Polygons — area, centroid, and containment

Points and lines cover most location queries, but the moment you need to answer "is this delivery address inside our service zone?" or "what's the area of this selected region?" you need polygon math. This chapter covers three fundamental operations: containment testing, area calculation, and centroid finding.

Point-in-polygon (ray casting)

Cast a horizontal ray from the test point and count how many times it crosses polygon edges. Odd = inside, even = outside. Click anywhere in the diagram to test a point.

The algorithm works for any simple (non-self-intersecting) polygon, including concave shapes. It can fail for points exactly on an edge — if that's a concern, add a small epsilon check.

function pointInPolygon(point, polygon) {
  const [px, py] = point;
  let inside = false;
  for (let i = 0, j = polygon.length - 1; i < polygon.length; j = i++) {
    const [xi, yi] = polygon[i];
    const [xj, yj] = polygon[j];
    const intersect = ((yi > py) !== (yj > py)) &&
                      (px < (xj - xi) * (py - yi) / (yj - yi) + xi);
    if (intersect) inside = !inside;
  }
  return inside;
}

For small polygons, treating lon/lat as planar (x, y) is fine. For polygons spanning continents, project first.

Polygon area on a plane (Shoelace)

For projected coordinates:

$$A = \tfrac{1}{2} \left| \sum_{i} (x_i \cdot y_{i+1} - x_{i+1} \cdot y_i) \right|$$

(indices wrap around)

Spherical polygon area

For lat/lon polygons, Shoelace is wrong. The Shoelace formula assumes a flat plane with equal-area spacing — but a 1° × 1° box at the equator is ~12,300 km², while the same angular box at 80° N is only ~210 km². Applying Shoelace to raw lat/lon coordinates produces wildly incorrect areas for any polygon that spans more than a few degrees.

Use the spherical excess formula instead:

$$A = \frac{R^2}{2} \left| \sum_{i} (\lambda_{i+1} - \lambda_i) \cdot (2 + \sin\varphi_i + \sin\varphi_{i+1}) \right|$$

This gives area in m² when R is in meters. Vertices must be in radians and ordered consistently (CCW for "outside").