backend/src/physics.rs

@@ -5,11 +5,16 @@ use crate::protocol::*;
 const MAX_SIM_TIME: f32 = 30.0;
 const REST_SPEED: f32 = 0.04;
 const REST_ANGULAR_SPEED: f32 = 0.05;
-// Rotation rate of the velocity vector, in rad/s.
-// Positive curl_sign = right curl -> curves toward +x when moving up-sheet.
-const CURL_RATE: f32 = 0.010;
 const G: f32 = 9.80665;
 
+/// Initial |ω| for full curl: 5 rotations over 14 s (rad/s).
+/// Sign follows curl_sign; clockwise (curl>0) uses +ω0 in spawn (see apply_curl).
+pub const INITIAL_OMEGA: f32 = 5.0 * 2.0 * std::f32::consts::PI / 14.0;
+
+/// Lateral speed scale: v_lat = curl_sign * CURL_LAT_K / max(v_forward, ε).
+/// Clockwise curl → right (+x when moving +y). Calibrated for clockwise_curl_moves_right.
+pub const CURL_LAT_K: f32 = 0.06;
+
 /// Calibrated initial speed (m/s) for a mid draw that stops near the tee line
 /// with friction_scalar = 1.0, curl = 0, broom aimed at HOUSE_CENTER.
 pub const DRAW_VELOCITY: f32 = 2.38;
@@ -160,10 +165,12 @@ impl PhysicsWorld {
         let id = self.next_stone_id;
         self.next_stone_id += 1;
 
+        // Clockwise curl (curl_sign > 0) → positive ω0; lateral model maps that to +x.
+        let omega0 = curl_sign as f32 * INITIAL_OMEGA;
         let body = RigidBodyBuilder::dynamic()
             .translation(Vector::new(x, y))
             .linvel(Vector::new(vx, vy))
-            .angvel(0.0)
+            .angvel(omega0)
             .linear_damping(0.0)
             .angular_damping(0.0)
             .can_sleep(false)
@@ -187,11 +194,9 @@ impl PhysicsWorld {
         self.simulate_until_rest(id)
     }
 
-    fn simulate_until_rest(&mut self, thrown_id: u32) -> Result<Vec<StoneTrajectory>, String> {
-        // All paths share the thrown stone's release instant as t=0. This keeps the
-        // frontend's existing trajectory helpers (which expect the thrown stone to
-        // start at x=0, y=HACK_Y with t=0) working unchanged while also giving every
-        // other stone a consistent timeline.
+    fn simulate_until_rest(&mut self, _thrown_id: u32) -> Result<Vec<StoneTrajectory>, String> {
+        // Path samples are (x, y, theta). Client time is sample_index / SAMPLE_RATE_HZ.
+        // All stones share the same sample clock from the thrown stone's release.
         let sample_step = 1.0 / SAMPLE_RATE_HZ as f32;
         let mut sample_accum: f32 = 0.0;
         let mut time: f32 = 0.0;
@@ -203,11 +208,12 @@ impl PhysicsWorld {
             .map(|(id, handle, _, _, _)| (*id, *handle, Vec::new()))
             .collect();
 
-        // Record the initial sample at t=0 for every stone.
+        // Record the initial sample for every stone.
         for (id, handle, path) in &mut paths {
             if let Some(body) = self.bodies.get(*handle) {
                 let pos = body.translation();
-                path.push((pos.x, pos.y, time));
+                let theta = body.rotation().angle();
+                path.push((pos.x, pos.y, theta));
             } else {
                 // Body missing for an tracked stone; this should not happen.
                 return Err(format!("stone {} has no rigid body", id));
@@ -226,7 +232,8 @@ impl PhysicsWorld {
                 for (_, handle, path) in &mut paths {
                     if let Some(body) = self.bodies.get(*handle) {
                         let pos = body.translation();
-                        path.push((pos.x, pos.y, time));
+                        let theta = body.rotation().angle();
+                        path.push((pos.x, pos.y, theta));
                     }
                 }
             }
@@ -238,26 +245,9 @@ impl PhysicsWorld {
 
         self.prune_out_of_play();
 
-        // The thrown stone is released at (0.0, HACK_Y). Shift every path in time so
-        // that t=0 corresponds to that release instant. Because we already started
-        // sampling at the release instant, the first sample time is 0.0 and no shift
-        // is required; this comment documents the invariant.
-        let thrown_first_t = paths
-            .iter()
-            .find(|(id, _, _)| *id == thrown_id)
-            .and_then(|(_, _, path)| path.first().map(|(_, _, t)| *t))
-            .unwrap_or(0.0);
-
         Ok(paths
             .into_iter()
-            .map(|(id, _, mut path)| {
-                if thrown_first_t != 0.0 {
-                    for (_, _, t) in &mut path {
-                        *t -= thrown_first_t;
-                    }
-                }
-                StoneTrajectory { stone_id: id, path }
-            })
+            .map(|(id, _, path)| StoneTrajectory { stone_id: id, path })
             .collect())
     }
 
@@ -289,26 +279,50 @@ impl PhysicsWorld {
         }
     }
 
-    // Rotate each stone's velocity slightly based on its selected curl direction.
-    // Right curl (curl_sign = +1) curves toward +x when moving up-sheet (positive y).
+    /// Spin-curl model after drag:
+    /// - Angular damping from µ(v)*friction_scalar (slower → larger µ → more damp)
+    /// - Lateral speed target v_lat = curl_sign * CURL_LAT_K / max(v_forward, ε)
+    /// - Clockwise (curl_sign > 0) → right: +x when moving +y
     fn apply_curl(&mut self) {
-        for (_, handle, _, curl_sign, _) in &self.stone_handles {
+        const EPS: f32 = 1e-3;
+
+        for (_, handle, _, curl_sign, friction_scalar) in &self.stone_handles {
             let body = match self.bodies.get_mut(*handle) {
                 Some(b) => b,
                 None => continue,
             };
 
             let v = body.linvel();
-            let speed_sq = v.x * v.x + v.y * v.y;
-            let speed = speed_sq.sqrt();
-            if speed < 1e-4 {
+            let speed = (v.x * v.x + v.y * v.y).sqrt();
+            let mu_eff = mu(speed) * *friction_scalar;
+
+            // Angular damping: same µ_eff basis as linear ice friction.
+            let omega = body.angvel();
+            if omega.abs() > 1e-8 {
+                let domega = mu_eff * G / STONE_RADIUS * PHYSICS_DT;
+                let new_omega = if domega >= omega.abs() {
+                    0.0
+                } else {
+                    omega - omega.signum() * domega
+                };
+                body.set_angvel(new_omega, true);
+            }
+
+            if *curl_sign == 0 || speed < 1e-4 {
                 continue;
             }
 
-            let angle = -(*curl_sign as f32) * CURL_RATE * PHYSICS_DT;
-            let cos = angle.cos();
-            let sin = angle.sin();
-            let new_v = Vector::new(v.x * cos - v.y * sin, v.x * sin + v.y * cos);
+            // Unit forward and right (CW 90° from forward): (ux,uy) → (uy, -ux).
+            // Moving +y → right = +x.
+            let ux = v.x / speed;
+            let uy = v.y / speed;
+            let rx = uy;
+            let ry = -ux;
+
+            let v_forward = speed;
+            let v_lat_target = (*curl_sign as f32) * CURL_LAT_K / v_forward.max(EPS);
+            let v_fwd = v.x * ux + v.y * uy;
+            let new_v = Vector::new(v_fwd * ux + v_lat_target * rx, v_fwd * uy + v_lat_target * ry);
             body.set_linvel(new_v, true);
         }
     }
@@ -526,6 +540,75 @@ mod tests {
         );
     }
 
+    #[test]
+    fn initial_angvel_magnitude_matches_5_rot_per_14s() {
+        let expected = 5.0 * 2.0 * std::f32::consts::PI / 14.0;
+        assert!(
+            (INITIAL_OMEGA - expected).abs() < 1e-5,
+            "INITIAL_OMEGA={} expected {}",
+            INITIAL_OMEGA,
+            expected
+        );
+
+        // Early path dθ/dt should be near |ω0| before damping eats much spin.
+        let mut world = PhysicsWorld::new();
+        let traj = world
+            .throw(Team::Red, 0.0, HOUSE_CENTER.1, DRAW_VELOCITY, 1, 1.0)
+            .unwrap();
+        let path = &traj[0].path;
+        assert!(path.len() >= 3, "need samples to estimate ω");
+        let dt = 1.0 / SAMPLE_RATE_HZ as f32;
+        let omega_est = (path[1].2 - path[0].2) / dt;
+        assert!(
+            (omega_est.abs() - expected).abs() < expected * 0.35,
+            "early |ω|≈{} should be near {} (5 rot / 14s)",
+            omega_est.abs(),
+            expected
+        );
+    }
+
+    #[test]
+    fn clockwise_curl_moves_right() {
+        // Clockwise curl (curl > 0) must finish to the right of counterclockwise.
+        let mut cw = PhysicsWorld::new();
+        let cw_traj = cw
+            .throw(Team::Red, 0.0, HOUSE_CENTER.1, DRAW_VELOCITY, 1, 1.0)
+            .unwrap();
+        let right_x = cw_traj[0].path.last().map(|p| p.0).unwrap_or(f32::NAN);
+
+        let mut ccw = PhysicsWorld::new();
+        let ccw_traj = ccw
+            .throw(Team::Red, 0.0, HOUSE_CENTER.1, DRAW_VELOCITY, -1, 1.0)
+            .unwrap();
+        let left_x = ccw_traj[0].path.last().map(|p| p.0).unwrap_or(f32::NAN);
+
+        println!("clockwise final x={} counterclockwise final x={}", right_x, left_x);
+        assert!(
+            right_x > left_x + 0.05,
+            "clockwise curl should move right: right_x={} left_x={}",
+            right_x,
+            left_x
+        );
+    }
+
+    #[test]
+    fn path_samples_include_nonzero_theta_when_spinning() {
+        let mut world = PhysicsWorld::new();
+        let traj = world
+            .throw(Team::Red, 0.0, HOUSE_CENTER.1, DRAW_VELOCITY, 1, 1.0)
+            .unwrap();
+        let path = &traj[0].path;
+        let max_abs_theta = path
+            .iter()
+            .map(|(_, _, theta)| theta.abs())
+            .fold(0.0_f32, f32::max);
+        assert!(
+            max_abs_theta > 0.05,
+            "spinning stone path should include nonzero theta, max|θ|={}",
+            max_abs_theta
+        );
+    }
+
     #[test]
     fn stones_persist_after_multiple_throws() {
         let mut world = PhysicsWorld::new();
@@ -614,11 +697,14 @@ mod tests {
             second_path.len()
         );
 
-        // Both paths should share the same t=0 reference (the thrown stone's release).
-        assert_eq!(first_path[0].2, 0.0, "first stone path should start at t=0");
-        assert_eq!(
-            second_path[0].2, 0.0,
-            "thrown stone path should start at t=0"
+        // Paths are (x, y, theta); both stones must have a sample at release (index 0).
+        assert!(
+            first_path[0].2.is_finite(),
+            "first stone path should include finite theta"
+        );
+        assert!(
+            second_path[0].2.is_finite(),
+            "thrown stone path should include finite theta"
         );
     }
 }

backend/src/protocol.rs

@@ -109,6 +109,7 @@ pub enum Phase {
 #[derive(Debug, Clone, Serialize, Deserialize)]
 pub struct StoneTrajectory {
     pub stone_id: u32,
+    /// Samples as (x, y, theta). Time is sample_index / SAMPLE_RATE_HZ on the client.
     pub path: Vec<(f32, f32, f32)>,
 }
 

frontend/src/game-helpers.test.ts

@@ -1,33 +1,57 @@
 import { describe, expect, it } from 'vitest'
 import {
+  hogTrimStartIndex,
+  sampleTime,
   trimPathToStartAtHogLine,
   velocityToWeight,
   weightToVelocity,
 } from './game-helpers'
+import { SAMPLE_RATE_HZ } from './protocol'
 
 describe('trimPathToStartAtHogLine', () => {
-  it('trims path at the first hog-line crossing and zeroes time', () => {
+  it('trims path at the first hog-line crossing and preserves theta', () => {
+    // (x, y, theta) — time is derived from index after trim
     const path: [number, number, number][] = [
-      [0, 2, 0],
-      [0, 20, 1],
-      [0, 21.5, 2],
-      [0, 30, 3],
+      [0, 2, 0.1],
+      [0, 20, 0.2],
+      [0, 21.5, 0.3],
+      [0, 30, 0.4],
     ]
     const trimmed = trimPathToStartAtHogLine(path)
     expect(trimmed[0][1]).toBe(20)
-    expect(trimmed[0][2]).toBe(0)
-    expect(trimmed[trimmed.length - 1][2]).toBe(2)
+    expect(trimmed[0][2]).toBe(0.2)
+    expect(trimmed[trimmed.length - 1][2]).toBe(0.4)
+    expect(trimmed).toHaveLength(3)
   })
 
   it('returns full path when hog line is never reached', () => {
     const path: [number, number, number][] = [
       [0, 2, 0],
-      [0, 10, 1],
+      [0, 10, 0.5],
     ]
     expect(trimPathToStartAtHogLine(path)).toEqual(path)
   })
 })
 
+describe('sampleTime', () => {
+  it('is index / SAMPLE_RATE_HZ', () => {
+    expect(sampleTime(0)).toBe(0)
+    expect(sampleTime(SAMPLE_RATE_HZ)).toBe(1)
+    expect(sampleTime(1)).toBeCloseTo(1 / SAMPLE_RATE_HZ)
+  })
+})
+
+describe('hogTrimStartIndex', () => {
+  it('returns index just before hog crossing', () => {
+    const path: [number, number, number][] = [
+      [0, 2, 0],
+      [0, 20, 0],
+      [0, 21.5, 0],
+    ]
+    expect(hogTrimStartIndex(path)).toBe(1)
+  })
+})
+
 describe('velocity ↔ weight', () => {
   it('maps endpoints correctly', () => {
     expect(velocityToWeight(3.0)).toBe(1)
@@ -43,4 +67,3 @@ describe('velocity ↔ weight', () => {
     expect(weightToVelocity(11)).toBe(6.45)
   })
 })
-

frontend/src/game-helpers.ts

@@ -1,4 +1,9 @@
-import { HOG_LINE_Y, MAX_SPEED, MIN_SPEED } from './protocol'
+import { HOG_LINE_Y, MAX_SPEED, MIN_SPEED, SAMPLE_RATE_HZ } from './protocol'
+
+/** Path samples are (x, y, theta). Time is sample index / SAMPLE_RATE_HZ. */
+export function sampleTime(index: number): number {
+  return index / SAMPLE_RATE_HZ
+}
 
 export function trimPathToStartAtHogLine(
   path: [number, number, number][],
@@ -7,9 +12,17 @@ export function trimPathToStartAtHogLine(
   const idx = path.findIndex(([, y]) => y >= HOG_LINE_Y)
   if (idx < 0) return path
   // Start just before the hog line crossing so the stone enters smoothly.
+  // Theta is preserved; time is rebased via sample index on the trimmed array.
   const start = Math.max(0, idx - 1)
-  const t0 = path[start][2]
-  return path.slice(start).map(([x, y, t]) => [x, y, t - t0])
+  return path.slice(start)
+}
+
+/** Index at which a path should start for hog-line sync (same as trim start). */
+export function hogTrimStartIndex(path: [number, number, number][]): number {
+  if (path.length < 2) return 0
+  const idx = path.findIndex(([, y]) => y >= HOG_LINE_Y)
+  if (idx < 0) return 0
+  return Math.max(0, idx - 1)
 }
 
 export function velocityToWeight(velocity: number): number {
@@ -23,4 +36,3 @@ export function weightToVelocity(weight: number): number {
   const t = (clamped - 1) / 9
   return MIN_SPEED + t * (MAX_SPEED - MIN_SPEED)
 }
-

frontend/src/game-model.test.ts

@@ -1,6 +1,7 @@
 import { describe, expect, it } from 'vitest'
 import { GameModel } from './game-model'
 import type { ServerStoneTrajectory, StoneState } from './protocol'
+import { SAMPLE_RATE_HZ } from './protocol'
 
 function stone(partial: Partial<StoneState> & Pick<StoneState, 'id' | 'team'>): StoneState {
   return {
@@ -12,40 +13,42 @@ function stone(partial: Partial<StoneState> & Pick<StoneState, 'id' | 'team'>):
   }
 }
 
+/** Build (x,y,theta) samples; time comes from index / SAMPLE_RATE_HZ. */
+function pathSamples(
+  points: { x: number; y: number; theta?: number }[],
+): [number, number, number][] {
+  return points.map((p) => [p.x, p.y, p.theta ?? 0])
+}
+
 describe('GameModel multi-path trajectory animation', () => {
   it('returns two DrawableStones in parallel mid-trajectory', () => {
     const model = new GameModel()
     model.state.stones = [stone({ id: 1, team: 'red' }), stone({ id: 2, team: 'yellow' })]
     model.state.turnTeam = 'red'
 
+    // 21 samples → duration 20/40 = 0.5s; mid at 0.25s is sample 10 → y=11
+    const n = 21
+    const path1 = pathSamples(
+      Array.from({ length: n }, (_, i) => ({ x: 0, y: 10 + i * 0.1, theta: 0 })),
+    )
+    const path2 = pathSamples(
+      Array.from({ length: n }, (_, i) => ({ x: 1, y: 10 + i * 0.1, theta: 0 })),
+    )
+
     const paths: ServerStoneTrajectory[] = [
-      {
-        stone_id: 1,
-        path: [
-          [0, 10, 0],
-          [0, 11, 0.5],
-          [0, 12, 1.0],
-        ],
-      },
-      {
-        stone_id: 2,
-        path: [
-          [1, 10, 0],
-          [1, 11, 0.5],
-          [1, 12, 1.0],
-        ],
-      },
+      { stone_id: 1, path: path1 },
+      { stone_id: 2, path: path2 },
     ]
 
     model.startTrajectory(paths)
     expect(model.state.animating).toBe(true)
 
-    const mid = performance.now() + 500
+    const mid = performance.now() + 250
     const drawn = model.tick(mid)
 
     expect(drawn).toHaveLength(2)
     expect(drawn.map((d) => d.team).sort()).toEqual(['red', 'yellow'])
-    // Mid-sample y ≈ 11 for both paths (y never reaches hog line → no trim shift)
+    // Mid-sample y ≈ 11 for both paths (y never reaches hog line → no trim)
     for (const d of drawn) {
       expect(d.y).toBeCloseTo(11, 0)
     }
@@ -56,19 +59,20 @@ describe('GameModel multi-path trajectory animation', () => {
     model.state.stones = [stone({ id: 1, team: 'red' })]
     model.state.turnTeam = 'red'
 
+    // 3 samples → max t = 2/40 = 0.05s
     model.startTrajectory([
       {
         stone_id: 1,
-        path: [
-          [0, 10, 0],
-          [0, 10.5, 0.2],
-          [0, 11, 0.4],
-        ],
+        path: pathSamples([
+          { x: 0, y: 10 },
+          { x: 0, y: 10.5 },
+          { x: 0, y: 11 },
+        ]),
       },
     ])
     expect(model.state.animating).toBe(true)
 
-    const afterEnd = performance.now() + 500
+    const afterEnd = performance.now() + 200
     const drawn = model.tick(afterEnd)
 
     expect(drawn).toEqual([])
@@ -81,17 +85,17 @@ describe('GameModel multi-path trajectory animation', () => {
     model.state.stones = [stone({ id: 1, team: 'yellow', x: 0.5, y: 35 })]
     model.state.turnTeam = 'red'
 
-    const thrownPath: [number, number, number][] = [
-      [0, 2, 0],
-      [0, 20, 1],
-      [0, 21.5, 2],
-      [0, 30, 3],
-    ]
+    const thrownPath = pathSamples([
+      { x: 0, y: 2, theta: 0.1 },
+      { x: 0, y: 20, theta: 0.2 },
+      { x: 0, y: 21.5, theta: 0.3 },
+      { x: 0, y: 30, theta: 0.4 },
+    ])
 
     model.startTrajectory([{ stone_id: 2, path: thrownPath }])
     expect(model.state.animating).toBe(true)
 
-    // Immediately after start: hog-trimmed path begins at y=20 (sample before hog), t=0
+    // Immediately after start: hog-trimmed path begins at y=20 (sample before hog)
     const atStart = performance.now()
     const drawn = model.tick(atStart)
 
@@ -100,5 +104,11 @@ describe('GameModel multi-path trajectory animation', () => {
     expect(drawn[0].y).toBeCloseTo(20, 0)
     // Must not still be at the hack (y=2)
     expect(drawn[0].y).toBeGreaterThan(15)
+    // Uses body theta from path (small elapsed may blend toward next sample)
+    expect(drawn[0].rotation).toBeCloseTo(0.2, 2)
+  })
+
+  it('uses sample index for time (SAMPLE_RATE_HZ)', () => {
+    expect(SAMPLE_RATE_HZ).toBe(40)
   })
 })

frontend/src/game-model.ts

@@ -1,5 +1,5 @@
-import { HOG_LINE_Y, HOUSE_CENTER, type DrawableStone, type Phase, type ServerGameStateMessage, type ServerStoneTrajectory, type StoneState, type Team } from './protocol'
-import { trimPathToStartAtHogLine } from './game-helpers'
+import { HOUSE_CENTER, type DrawableStone, type Phase, type ServerGameStateMessage, type ServerStoneTrajectory, type StoneState, type Team } from './protocol'
+import { hogTrimStartIndex, sampleTime, trimPathToStartAtHogLine } from './game-helpers'
 
 export interface GameModelState {
   end: number
@@ -77,16 +77,12 @@ export class GameModel {
       thrownId = paths[0].stone_id
     }
 
-    let tRef = 0
+    // Shared sample-index trim so multi-stone paths stay on one clock.
+    // Path entries are (x, y, theta); t = index / SAMPLE_RATE_HZ after trim.
+    let startIdx = 0
     if (thrownId !== null) {
       const thrownPath = paths.find((p) => p.stone_id === thrownId)?.path ?? []
-      if (thrownPath.length >= 2) {
-        const idx = thrownPath.findIndex(([, y]) => y >= HOG_LINE_Y)
-        if (idx >= 0) {
-          const start = Math.max(0, idx - 1)
-          tRef = thrownPath[start][2]
-        }
-      }
+      startIdx = hogTrimStartIndex(thrownPath)
     }
 
     const pathMap = new Map<number, [number, number, number][]>()
@@ -94,10 +90,7 @@ export class GameModel {
       if (stone_id === thrownId) {
         pathMap.set(stone_id, trimPathToStartAtHogLine(path))
       } else {
-        const shifted = path
-          .map(([x, y, t]) => [x, y, t - tRef] as [number, number, number])
-          .filter(([, , t]) => t >= 0)
-        pathMap.set(stone_id, shifted)
+        pathMap.set(stone_id, path.slice(startIdx))
       }
     }
 
@@ -115,7 +108,9 @@ export class GameModel {
     const elapsed = (now - this.animationStartTime) / 1000
     const maxTotal = Math.max(
       0,
-      ...Array.from(this.activePaths.values()).map((p) => (p.length > 0 ? p[p.length - 1][2] : 0)),
+      ...Array.from(this.activePaths.values()).map((p) =>
+        p.length > 0 ? sampleTime(p.length - 1) : 0,
+      ),
     )
 
     if (elapsed >= maxTotal) {
@@ -143,31 +138,33 @@ export class GameModel {
   ): { x: number; y: number; rotation: number } | null {
     if (path.length === 0) return null
     if (path.length === 1) {
-      const [x, y] = path[0]
-      return { x, y, rotation: 0 }
+      const [x, y, theta] = path[0]
+      return { x, y, rotation: theta }
     }
 
-    if (elapsed >= path[path.length - 1][2]) {
+    const lastT = sampleTime(path.length - 1)
+    if (elapsed >= lastT) {
       const last = path[path.length - 1]
-      const prev = path[path.length - 2]
-      const dx = last[0] - prev[0]
-      const dy = last[1] - prev[1]
-      return { x: last[0], y: last[1], rotation: Math.atan2(dy, dx) * 2 }
+      return { x: last[0], y: last[1], rotation: last[2] }
     }
 
+    // Find segment where sampleTime(i) <= elapsed < sampleTime(i+1)
     let i = 0
-    while (i + 1 < path.length && path[i + 1][2] < elapsed) i++
+    while (i + 1 < path.length && sampleTime(i + 1) < elapsed) i++
     const p0 = path[i]
     const p1 = path[i + 1] ?? p0
-    const t0 = path[Math.max(i - 1, 0)]
-    const t2 = path[Math.min(i + 2, path.length - 1)]
-    const dt = p1[2] - p0[2]
-    const t = dt > 0 ? (elapsed - p0[2]) / dt : 0
+    const t0 = sampleTime(i)
+    const t1 = sampleTime(i + 1)
+    const dt = t1 - t0
+    const t = dt > 0 ? (elapsed - t0) / dt : 0
     const x = p0[0] + (p1[0] - p0[0]) * t
     const y = p0[1] + (p1[1] - p0[1]) * t
-    const dx = t2[0] - t0[0]
-    const dy = t2[1] - t0[1]
-    const rotation = Math.atan2(dy, dx) * 2
+    // Interpolate body rotation (theta) from path samples
+    let dTheta = p1[2] - p0[2]
+    // Unwrap shortest path across ±π
+    if (dTheta > Math.PI) dTheta -= 2 * Math.PI
+    if (dTheta < -Math.PI) dTheta += 2 * Math.PI
+    const rotation = p0[2] + dTheta * t
     return { x, y, rotation }
   }
 

frontend/src/protocol.ts

@@ -64,6 +64,7 @@ export interface ServerGameStateMessage {
   phase: Phase
 }
 
+/** Path samples are (x, y, theta). Time is sample_index / SAMPLE_RATE_HZ. */
 export interface ServerStoneTrajectory {
   stone_id: number
   path: [number, number, number][]