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]
  };
}