Haversine formula to find distance between two points on a sphere

The Haversine formula calculates the shortest distance between two points on a sphere using their latitudes and longitudes measured along the surface. It is important for use in navigation. The haversine can be expressed in trignometric function as:
$haversine(\theta)=sin^2\Big(\frac{\theta}{2}\Big)$

The haversine of the central angle (which is d/r) is calculated by the following formula:

$\largehaversine\Big(\frac{d}{r}\Big)=haversine(\Phi_2-\Phi_1)+ cos(\Phi_1)cos(\Phi_2)haversine(\lambda_2-\lambda_1)$
where r is the radius of earth(6371 km), d is the distance between two points, $\phi_1, \phi_2$ is latitude of the two points and $\lambda_1, \lambda_2$ is longitude of the two points respectively.

Solving d by applying the inverse haversine or by using the inverse sine function, we get:

$d = r hav^{-1}(h) = 2r sin^{-1}(\sqrt{h})$

$d = 2r sin^{-1}\bigg(\sqrt{sin^2\Big(\frac{\Phi_2-\Phi_1}{2}\Big)+cos(\Phi_1)cos(\Phi_2)sin^2\Big(\frac{\lambda_2-\lambda_1}{2}\Big)}\ \bigg)$

The distance between Big Ben in London (51.5007° N, 0.1246° W) and The Statue of Liberty in
New York (40.6892° N, 74.0445° W) is 5574.8 km. This is not the exact measurement because the
formula assumes that the Earth is a perfect sphere when in fact it is an oblate spheroid.

Below is the implementation of the above formulae:

C++

// C++ program for the haversine formula 
// C++ program for the 
// haversine formula 
#include <iostream> 
#include <cmath> 
using namespace std; 
  
static double haversine(double lat1, double lon1, 
                        double lat2, double lon2) 
    { 
        // distance between latitudes 
        // and longitudes 
        double dLat = (lat2 - lat1) * 
                      M_PI / 180.0; 
        double dLon = (lon2 - lon1) *  
                      M_PI / 180.0; 
  
        // convert to radians 
        lat1 = (lat1) * M_PI / 180.0; 
        lat2 = (lat2) * M_PI / 180.0; 
  
        // apply formulae 
        double a = pow(sin(dLat / 2), 2) +  
                   pow(sin(dLon / 2), 2) *  
                   cos(lat1) * cos(lat2); 
        double rad = 6371; 
        double c = 2 * asin(sqrt(a)); 
        return rad * c; 
    } 
  
// Driver code 
int main() 
{ 
    double lat1 = 51.5007; 
    double lon1 = 0.1246; 
    double lat2 = 40.6892; 
    double lon2 = 74.0445; 
      
    cout << haversine(lat1, lon1, 
                      lat2, lon2) << " K.M."; 
    return 0; 
} 
  
// This code is contributed 
// by Mahadev. 

Java

// Java program for the haversine formula 
public class Haversine { 
  
    static double haversine(double lat1, double lon1, 
                            double lat2, double lon2) 
    { 
        // distance between latitudes and longitudes 
        double dLat = Math.toRadians(lat2 - lat1); 
        double dLon = Math.toRadians(lon2 - lon1); 
  
        // convert to radians 
        lat1 = Math.toRadians(lat1); 
        lat2 = Math.toRadians(lat2); 
  
        // apply formulae 
        double a = Math.pow(Math.sin(dLat / 2), 2) +  
                   Math.pow(Math.sin(dLon / 2), 2) *  
                   Math.cos(lat1) *  
                   Math.cos(lat2); 
        double rad = 6371; 
        double c = 2 * Math.asin(Math.sqrt(a)); 
        return rad * c; 
    } 
  
    // Driver Code 
    public static void main(String[] args) 
    { 
        double lat1 = 51.5007; 
        double lon1 = 0.1246; 
        double lat2 = 40.6892; 
        double lon2 = 74.0445; 
        System.out.println(haversine(lat1, lon1, lat2, lon2) + " K.M."); 
    } 
} 

Python 3

# Python 3 program for the  
# haversine formula 
import math 
  
# Python 3 program for the 
# haversine formula 
def haversine(lat1, lon1, lat2, lon2): 
      
    # distance between latitudes 
    # and longitudes 
    dLat = (lat2 - lat1) * math.pi / 180.0
    dLon = (lon2 - lon1) * math.pi / 180.0
  
    # convert to radians 
    lat1 = (lat1) * math.pi / 180.0
    lat2 = (lat2) * math.pi / 180.0
  
    # apply formulae 
    a = (pow(math.sin(dLat / 2), 2) + 
         pow(math.sin(dLon / 2), 2) * 
             math.cos(lat1) * math.cos(lat2)); 
    rad = 6371
    c = 2 * math.asin(math.sqrt(a)) 
    return rad * c 
  
# Driver code 
if __name__ == "__main__": 
    lat1 = 51.5007
    lon1 = 0.1246
    lat2 = 40.6892
    lon2 = 74.0445
      
    print(haversine(lat1, lon1,lat2, lon2), "K.M.") 
  
# This code is contributed  
# by ChitraNayal 

C#

// C# program for the haversine formula 
using System; 
class GFG 
{ 
  
static double haversine(double lat1, double lon1, 
                        double lat2, double lon2) 
{ 
    // distance between latitudes and longitudes 
    double dLat = (Math.PI / 180) * (lat2 - lat1); 
    double dLon = (Math.PI / 180) * (lon2 - lon1); 
  
    // convert to radians 
    lat1 = (Math.PI / 180) * (lat1); 
    lat2 = (Math.PI / 180) * (lat2); 
  
    // apply formulae 
    double a = Math.Pow(Math.Sin(dLat / 2), 2) +  
               Math.Pow(Math.Sin(dLon / 2), 2) *  
               Math.Cos(lat1) * Math.Cos(lat2); 
    double rad = 6371; 
    double c = 2 * Math.Asin(Math.Sqrt(a)); 
    return rad * c; 
} 
  
// Driver Code 
public static void Main() 
{ 
    double lat1 = 51.5007; 
    double lon1 = 0.1246; 
    double lat2 = 40.6892; 
    double lon2 = 74.0445; 
    Console.WriteLine(haversine(lat1, lon1,  
                                lat2, lon2) + " K.M."); 
} 
} 
  
// This code is contributed  
// by Akanksha Rai(Abby_akku) 

PHP

<?php 
// PHP program for the 
// haversine formula 
  
function haversine($lat1, $lon1, 
                   $lat2, $lon2) 
{ 
    // distance between latitudes 
    // and longitudes 
    $dLat = ($lat2 - $lat1) * 
                M_PI / 180.0; 
    $dLon = ($lon2 - $lon1) *  
                M_PI / 180.0; 
  
    // convert to radians 
    $lat1 = ($lat1) * M_PI / 180.0; 
    $lat2 = ($lat2) * M_PI / 180.0; 
  
    // apply formulae 
    $a = pow(sin($dLat / 2), 2) +  
         pow(sin($dLon / 2), 2) *  
             cos($lat1) * cos($lat2); 
    $rad = 6371; 
    $c = 2 * asin(sqrt($a)); 
    return $rad * $c; 
} 
  
// Driver code 
$lat1 = 51.5007; 
$lon1 = 0.1246; 
$lat2 = 40.6892; 
$lon2 = 74.0445; 
  
echo haversine($lat1, $lon1, 
               $lat2, $lon2) .  
                      " K.M."; 
                  
// This code is contributed 
// by Akanksha Rai(Abby_akku) 
?> 

Output:

5574.840456848555 K.M.

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My Personal Notes

C++

Java

Python 3

C#

PHP

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