Bounding boxes and spatial queries

Spatial proximity queries — "find all restaurants within 2 km of this point" — are one of the most common operations in location-aware apps. Running Haversine against every row in a million-record database is too slow. The solution is a two-step pattern: use a bounding box to filter cheaply, then Haversine to filter precisely.

Bounding box from a point and radius

A common task: "find all venues within X km of me." Database indexes can't easily do "within X km on a sphere," but they handle "lat between A and B AND lon between C and D" instantly. So we compute a bounding box, query the box, then filter precisely.

$$\Delta\varphi = \frac{d}{R}$$ $$\Delta\lambda = \frac{d}{R \cdot \cos\varphi}$$ $$\text{minLat} = \varphi - \Delta\varphi, \quad \text{maxLat} = \varphi + \Delta\varphi$$ $$\text{minLon} = \lambda - \Delta\lambda, \quad \text{maxLon} = \lambda + \Delta\lambda$$

(All angles in radians, then convert back to degrees for the query.)

The Δφ formula converts a metre distance to a latitude offset by dividing by Earth's radius — this works because latitude lines are evenly spaced. The Δλ formula includes the cos(φ) correction because longitude lines narrow as you move toward the poles: the same angular step represents fewer metres at high latitudes.

Edge cases: near the poles, $\cos\varphi$ is tiny so $\Delta\lambda$ blows up — use $[-180°, 180°]$ for longitude. Near the antimeridian (±180°), the box wraps; you may need two queries.

Worked example

Q: Bounding box for a 5 km radius around Lahore (31.5204, 74.3587).

$$d = 5000 \text{ m}, \quad R = 6{,}371{,}008.8$$ $$\Delta\varphi = 5000 / 6{,}371{,}008.8 = 0.000785 \text{ rad} \approx 0.04497°$$ $$\Delta\lambda = 0.000785 / \cos(0.55013) = 0.000921 \text{ rad} \approx 0.05279°$$

Box: lat $[31.4754, 31.5654]$, lon $[74.3059, 74.4115]$

function boundingBox(lat, lon, radiusMeters) {
  const R = 6371008.8;
  const φ = lat * Math.PI / 180;

  const Δφ = (radiusMeters / R) * 180 / Math.PI;
  const Δλ = (radiusMeters / (R * Math.cos(φ))) * 180 / Math.PI;

  return {
    minLat: lat - Δφ,
    maxLat: lat + Δφ,
    minLon: lon - Δλ,
    maxLon: lon + Δλ,
  };
}