Distance Between Coordinates

Fórmula de Haversine

A fórmula de Haversine calcula a distância em arco (great-circle) sobre uma esfera. É a distância em linha reta entre dois pontos na superfície da Terra — mais curta que percorrer a curvatura, mas representa o caminho mais curto possível voando.

Usa raio médio da Terra de 6371 km. Erro típico de 0.5% para distâncias até 1000 km — suficiente para a maioria dos casos. Para precisão extrema, use Vincenty (que considera o achatamento da Terra).

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Distance between coordinates: Pythagoras and Haversine

For two points on the Cartesian plane, distance is the Pythagorean theorem: d = √((x₂−x₁)² + (y₂−y₁)²). For points on the sphere (Earth, given as latitude/longitude), use the Haversine formula: d = 2R · asin(√(sin²(Δφ/2) + cos φ₁ · cos φ₂ · sin²(Δλ/2))), with R = 6,371 km (mean Earth radius). Haversine has typical error below 0.5% — enough for most uses. The Vincenty formula models Earth as an oblate ellipsoid and is more accurate over long distances, at the cost of complexity. Other metrics include Manhattan distance |x₁−x₂| + |y₁−y₂| (used in grid-based routing) and Chebyshev distance max(|Δx|, |Δy|) (chess king moves). Example: São Paulo (−23.55, −46.63) to Rio (−22.91, −43.17) via Haversine ≈ 360 km in a straight line, while road distance is ~430 km.

Applications: GPS, geofencing and clustering

Used by GPS devices for waypoint distance, geofencing (alert when a vehicle leaves a zone), radius search on Uber/iFood/Rappi ("drivers within 5 km"), k-NN clustering in machine learning, and routing algorithms. Note: door-to-door travel time in Google Maps or Waze uses a street graph with Dijkstra/A*, not raw Haversine — the great-circle distance is only the lower bound. For drone delivery or air routes, Haversine is the right model.

FAQ

Haversine or Vincenty? Haversine is fine for almost everything (~0.5% error, easy to implement). Use Vincenty for surveying, aviation flight planning over thousands of km, or any case where ellipsoidal precision matters.

Why does Google Maps show a longer distance? Maps shows the road distance (graph of streets); this tool shows the great-circle distance "as the crow flies", which is always shorter.

What's the difference between km, miles and nautical miles? 1 nautical mile = 1.852 km = 1.151 statute miles, and equals 1 arc-minute of latitude on Earth. Aviation and shipping use nautical miles.

What is the bearing? The initial compass direction from A to B (0° = north, 90° = east). On long routes following a great circle, the bearing changes along the path.