diff --git a/src/layers/awareness/rangeRing.test.mjs b/src/layers/awareness/rangeRing.test.mjs
index 074d449fe..b17651090 100644
--- a/src/layers/awareness/rangeRing.test.mjs
+++ b/src/layers/awareness/rangeRing.test.mjs
@@ -81,6 +81,41 @@ for (const [place, lat, lon] of [
   });
 }
 
+for (const [place, lat, lon] of [
+  ['the North Pole', 90, 0],
+  ['the South Pole', -90, 0],
+  ['1e-7° from the North Pole', 90 - 1e-7, 45],
+  ['1e-8° from the South Pole', -90 + 1e-8, -120],
+]) {
+  test(`the ring stays a full circle at ${place}`, () => {
+    const center = Cesium.Cartesian3.fromDegrees(lon, lat, 0);
+    const segments = AWARENESS_RANGE_RING_STYLE.segments;
+    const positions = awarenessRangeRingPositions(
+      center,
+      AWARENESS_RADIUS_M,
+      CONTEXT_RIM_HEIGHT_M,
+    );
+    assert.equal(positions.length, segments + 1);
+    assert.ok(
+      Cesium.Cartesian3.equalsEpsilon(positions[0], positions.at(-1), 0, 1e-6),
+      'the ring closes on itself',
+    );
+    const distinct = new Set(
+      positions
+        .slice(0, segments)
+        .map(({ x, y, z }) => [x, y, z].map((v) => v.toFixed(1)).join()),
+    );
+    assert.equal(distinct.size, segments, 'every vertex is its own point');
+    // Ordinary chords are 2π × 250 km / 128 ≈ 12.3 km long.
+    for (let i = 0; i < segments; i++) {
+      const chord = Cesium.Cartesian3.distance(positions[i], positions[i + 1]);
+      assert.ok(chord > 11_500 && chord < 13_000, `chord ${i}: ${chord} m`);
+    }
+    for (const distance of surfaceDistances(center, positions))
+      assert.ok(Math.abs(distance - AWARENESS_RADIUS_M) < 100, `${distance} m`);
+  });
+}
+
 test('the ring core and halo contrast on light and dark ground', () => {
   const core = Cesium.Color.fromCssColorString(
     AWARENESS_RANGE_RING_STYLE.color,
diff --git a/src/layers/awareness/rendering.js b/src/layers/awareness/rendering.js
index 8cee87d80..36d920850 100644
--- a/src/layers/awareness/rendering.js
+++ b/src/layers/awareness/rendering.js
@@ -40,9 +40,17 @@ export function awarenessRangeRingPositions(
   const e2 = 1 - (b * b) / (a * a);
   const sinLat = Math.sin(latitude);
   const cosLat = Math.cos(latitude);
+  const sinLon = Math.sin(longitude);
+  const cosLon = Math.cos(longitude);
   const w = 1 - e2 * sinLat * sinLat;
   const meridional = (a * (1 - e2)) / (w * Math.sqrt(w));
   const primeVertical = a / Math.sqrt(w);
+  // Step along the great circle in 3D from the centre's up, north and east
+  // unit vectors. Unlike the latitude/longitude destination formula, this
+  // stays well conditioned at and next to the poles.
+  const up = [cosLat * cosLon, cosLat * sinLon, sinLat];
+  const north = [-sinLat * cosLon, -sinLat * sinLon, cosLat];
+  const east = [-sinLon, cosLon, 0];
   const positions = [];
   for (let i = 0; i <= segments; i++) {
     const bearing = ((i % segments) / segments) * Cesium.Math.TWO_PI;
@@ -50,15 +58,13 @@ export function awarenessRangeRingPositions(
     const sinB = Math.sin(bearing);
     const angular =
       radiusM * ((cosB * cosB) / meridional + (sinB * sinB) / primeVertical);
-    const lat = Math.asin(
-      sinLat * Math.cos(angular) + cosLat * Math.sin(angular) * cosB,
+    const cosA = Math.cos(angular);
+    const sinA = Math.sin(angular);
+    const [x, y, z] = up.map(
+      (u, k) => u * cosA + (north[k] * cosB + east[k] * sinB) * sinA,
     );
-    const lon =
-      longitude +
-      Math.atan2(
-        sinB * Math.sin(angular) * cosLat,
-        Math.cos(angular) - sinLat * Math.sin(lat),
-      );
+    const lat = Math.atan2(z, Math.hypot(x, y));
+    const lon = Math.atan2(y, x);
     positions.push(Cesium.Cartesian3.fromRadians(lon, lat, heightM));
   }
   return positions;