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.");
    }
}

Output:

5574.840456848555 K.M.

My Personal Notes

prakhar7

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Improved By : Mahadev99

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