diff --git a/docs/CURRENT-STATE.md b/docs/CURRENT-STATE.md
index bcb1de592..63041c29a 100644
--- a/docs/CURRENT-STATE.md
+++ b/docs/CURRENT-STATE.md
@@ -1709,8 +1709,8 @@ time. TRACK ACQUIRED stays first. Then:
 - **Sites**: the Mapped Installations layer's loaded records around the
   sub-satellite point (`getNearby`, asked only as far as the reach extends
   over the ground, and never past 3,000 km, so from a high orbit only the
-  sites within 3,000 km of the point below are listed) that face the satellite and lie within the off-nadir
-  reach, nearest to nadir first, the focused site always among them. A row
+  sites within 3,000 km of the point below are listed) that face the
+  satellite and lie within the off-nadir reach, nearest to nadir first, the focused site always among them. A row
   reads the site's name over its class and off-nadir angle; its button
   (`Keep the camera on …`) focuses the site, and on the focused site's row,
   marked as the subject is (`FOCUS · …`, `Release the camera from …`), ends
diff --git a/src/cockpitOrbitalSatellites.js b/src/cockpitOrbitalSatellites.js
index b52d4db5c..343530f42 100644
--- a/src/cockpitOrbitalSatellites.js
+++ b/src/cockpitOrbitalSatellites.js
@@ -22,8 +22,10 @@ function finiteVector(v) {
 /**
  * Where the camera is part-way through a hop: along the great circle
  * between the two positions (never through the Earth), its distance from the
- * centre interpolated linearly. Opposite ends turn about a fixed axis
- * square to them, so even then the path stays off the Earth.
+ * centre interpolated linearly. The circle is the one through both ends, so
+ * the path moves smoothly as the destination does, however far round it
+ * lies; only exactly opposite ends, which no single circle joins, turn about
+ * a fixed axis square to them.
  * @param {{x: number, y: number, z: number}} from
  * @param {{x: number, y: number, z: number}} to
  * @param {number} progress Eased, 0..1.
@@ -37,39 +39,36 @@ export function hopPosition(from, to, progress) {
   if (!(ra > 0) || !(rb > 0)) return null;
   const a = { x: from.x / ra, y: from.y / ra, z: from.z / ra };
   const b = { x: to.x / rb, y: to.y / rb, z: to.z / rb };
-  const cosine = Math.min(1, Math.max(-1, a.x * b.x + a.y * b.y + a.z * b.z));
-  const omega = Math.acos(cosine);
+  const cosine = a.x * b.x + a.y * b.y + a.z * b.z;
+  // The direction the circle leaves `a` in: `b` less its part along `a`.
+  let c = {
+    x: b.x - cosine * a.x,
+    y: b.y - cosine * a.y,
+    z: b.z - cosine * a.z,
+  };
+  let rc = Math.hypot(c.x, c.y, c.z);
   const r = ra + (rb - ra) * p;
-  if (Math.PI - omega < 1e-6) {
-    // No single great circle joins opposite points: turn about the axis
-    // square to `a` that is least aligned with the Earth's.
+  const omega = Math.atan2(rc, cosine);
+  if (!(rc > 1e-12)) {
+    if (cosine > 0) return { x: a.x * r, y: a.y * r, z: a.z * r };
+    // Opposite ends: set off square to `a`, northward (or, near a pole,
+    // toward the prime meridian).
     const pole =
       Math.abs(a.z) < 0.9 ? { x: 0, y: 0, z: 1 } : { x: 1, y: 0, z: 0 };
     const along = pole.x * a.x + pole.y * a.y + pole.z * a.z;
-    const c = {
+    c = {
       x: pole.x - along * a.x,
       y: pole.y - along * a.y,
       z: pole.z - along * a.z,
     };
-    const rc = Math.hypot(c.x, c.y, c.z);
-    const cos = Math.cos(p * Math.PI);
-    const sin = Math.sin(p * Math.PI);
-    return {
-      x: (a.x * cos + (c.x / rc) * sin) * r,
-      y: (a.y * cos + (c.y / rc) * sin) * r,
-      z: (a.z * cos + (c.z / rc) * sin) * r,
-    };
-  }
-  let ka = 1 - p;
-  let kb = p;
-  if (omega > 1e-9) {
-    const sine = Math.sin(omega);
-    ka = Math.sin((1 - p) * omega) / sine;
-    kb = Math.sin(p * omega) / sine;
+    rc = Math.hypot(c.x, c.y, c.z);
   }
+  const angle = p * omega;
+  const cos = Math.cos(angle);
+  const sin = Math.sin(angle) / rc;
   return {
-    x: (a.x * ka + b.x * kb) * r,
-    y: (a.y * ka + b.y * kb) * r,
-    z: (a.z * ka + b.z * kb) * r,
+    x: (a.x * cos + c.x * sin) * r,
+    y: (a.y * cos + c.y * sin) * r,
+    z: (a.z * cos + c.z * sin) * r,
   };
 }
diff --git a/src/cockpitOrbitalSatellites.test.mjs b/src/cockpitOrbitalSatellites.test.mjs
index 3a4902b11..60156b89e 100644
--- a/src/cockpitOrbitalSatellites.test.mjs
+++ b/src/cockpitOrbitalSatellites.test.mjs
@@ -33,9 +33,41 @@ test('a hop to the opposite side goes round the Earth, not through it', () => {
   const radius = Cesium.Cartesian3.magnitude(from);
   for (const p of [0.25, 0.5, 0.75]) {
     const point = hopPosition(from, to, p);
-    assert.ok(Math.abs(Math.hypot(point.x, point.y, point.z) - radius) < 1e-3, `p=${p}`);
+    assert.ok(
+      Math.abs(Math.hypot(point.x, point.y, point.z) - radius) < 1e-3,
+      `p=${p}`,
+    );
   }
   const end = hopPosition(from, to, 1);
   for (const axis of ['x', 'y', 'z'])
     assert.ok(Math.abs(end[axis] - to[axis]) < 1e-3, axis);
 });
+
+test('a hop to nearly the opposite side moves smoothly as the destination does', () => {
+  const r = 6_778_137;
+  const from = { x: r, y: 0, z: 0 };
+  const near = (delta) => ({
+    x: -r * Math.cos(delta),
+    y: r * Math.sin(delta),
+    z: 0,
+  });
+  const mid = (delta) => hopPosition(from, near(delta), 0.5);
+  const [a, b] = [mid(0.999e-6), mid(1.001e-6)];
+  assert.ok(
+    Math.hypot(a.x - b.x, a.y - b.y, a.z - b.z) < 1,
+    'no jump between neighbours',
+  );
+  for (const delta of [1e-3, 1e-6, 1e-9]) {
+    const to = near(delta);
+    const end = hopPosition(from, to, 1);
+    assert.ok(
+      Math.hypot(end.x - to.x, end.y - to.y, end.z - to.z) < 1e-3,
+      `lands at ${delta}`,
+    );
+    const half = mid(delta);
+    assert.ok(
+      Math.abs(Math.hypot(half.x, half.y, half.z) - r) < 1e-3,
+      `round at ${delta}`,
+    );
+  }
+});