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Haversine Formula

The Haversine formula calculates the distance between two points on the Earth's surface taking into account its spherical shape.

Key Points

• The Earth is approximately spherical.
• The formula computes the great-circle distance — the shortest path between two points on a sphere.
• Latitude and longitude coordinates are used.

Formulas

R = 6371 km (Earth's radius)
Δlat = lat2 - lat1
Δlon = lon2 - lon1
a = sin²(Δlat/2) + cos(lat1) * cos(lat2) * sin²(Δlon/2)
c = 2 * arctan2(√a, √(1-a))
d = R * c
    

Python Example

import math

def haversine(lat1, lon1, lat2, lon2):
    R = 6371  # Earth's radius in kilometers
    # Convert degrees to radians
    lat1, lon1, lat2, lon2 = map(math.radians, [lat1, lon1, lat2, lon2])

    dlat = lat2 - lat1
    dlon = lon2 - lon1

    a = math.sin(dlat / 2) ** 2 + math.cos(lat1) * math.cos(lat2) * math.sin(dlon / 2) ** 2
    c = 2 * math.atan2(math.sqrt(a), math.sqrt(1 - a))

    distance = R * c
    return distance


# Example usage:
berlin = (52.52, 13.4)
paris = (48.85, 2.35)

dist = haversine(berlin[0], berlin[1], paris[0], paris[1])
print(f"Distance between Berlin and Paris: {dist:.2f} km")

# Distance between Berlin and Paris: 877.68 km