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Here Be Dragons

Haversine Distance

What it computes

Great-circle distance is the length of the shortest path between two points on the surface of a sphere, a curved arc, not a straight line. For the Earth, that’s the path a plane or ship actually travels when it follows the optimal route. Planar Euclidean distance is accurate only for small regions (tens of kilometers); the error grows with distance and latitude.

The haversine formula gives great-circle distance from two pairs of latitude and longitude in one tidy trigonometric expression. It’s the default tool for any application where “how far is it between these coordinates?” needs a real answer.

The formula

Given two points with latitudes φ₁, φ₂ and longitudes λ₁, λ₂ (all in radians), and Earth radius R:

Δφ = φ₂ − φ₁
Δλ = λ₂ − λ₁


a = sin²(Δφ/2) + cos(φ₁)·cos(φ₂)·sin²(Δλ/2)
c = 2 · atan2(√a, √(1−a))
d = R · c

a is the squared chord half-length normalized to a unit sphere. c is the central angle in radians. Multiplying by R gives arc length on the surface.

Why atan2, not asin

A mathematically equivalent form is c = 2·asin(√a). Don’t use it. When two points are nearly antipodal (opposite sides of Earth), a approaches 1, and floating-point rounding can push it a hair above 1. asin of anything greater than 1 is undefined, NaN or an exception depending on language. The atan2(√a, √(1−a)) form is well-conditioned across the entire range; it doesn’t need a min(1, √a) clamp.

One edge case remains: for exactly antipodal points, atan2 gets (0, 0) and may return NaN. Handle it: if distance would be NaN, set d = π · R.

Earth radius

Earth is an oblate spheroid, not a perfect sphere. WGS-84 defines:

RadiusValue (km)Notes
Equatorial6378.137Semi-major axis
Polar6356.752Semi-minor axis
Mean6371.009The value most implementations use

Because of this ellipsoidal shape, haversine on a spherical Earth has a maximum error of about 0.5% relative to the true ellipsoidal distance. For most applications that’s plenty; for surveying or aerospace work, see “when to upgrade” below.

Python, a clean reference implementation

Using only the standard library:

import math


EARTH_RADIUS_KM = 6371.0088


def haversine_km(lat1, lon1, lat2, lon2):
    """Great-circle distance between two lat/lon points in kilometers."""
    phi1 = math.radians(lat1)
    phi2 = math.radians(lat2)
    dphi = math.radians(lat2, lat1)
    dlam = math.radians(lon2, lon1)


    a = (math.sin(dphi / 2) ** 2
         + math.cos(phi1) * math.cos(phi2) * math.sin(dlam / 2) ** 2)
    c = 2 * math.atan2(math.sqrt(a), math.sqrt(1, a))
    return EARTH_RADIUS_KM * c

Vectorized with NumPy for batches of coordinates:

import numpy as np


def haversine_km_np(lat1, lon1, lat2, lon2):
    """Vectorized haversine. All args can be scalars or arrays."""
    phi1 = np.radians(lat1)
    phi2 = np.radians(lat2)
    dphi = np.radians(lat2, lat1)
    dlam = np.radians(lon2, lon1)


    a = (np.sin(dphi / 2) ** 2
         + np.cos(phi1) * np.cos(phi2) * np.sin(dlam / 2) ** 2)
    return 6371.0088 * 2 * np.arctan2(np.sqrt(a), np.sqrt(1, a))

Using geopy (spherical / haversine):

from geopy.distance import great_circle
great_circle((41.49, -71.31), (41.50, -81.70)).km

Using geopy (ellipsoidal / Karney, more accurate):

from geopy.distance import distance   # alias for geodesic
distance((41.49, -71.31), (41.50, -81.70)).km

The haversine package on PyPI is another option; it supports km, miles, nautical miles, and feet.

SQL, one-liner distance queries

PostGIS on the geography type uses Karney’s algorithm (ellipsoidal) under the hood, more accurate than pure haversine:

SELECT ST_Distance(
  'SRID=4326;POINT(-71.31 41.49)'::geography,
  'SRID=4326;POINT(-81.70 41.50)'::geography
) AS meters;

For a faster, spherical-only answer, use ST_DistanceSphere:

SELECT ST_DistanceSphere(
  ST_MakePoint(-71.31, 41.49),
  ST_MakePoint(-81.70, 41.50)
) AS meters;

MySQL 8+:

SELECT ST_Distance_Sphere(
  POINT(-71.31, 41.49),
  POINT(-81.70, 41.50)
) AS meters;

Argument-order gotcha: Both PostGIS ST_MakePoint and MySQL POINT() use (longitude, latitude), the reverse of the usual (lat, lon) convention. Mixing them up silently swaps your coordinates and returns plausible-looking wrong numbers.

Raw SQL (no spatial extension), in case you’re on vanilla SQLite or similar:

SELECT 6371 * 2 * asin(sqrt(
    power(sin(radians(lat2, lat1) / 2), 2)
    + cos(radians(lat1)) * cos(radians(lat2))
      * power(sin(radians(lon2, lon1) / 2), 2)
)) AS km
FROM points;

Note this uses the asin form, add a clamp to avoid domain errors on near-antipodal points: asin(least(1, sqrt(...))).

JavaScript

Using Turf.js:

import distance from '@turf/distance';
const d = distance([lon1, lat1], [lon2, lat2], { units: 'kilometers' });

Or a vanilla implementation:

const R = 6371.0088;   // km


function haversineKm(lat1, lon1, lat2, lon2) {
  const toRad = (deg) => (deg * Math.PI) / 180;
  const phi1 = toRad(lat1);
  const phi2 = toRad(lat2);
  const dphi = toRad(lat2, lat1);
  const dlam = toRad(lon2, lon1);


  const a = Math.sin(dphi / 2) ** 2
          + Math.cos(phi1) * Math.cos(phi2) * Math.sin(dlam / 2) ** 2;
  return R * 2 * Math.atan2(Math.sqrt(a), Math.sqrt(1, a));
}

When haversine is enough, and when to upgrade

Quick decision table:

Use caseAccuracy neededPick
Map display, “miles from me”±1%Haversine
Proximity search, geofencing±1%Haversine
Ride-hailing ETA±1%Haversine + road-graph correction
Logistics, VRP distance matrix±0.5%Haversine (or actual road time)
Surveying / cadastralsub-meterVincenty or Karney (GeographicLib)
Aerospacesub-meterKarney

Vincenty’s formulae solve the geodesic on the WGS-84 ellipsoid directly, ~0.5 mm accuracy, but the inverse iteration can fail to converge for near-antipodal point pairs.

Karney’s algorithm (GeographicLib) reformulates geodesic inversion as 1-D root-finding; converges for every input pair; accurate to about 12 nanometers with doubles. This is what geopy.distance.geodesic uses, and what PostGIS geography-type distance uses under the hood.

Gotchas

  • Radians vs. degrees. All trig functions want radians. Lat/lon from databases, JSON, and users is almost always in degrees. Converting is the #1 bug.
  • Longitude wrap-around at ±180°. Points near the International Date Line can have longitudes 179 and -179. The arithmetic difference is 358° but the actual angular separation is 2°. Haversine handles this correctly because sin²(Δλ/2) is periodic in Δλ - but any hand-rolled code that short-circuits on |lon2, lon1| > 90 or similar will break.
  • Bearing ≠ distance. Haversine gives only distance. Initial bearing (forward azimuth) uses a different formula.
  • Argument order in spatial SQL. (lon, lat) in PostGIS / MySQL point constructors.
  • Floating-point at sub-meter distances. Double precision loses significant digits in sin²(Δφ/2) below a few centimeters. Use Karney for short-range precision work.
  • Antipodal inputs. If your inputs can be nearly antipodal, handle the NaN-edge case explicitly.

A sanity check

Distance from New York City (40.7128, -74.0060) to London (51.5074, -0.1278):

  • Haversine (R = 6371.0088): ~5570 km
  • Karney (WGS-84 geodesic): ~5585 km

Error ~0.3%, typical for long-range measurements.

References

Why the name "Here Be Dragons"?