Fórmula Haversine para encontrar la distancia entre dos puntos en una esfera

Posted on by Rudeus Greyrat

La fórmula de Haversine calcula la distancia más corta entre dos puntos en una esfera utilizando sus latitudes y longitudes medidas a lo largo de la superficie. Es importante para su uso en la navegación. El haversine se puede expresar en función trigonométrica como: 
haversine(\theta)=sin^2\Big(\frac{\theta}{2}\Big)
El haversine del ángulo central (que es d/r) se calcula mediante la siguiente fórmula:
\largehaversine\Big(\frac{d}{r}\Big)=haversine(\Phi_2-\Phi_1)+ cos(\Phi_1)cos(\Phi_2)haversine(\lambda_2-\lambda_1)
donde r es el radio de la tierra (6371 km), d es la distancia entre dos puntos ,  \phi_1, \phi_2 es la latitud de los dos puntos, y  \lambda_1, \lambda_2 es la longitud de los dos puntos respectivamente.
Resolviendo d aplicando el haversine inverso o usando la función seno inversa, obtenemos: 
 d = r hav^{-1}(h) = 2r sin^{-1}(\sqrt{h})

o

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)
?>

Javascript

<script>
 
// Javascript program for the haversine formula
     
    function haversine(lat1, lon1, lat2, lon2)
    {
        // distance between latitudes
        // and longitudes
        let dLat = (lat2 - lat1) * Math.PI / 180.0;
        let dLon = (lon2 - lon1) * Math.PI / 180.0;
           
        // convert to radiansa
        lat1 = (lat1) * Math.PI / 180.0;
        lat2 = (lat2) * Math.PI / 180.0;
         
        // apply formulae
        let a = Math.pow(Math.sin(dLat / 2), 2) +
                   Math.pow(Math.sin(dLon / 2), 2) *
                   Math.cos(lat1) *
                   Math.cos(lat2);
        let rad = 6371;
        let c = 2 * Math.asin(Math.sqrt(a));
        return rad * c;
         
    }
     // Driver Code
     let lat1 = 51.5007;
     let lon1 = 0.1246;
     let lat2 = 40.6892;
     let lon2 = 74.0445;
     document.write(haversine(lat1, lon1, lat2, lon2) + " K.M.");
 
 
// This code is contributed by avanitrachhadiya2155
</script>
Producción: 

5574.840456848555 K.M.

 

Complejidad de Tiempo : O(1)
Espacio Auxiliar : O(1)

Publicación traducida automáticamente

Artículo escrito por prakhar7 y traducido por Barcelona Geeks. The original can be accessed here. Licence: CCBY-SA

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