Destination points — moving on a sphere

The inverse of distance/bearing: given a starting point, a bearing, and a distance, find the destination on the sphere. This operation is called dead reckoning in navigation — it's how ships historically computed position before GPS. In modern apps it's used for geofence construction, range rings, path simulation, and anywhere you need to place a point at a known offset from another.

Try it live before diving into the derivation.

The formula

Let $\delta = d / R$ be the angular distance in radians. This normalises the linear distance by Earth's radius to get the central angle subtended by the arc.

$$\varphi_2 = \arcsin\!\big(\sin\varphi_1 \cos\delta + \cos\varphi_1 \sin\delta \cos\theta\big)$$ $$\lambda_2 = \lambda_1 + \arctan2\!\big(\sin\theta \sin\delta \cos\varphi_1,\; \cos\delta - \sin\varphi_1 \sin\varphi_2\big)$$

The first equation derives the new latitude by rotating the starting point through angle δ in the direction θ. The second computes the longitude offset using arctan2 to handle the full 360° range correctly.

function destination(lat1, lon1, bearingDeg, distanceMeters) {
  const R = 6371008.8;
  const δ = distanceMeters / R;
  const θ = bearingDeg * Math.PI / 180;
  const φ1 = lat1 * Math.PI / 180;
  const λ1 = lon1 * Math.PI / 180;

  const φ2 = Math.asin(
    Math.sin(φ1) * Math.cos(δ) +
    Math.cos(φ1) * Math.sin(δ) * Math.cos(θ)
  );
  const λ2 = λ1 + Math.atan2(
    Math.sin(θ) * Math.sin(δ) * Math.cos(φ1),
    Math.cos(δ) - Math.sin(φ1) * Math.sin(φ2)
  );

  return {
    lat: φ2 * 180 / Math.PI,
    lon: ((λ2 * 180 / Math.PI) + 540) % 360 - 180   // normalize to [-180, 180]
  };
}