The hook turns as hard as the geometry needs, or not at all

A slice hooking onto a body two metres to the side flew a closed loop around
it forever: 285 frames of the same circuit, never arriving, never falling.
A constant turn rate has a constant turn radius (v/w = 2 m), and anything
inside that circle cannot be reached however long the ball steers — while
the hook pins vy at 0, so nothing else ends it either. A crowd is nothing
but close bodies, which made the one place this weapon should shine the one
place it hung.

The rate is no longer a constant. An arc leaving along the current heading
and passing through a body at distance d, bearing a, has radius d/(2 sin a),
so the ball needs 2 v sin(a) / d — solved every frame, because the body is
walking while the ball travels. sliceTurn stays as the FLOOR, which is what
it was always really for: how wide the curl looks when there is room.
Turning faster than needed is safe, the aim being re-solved each frame.

Two things the maths cannot cover, so they are ruled out instead: a body
behind the ball, which no tangent arc reaches at any rate (sliceArc), and a
target that walks out of reach mid-hook (sliceHookT, a hard 0.7 s licence to
steer). An endless orbit is now impossible by construction rather than by
the geometry happening to work out.

Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>

config.js

@@ -247,8 +247,19 @@ export const TUNE = {
   sliceHops: 2,     // extra bodies one perfect serve hooks to. 2 => TRIPLE KILL is the ceiling
   sliceR: 6.5,      // m the hook reaches for the next body. Wider reads as the game aiming for you;
                     // much narrower and a spread-out crowd never chains at all
-  sliceTurn: 11,    // rad/s the ball swings onto the next one, and THE knob for how it looks: past
+  sliceTurn: 11,    // rad/s the ball swings onto the next one: a FLOOR, not the rate. It is the knob
-                    // ~30 it snaps like a ricochet, under ~6 it sails wide before finishing the turn
+                    // for how the shot LOOKS when there is room — past ~30 it snaps like a ricochet,
+                    // under ~6 it sails wide. The rate actually used is whatever the geometry needs
+                    // (see the hook), because a fixed rate has a fixed turn radius (v/w = 2 m here)
+                    // and ANY body inside that circle is unreachable: the ball flew a closed loop
+                    // around it forever, never arriving, never falling. A crowd is all close bodies,
+                    // so the one place the weapon should shine was the one place it hung.
+  sliceArc: 120,    // deg: the widest bearing off the ball's heading that can be hooked at all. No
+                    // single tangent arc reaches a body directly BEHIND the ball at any turn rate,
+                    // so past this the chain just ends and it plows on
+  sliceHookT: 0.7,  // s a hook may steer before it lets go. The geometry above is solved fresh every
+                    // frame and still cannot promise an arrival when the target WALKS, so this is the
+                    // backstop that makes an endless orbit impossible by construction, not by maths
   // The schedule calls the DUEL, not the tin: weapons are only ever won, but a duel comes round
   // on its own so there is always a chance at one. Fires off the next hand-cleared wave.
   duelFirst: 2,     // game after which the first scheduled sudden-death duel is called

gameplay.js

@@ -1581,12 +1581,23 @@ export function update(dt) {
     // 1 px, so the ball stays the bright thing and the line reads as the cut, not as a second ball.
     if (ball.hook) {
       const dx = ball.hook.x - ball.x, dz = ball.hook.z - ball.z;
-      if (dx * dx + dz * dz < 0.09) ball.hook = null; // arrived (or it moved) — gravity has it back
+      ball.hook.t -= dt;
+      // arrived, or out of time. The timer is not a polish detail: without it a ball that cannot
+      // close on a walking body steers forever, and with vy pinned at 0 nothing else ever ends it.
+      if (dx * dx + dz * dz < 0.09 || ball.hook.t <= 0) ball.hook = null; // gravity has it back
       else {
         const sp = Math.hypot(ball.vx, ball.vz) || 1;
         const cur = Math.atan2(ball.vx, ball.vz), want = Math.atan2(dx, dz);
         const da = ((want - cur + Math.PI * 3) % (Math.PI * 2)) - Math.PI; // shortest way round
-        const a = cur + Math.sign(da) * Math.min(Math.abs(da), TUNE.sliceTurn * dt);
+        // THE RATE COMES FROM THE GEOMETRY. An arc that starts along the current heading and passes
+        // through a body at distance d, bearing da, has radius d / (2 sin da) — so the ball must be
+        // able to turn at 2 v sin(da) / d or it can never arrive, however long it steers. sliceTurn
+        // is only the floor (turning FASTER than needed is safe: the aim is re-solved every frame,
+        // so it simply finishes the turn early and flies straight in). Solved live rather than
+        // frozen at the kill, because the body it is hooking to is walking while the ball travels.
+        const d = Math.max(0.6, Math.hypot(dx, dz)); // floored: d -> 0 would demand an infinite rate
+        const w = Math.max(TUNE.sliceTurn, 2 * sp * Math.abs(Math.sin(da)) / d);
+        const a = cur + Math.sign(da) * Math.min(Math.abs(da), w * dt);
         ball.vx = Math.sin(a) * sp; ball.vz = Math.cos(a) * sp; ball.vy = 0;
         parts.push({ x: ball.x, y: ball.y, z: ball.z, vx: 0, vy: 0, vz: 0,
           t: 0.28, fine: true, spark: '#f2f2e6' });

zombies.js

@@ -637,12 +637,14 @@ function splatter(i, fanfare = true) { // the MAIN ball hit a zombie
   // ball pays for plowing through meat, and a cut ball does not plow — it leaves. Only a kill gets
   // here, so a wound ends the chain on its own and no brute, steward or boss ever extends one.
   if (ball.chain > 0) {
-    const nx = nearestTo(zx, zz, TUNE.sliceR);
+    const nx = nearestTo(zx, zz, TUNE.sliceR, ball, TUNE.sliceArc * Math.PI / 180);
     if (nx) {
       ball.chain--;
       const k = sp0 / (Math.hypot(ball.vx, ball.vz) || 1);
       ball.vx *= k; ball.vz *= k;
-      ball.hook = { x: nx.x, z: nx.z }; // a POINT, not the body: the hook must not pin a corpse alive
+      // a POINT, not the body: the hook must not pin a corpse alive. `t` is its licence to steer —
+      // see the hook in gameplay, and sliceHookT for why it cannot be left open-ended.
+      ball.hook = { x: nx.x, z: nx.z, t: TUNE.sliceHookT };
       G.trail = null;   // the fired band is the parabola this ball has just STOPPED flying — leaving
                         // it up draws a streak peeling away from the ball, which is the one thing
                         // the band exists to avoid. From here the chalk thread is the trail.
@@ -711,15 +713,22 @@ export function firePellet(w = WEAPONS.gun) {
   else if (++mgTick % 2 === 0) pock(0.25, w.ignite ? 300 : 1400);
 }
 const HUES = ['#ff6b6b', '#ffd166', '#7bffa0', '#6bd5ff', '#c08bff', '#ff8bd5'];
-// the next body a slice hooks onto: nearest live thing to the kill it just made, in any direction —
+// The next body a slice hooks onto: nearest live thing to the kill it just made, and it may well be
-// a cut ball can come BACK down-court, and the shot that curls round into something behind the body
+// off to the side or short of it — that curl is the whole shot. What it may NOT be is something the
-// you aimed at is the one worth earning. Furniture and barrels are not bodies and never a target.
+// ball cannot physically turn onto. A ball travelling at `b` can only reach a body within `arc` of
-function nearestTo(x, z, maxD) {
+// its heading; past that there is no tangent arc at any turn rate, and steering at one anyway is
+// what made the ball fly circles round a body two metres away instead of ever hitting it.
+// Furniture and barrels are not bodies and are never a target.
+function nearestTo(x, z, maxD, b, arc) {
+  const head = Math.atan2(b.vx, b.vz);
   let best = null, bd = maxD;
   for (const zb of zombies) {
     if ('mo'.includes(zb.variant)) continue;
     const d = Math.hypot(zb.x - x, zb.z - z);
-    if (d < bd) { bd = d; best = zb; }
+    if (d >= bd) continue;
+    const da = ((Math.atan2(zb.x - b.x, zb.z - b.z) - head + Math.PI * 3) % (Math.PI * 2)) - Math.PI;
+    if (Math.abs(da) > arc) continue;   // behind the ball: no arc gets there, so the chain ends
+    bd = d; best = zb;
   }
   return best;
 }