## 1. Overview

In this quick tutorial, we’ll implement methods to calculate the distance between two geographical coordinates.

In particular, we’ll start by implementing an approximation of the distance first. Then, we’ll look at the Haversine and Vincenty formulas, which provide more accuracy.

## 2. Equirectangular Distance Approximation

Let’s start by implementing the equirectangular approximation. In detail, since this formula uses the least mathematical operations, it’s very fast:

```
double calculateDistance(double lat1, double lon1, double lat2, double lon2) {
double lat1Rad = Math.toRadians(lat1);
double lat2Rad = Math.toRadians(lat2);
double lon1Rad = Math.toRadians(lon1);
double lon2Rad = Math.toRadians(lon2);
double x = (lon2Rad - lon1Rad) * Math.cos((lat1Rad + lat2Rad) / 2);
double y = (lat2Rad - lat1Rad);
double distance = Math.sqrt(x * x + y * y) * EARTH_RADIUS;
return distance;
}
```

Above, the *EARTH_RADIUS* is a constant equal to 6371, which is a good approximation of the Earth’s radius in kilometers.

Even though it seems a simple formula, **equirectangular approximation isn’t very accurate when calculating long distances**. In fact, it treats the Earth as a perfect sphere and maps the sphere to a rectangular grid.

Next, we’ll take a look at the Haversine formula. Again, it views the Earth as a perfect sphere. Nonetheless, it’s more accurate in calculating the distance between long distances.

Moreover, **the Haversine Formula is based on the spherical law of haversines**:

```
double haversine(double val) {
return Math.pow(Math.sin(val / 2), 2);
}
```

Then, using this helper function, we can implement the method to calculate the distance:

```
double calculateDistance(double startLat, double startLong, double endLat, double endLong) {
double dLat = Math.toRadians((endLat - startLat));
double dLong = Math.toRadians((endLong - startLong));
startLat = Math.toRadians(startLat);
endLat = Math.toRadians(endLat);
double a = haversine(dLat) + Math.cos(startLat) * Math.cos(endLat) * haversine(dLong);
double c = 2 * Math.atan2(Math.sqrt(a), Math.sqrt(1 - a));
return EARTH_RADIUS * c;
}
```

Although it improves the accuracy of the calculation, it still considers the Earth as a flattened shape.

Finally, we must use Vincenty’s formula if we want the highest precision. In detail, **Vincenty’s formula calculates the distance iteratively until the error reaches acceptable values**. Moreover, it also takes into consideration the elliptical shape of the Earth.

To begin with, the formula requires some constants that describe the ellipsoid model of the Earth:

```
double SEMI_MAJOR_AXIS_MT = 6378137;
double SEMI_MINOR_AXIS_MT = 6356752.314245;
double FLATTENING = 1 / 298.257223563;
double ERROR_TOLERANCE = 1e-12;
```

Obviously, the *ERROR_TOLERANCE* represents the error we’re willing to accept. Further, we’ll use these values in Vincenty’s formula:

```
double calculateDistance(double latitude1, double longitude1, double latitude2, double longitude2) {
double U1 = Math.atan((1 - FLATTENING) * Math.tan(Math.toRadians(latitude1)));
double U2 = Math.atan((1 - FLATTENING) * Math.tan(Math.toRadians(latitude2)));
double sinU1 = Math.sin(U1);
double cosU1 = Math.cos(U1);
double sinU2 = Math.sin(U2);
double cosU2 = Math.cos(U2);
double longitudeDifference = Math.toRadians(longitude2 - longitude1);
double previousLongitudeDifference;
double sinSigma, cosSigma, sigma, sinAlpha, cosSqAlpha, cos2SigmaM;
do {
sinSigma = Math.sqrt(Math.pow(cosU2 * Math.sin(longitudeDifference), 2) +
Math.pow(cosU1 * sinU2 - sinU1 * cosU2 * Math.cos(longitudeDifference), 2));
cosSigma = sinU1 * sinU2 + cosU1 * cosU2 * Math.cos(longitudeDifference);
sigma = Math.atan2(sinSigma, cosSigma);
sinAlpha = cosU1 * cosU2 * Math.sin(longitudeDifference) / sinSigma;
cosSqAlpha = 1 - Math.pow(sinAlpha, 2);
cos2SigmaM = cosSigma - 2 * sinU1 * sinU2 / cosSqAlpha;
if (Double.isNaN(cos2SigmaM)) {
cos2SigmaM = 0;
}
previousLongitudeDifference = longitudeDifference;
double C = FLATTENING / 16 * cosSqAlpha * (4 + FLATTENING * (4 - 3 * cosSqAlpha));
longitudeDifference = Math.toRadians(longitude2 - longitude1) + (1 - C) * FLATTENING * sinAlpha *
(sigma + C * sinSigma * (cos2SigmaM + C * cosSigma * (-1 + 2 * Math.pow(cos2SigmaM, 2))));
} while (Math.abs(longitudeDifference - previousLongitudeDifference) > ERROR_TOLERANCE);
double uSq = cosSqAlpha * (Math.pow(SEMI_MAJOR_AXIS_MT, 2) - Math.pow(SEMI_MINOR_AXIS_MT, 2)) / Math.pow(SEMI_MINOR_AXIS_MT, 2);
double A = 1 + uSq / 16384 * (4096 + uSq * (-768 + uSq * (320 - 175 * uSq)));
double B = uSq / 1024 * (256 + uSq * (-128 + uSq * (74 - 47 * uSq)));
double deltaSigma = B * sinSigma * (cos2SigmaM + B / 4 * (cosSigma * (-1 + 2 * Math.pow(cos2SigmaM, 2))
- B / 6 * cos2SigmaM * (-3 + 4 * Math.pow(sinSigma, 2)) * (-3 + 4 * Math.pow(cos2SigmaM, 2))));
double distanceMt = SEMI_MINOR_AXIS_MT * A * (sigma - deltaSigma);
return distanceMt / 1000;
}
```

This formula is compute-heavy. Therefore, we might want to use it when precision is a goal. Otherwise, we’ll stick to the Haversine formula.

## 5. Testing the Accuracy

At last, we can test the accuracy of all the methods we saw above:

```
double lat1 = 40.714268; // New York
double lon1 = -74.005974;
double lat2 = 34.0522; // Los Angeles
double lon2 = -118.2437;
double equirectangularDistance = EquirectangularApproximation.calculateDistance(lat1, lon1, lat2, lon2);
double haversineDistance = HaversineDistance.calculateDistance(lat1, lon1, lat2, lon2);
double vincentyDistance = VincentyDistance.calculateDistance(lat1, lon1, lat2, lon2);
double expectedDistance = 3944;
assertTrue(Math.abs(equirectangularDistance - expectedDistance) < 100);
assertTrue(Math.abs(haversineDistance - expectedDistance) < 10);
assertTrue(Math.abs(vincentyDistance - expectedDistance) < 0.5);
```

Above, we calculated the distance between New York and Los Angeles and then evaluated the accuracy in kilometers.

## 6. Conclusion

In this article, we saw three ways to calculate the distance between two geographical points in Java. We started with the least accurate equirectangular approximation. Then we looked at the more accurate Haversine formula. Finally, we used the most accurate Vincenty’s formula.

As always, the code used in the examples is available over on GitHub.