From 5b73b70d19ac30be19e1f2450111336a7820db73 Mon Sep 17 00:00:00 2001
From: Sameh Khamis <sameh@halfpixel.ai>
Date: Thu, 1 Oct 2026 14:55:43 -0700
Subject: [PATCH] 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       | 14 +++++++++++---
 src/tools/area.test.mjs | 30 ++++++++++++++++++++++++++++++
 2 files changed, 41 insertions(+), 3 deletions(-)

diff --git a/src/tools/area.js b/src/tools/area.js
index e494c6782..41b417482 100644
--- a/src/tools/area.js
+++ b/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)}`,
diff --git a/src/tools/area.test.mjs b/src/tools/area.test.mjs
index 99df85684..d22611282 100644
--- a/src/tools/area.test.mjs
+++ b/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}`);
+  }
+});