fix: bound radius areas by the exact spherical cap

The longitude span divided the angular radius by the center's cosine, which
is too narrow for large radii and near the poles. Areas now use the exact
tangent bound and span every longitude when the circle reaches a pole.

src/tools/area.js

@@ -82,10 +82,18 @@ export async function resolveArea(area, { services, signal } = {}) {
   };
 }
 
+/**
+ * The bounding box of a spherical cap. When the cap reaches a pole it spans
+ * every longitude; otherwise the longitude span is the exact tangent bound,
+ * asin(sin(d) / cos(lat)) for angular radius d.
+ */
 function circleArea(lat, lon, radiusKm) {
-  const latSpan = (radiusKm / EARTH_RADIUS_KM) * (180 / Math.PI);
-  const cosLat = Math.cos((lat * Math.PI) / 180);
-  const lonSpan = cosLat < 1e-6 ? 180 : Math.min(180, latSpan / cosLat);
+  const angular = radiusKm / EARTH_RADIUS_KM;
+  const latSpan = angular * (180 / Math.PI);
+  const reachesPole = lat + latSpan >= 90 || lat - latSpan <= -90;
+  const ratio = Math.sin(angular) / Math.cos((lat * Math.PI) / 180);
+  const lonSpan =
+    reachesPole || ratio >= 1 ? 180 : Math.asin(ratio) * (180 / Math.PI);
   const wrap = (value) => ((((value + 180) % 360) + 360) % 360) - 180;
   return {
     label: `${radiusKm} km around ${lat.toFixed(3)}, ${lon.toFixed(3)}`,