latitude and longitude. The first element of the tuple should be the longitude in the range of [-180, 180] and the second element is the latitude in the range of [-90, 90].

If this check fails, but it appears that the user has "flipped" the elements of the point (i.e. (lat, lon) is provided), a ValueError is thrown with information indicating the belief you have flipped the elements of the coordinates. If this is not true, ValueError is thrown with a different error message indicating the expected input.

And finally...

Package last updated October 2026. Readme last updated October 2026.

Change Log

• 2.1.0 - functions that take a point or return a point now have the parameter lat_lon. Set lat_lon=True to pass points in the form (lat, lon). Points that are passed with this lat_lon=True will also return their results as (lat, lon)
• 2.0.0 - Big version bump and I wanted to add a bunch of things I wanted to a few years ago plus some other things I've learned since then. The big thing was allowing custom units and custom radii into the formulas. I corrected a few typos here and there and added some type hints. I also added units are no longer a fixed list: register_unit(name, units_per_meter) lets you add any distance unit, and eligible_units() now reflects whatever's registered. Added a CelestialBody class plus predefined EARTH, MOON, MARS, VENUS, MERCURY bodies so distance_between_points and point_given_start_and_bearing can compute on other planets via body=; a one-off custom sphere can also be supplied directly with planet_radius=. intermediate_point was refactored to compute the central angle directly instead of going through distance_between_points/Earth's radius, so it's now body-agnostic by construction. Added type hints throughout (tuple[float, float], PEP 604 unions via from __future__ import annotations) targeting Python 3.9+.
• 1.3.1 - Put in a line of code when computing the haversine distance to ensure that (due to floating point errors) may be above 1
• 1.3.0 - Updated the code in the __error_checking.py file to, by default, throw errors rather than try to correct points. Also expanded the error messages here to be more clear.
• 1.2.0 - If the user sends a point that is decimal data type, it will convert to a tuple of float types. Updated the readme for clarity
• 1.1.0 - Changed haversine=True as the default for distance_between_points as it more accurately reflects the small distance calculation I am using in my projects.
• 1.0.2 - squished an error in the intermediate function. The number of errors has been embarrassing. I hope you won't judge me too harshly.
• 1.0.1.post1 - includes a domain checker for asin() and acos() because rounding errors can cause the function to be out of range.
• 1.0.post1 - I screwed up the numbering order, still new at this... please ignore.
• 1.0.1 - Fixed an error in the (point_given_start_and_bearing)[#point_given_start_and_bearing]
• 1.0.0 - First Edition, initial commit, etc.