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.test.mjs

@@ -117,3 +117,33 @@ test('a named place centers on its own point, not the middle of its bounds', asy
   const box = await resolveArea({ bbox: [0, 0, 10, 10] });
   assert.deepEqual(areaCenter(box), { lat: 5, lon: 5 });
 });
+
+test('radius areas include every point within the radius, near the poles too', async () => {
+  const polar = await resolveArea({ lat: 80, lon: 0, radius_km: 1500 });
+  const across = { lat: 89, lon: 180 };
+  assert.ok(distanceKm({ lat: 80, lon: 0 }, across) < 1500);
+  assert.ok(areaContains(polar, across));
+  assert.equal(polar.west, -180);
+  assert.equal(polar.east, 180);
+  assert.equal(polar.north, 90);
+  // Away from the poles, sample the circle's edge and check each point is kept.
+  const center = { lat: 60, lon: 20 };
+  const wide = await resolveArea({ ...center, radius_km: 2000 });
+  assert.ok(wide.east - wide.west < 360);
+  for (let bearing = 0; bearing < 360; bearing += 5) {
+    const d = 1999 / 6371.0088;
+    const b = (bearing * Math.PI) / 180;
+    const lat1 = (center.lat * Math.PI) / 180;
+    const lat2 = Math.asin(
+      Math.sin(lat1) * Math.cos(d) + Math.cos(lat1) * Math.sin(d) * Math.cos(b),
+    );
+    const lon2 =
+      (center.lon * Math.PI) / 180 +
+      Math.atan2(
+        Math.sin(b) * Math.sin(d) * Math.cos(lat1),
+        Math.cos(d) - Math.sin(lat1) * Math.sin(lat2),
+      );
+    const point = { lat: (lat2 * 180) / Math.PI, lon: (lon2 * 180) / Math.PI };
+    assert.ok(areaContains(wide, point), `bearing ${bearing}`);
+  }
+});