From a0863b290c6082c61044ac687a07e9934b9f5013 Mon Sep 17 00:00:00 2001
From: Harald Hoyer <harald@hoyer.xyz>
Date: Thu, 7 May 2026 19:54:18 +0200
Subject: [PATCH] feat(gameplay): add scripted flight paths for special stage
 enemies

Add Lissajous and cubic Bezier path types for enemies on special stages
(every 3rd level). Path-following enemies spawn with ScriptedPath marker
and FollowingPath state, move along parametric curves, and despawn when
their path ends off-screen.

Key changes:
- New flight_paths module with pure path evaluation (Lissajous + Bezier)
- ScriptedPath marker component and FollowingPath EnemyState variant
- flight_patterns field on StageConfigurations for composable path data
- Special stage spawn branch in spawn_enemies() with stagger delays
- Path-following movement in move_enemies() with delay gating and despawn
- Formation complete early return for special stages
- Without<ScriptedPath> filter on attacking_query to prevent double-processing
- 13 new unit tests for path evaluation and data contracts

Refs: GAL-40
---
 src/components.rs      |  12 ++
 src/enemy.rs           | 267 ++++++++++++++++++++++++++++++++++++++++-
 src/flight_paths.rs    | 141 ++++++++++++++++++++++
 src/game_state.rs      |   4 +-
 src/lib.rs             |   4 +-
 src/player.rs          |   8 +-
 src/resources.rs       |  95 +++++++++++++++
 src/systems.rs         |   6 +-
 tests/special_stage.rs |   5 +-
 9 files changed, 526 insertions(+), 16 deletions(-)
 create mode 100644 src/flight_paths.rs

diff --git a/src/components.rs b/src/components.rs
index c7fb877..c7a9e4c 100644
--- a/src/components.rs
+++ b/src/components.rs
@@ -1,5 +1,7 @@
 use bevy::prelude::*;
 
+use crate::flight_paths::PathType;
+
 #[derive(Component)]
 pub struct Player {
     pub speed: f32,
@@ -69,6 +71,12 @@ pub enum EnemyState {
     InFormation,
     Attacking(AttackPattern),
     ReturningWithCaptive,
+    FollowingPath {
+        path: PathType,
+        t: f32,
+        delay: f32,
+        elapsed: f32,
+    },
 }
 
 #[derive(Component)]
@@ -95,3 +103,7 @@ pub struct Star {
 pub struct Explosion {
     pub timer: Timer,
 }
+
+/// Marks an enemy as following a scripted flight path.
+#[derive(Component)]
+pub struct ScriptedPath;
diff --git a/src/enemy.rs b/src/enemy.rs
index 20f59c8..82ca360 100644
--- a/src/enemy.rs
+++ b/src/enemy.rs
@@ -3,16 +3,17 @@ use std::time::Duration;
 
 use crate::components::{
     AttackPattern, Captured, Enemy, EnemyBullet, EnemyState, EnemyType, FormationTarget,
-    OriginalFormationPosition, Player, TractorBeam, TractorBeamSprite,
+    OriginalFormationPosition, Player, ScriptedPath, TractorBeam, TractorBeamSprite,
 };
 use crate::constants::{
     BEAM_CORE_COLOR, BEAM_GLOW_COLOR, BEAM_GLOW_WIDTH, BEAM_PULSE_AMPLITUDE, BEAM_PULSE_FREQ,
     CAPTURE_DURATION, ENEMY_BULLET_SIZE, ENEMY_SIZE, ENEMY_SPEED, TRACTOR_BEAM_DURATION,
     TRACTOR_BEAM_WIDTH, WINDOW_HEIGHT, WINDOW_WIDTH,
 };
+use crate::flight_paths;
 use crate::resources::{
-    AttackDiveTimer, CurrentStage, EnemySpawnTimer, FormationState, StageConfigurations,
-    is_special_stage,
+    is_special_stage, AttackDiveTimer, CurrentStage, EnemySpawnTimer, FormationState,
+    StageConfigurations,
 };
 
 const BOSS_BASE_CHANCE: f32 = 0.30;
@@ -45,6 +46,72 @@ pub fn spawn_enemies(
         return;
     }
 
+    // Special stage: path-following enemies
+    if is_special_stage(stage.number) {
+        let flight_patterns = &stage_configs.flight_patterns;
+        if flight_patterns.is_empty() {
+            // Fall through to normal formation spawn
+        } else {
+            let slot = formation_state.next_slot_index;
+            if slot >= config.enemy_count {
+                return;
+            }
+            let pattern = &flight_patterns[slot % flight_patterns.len()];
+            let spawn_pos = flight_paths::evaluate(pattern, 0.0);
+            let delay = slot as f32 * 0.15;
+
+            let boss_chance = BOSS_BASE_CHANCE
+                + (stage.number as f32 * BOSS_CHANCE_PER_STAGE).min(BOSS_CHANCE_BONUS_CAP);
+            let enemy_type = if fastrand::f32() < boss_chance {
+                EnemyType::Boss
+            } else {
+                EnemyType::Grunt
+            };
+
+            let color = match enemy_type {
+                EnemyType::Grunt => Color::srgb(1.0, 0.2, 0.2),
+                EnemyType::Boss => Color::srgb(0.8, 0.2, 1.0),
+            };
+
+            commands.spawn((
+                Sprite {
+                    color,
+                    custom_size: Some(ENEMY_SIZE),
+                    ..default()
+                },
+                Transform::from_translation(spawn_pos),
+                Enemy {
+                    enemy_type,
+                    shoot_cooldown: Timer::new(
+                        Duration::from_secs_f32(config.enemy_shoot_interval),
+                        TimerMode::Once,
+                    ),
+                },
+                ScriptedPath,
+                EnemyState::FollowingPath {
+                    path: pattern.clone(),
+                    t: 0.0,
+                    delay,
+                    elapsed: 0.0,
+                },
+            ));
+
+            formation_state.next_slot_index = (slot + 1) % config.enemy_count;
+            formation_state.total_spawned_this_stage += 1;
+            stage.waiting_for_clear = true;
+
+            if formation_state.total_spawned_this_stage < config.enemy_count {
+                timer.timer.reset();
+            } else {
+                info!(
+                    "Stage {} special stage formation fully spawned",
+                    stage.number
+                );
+            }
+            return; // Skip normal spawn logic
+        }
+    }
+
     let slot = formation_state.next_slot_index;
     if slot >= config.formation_layout.positions.len() {
         warn!(
@@ -64,8 +131,8 @@ pub fn spawn_enemies(
         0.0,
     );
 
-    let boss_chance = BOSS_BASE_CHANCE
-        + (stage.number as f32 * BOSS_CHANCE_PER_STAGE).min(BOSS_CHANCE_BONUS_CAP);
+    let boss_chance =
+        BOSS_BASE_CHANCE + (stage.number as f32 * BOSS_CHANCE_PER_STAGE).min(BOSS_CHANCE_BONUS_CAP);
     let enemy_type = if fastrand::f32() < boss_chance {
         EnemyType::Boss
     } else {
@@ -111,11 +178,13 @@ pub fn spawn_enemies(
     }
 }
 
+#[allow(clippy::too_many_arguments)]
 pub fn move_enemies(
     mut entering_query: Query<
         (Entity, &mut Transform, &FormationTarget, &mut EnemyState),
         (With<Enemy>, With<FormationTarget>),
     >,
+    mut path_query: Query<(Entity, &mut Transform, &mut EnemyState), With<ScriptedPath>>,
     mut attacking_query: Query<
         (
             Entity,
@@ -124,7 +193,7 @@ pub fn move_enemies(
             &Enemy,
             Option<&OriginalFormationPosition>,
         ),
-        (With<Enemy>, Without<FormationTarget>),
+        (With<Enemy>, Without<FormationTarget>, Without<ScriptedPath>),
     >,
     time: Res<Time>,
     mut commands: Commands,
@@ -133,6 +202,37 @@ pub fn move_enemies(
     has_beam_query: Query<&TractorBeam>,
 ) {
     let dt = time.delta_secs();
+
+    // Path-following enemies (special stages)
+    let mut entities_to_despawn = Vec::new();
+    for (entity, mut transform, mut state) in path_query.iter_mut() {
+        if let EnemyState::FollowingPath {
+            path,
+            t,
+            delay,
+            elapsed,
+        } = state.as_mut()
+        {
+            let new_elapsed = *elapsed + dt;
+            if new_elapsed >= *delay {
+                let new_t = (*t + dt / path.duration()).clamp(0.0, 1.0);
+                let pos = flight_paths::evaluate(path, new_t);
+                transform.translation.x = pos.x;
+                transform.translation.y = pos.y;
+                *t = new_t;
+                *elapsed = new_elapsed;
+                if new_t >= 1.0 || transform.translation.y < -WINDOW_HEIGHT / 2.0 - ENEMY_SIZE.y {
+                    entities_to_despawn.push(entity);
+                }
+            } else {
+                *elapsed = new_elapsed;
+            }
+        }
+    }
+    for entity in entities_to_despawn {
+        commands.entity(entity).despawn();
+    }
+
     let speed_multiplier = stage_configs.for_stage(stage.number).enemy_speed_multiplier;
     let base_speed = ENEMY_SPEED * speed_multiplier;
     let attack_speed = base_speed * ATTACKER_SPEED_MULTIPLIER;
@@ -257,6 +357,9 @@ pub fn check_formation_complete(
     stage: Res<CurrentStage>,
     stage_configs: Res<StageConfigurations>,
 ) {
+    if is_special_stage(stage.number) {
+        return;
+    }
     if formation_state.formation_complete {
         return;
     }
@@ -513,6 +616,81 @@ pub fn update_tractor_beam_visual(
     }
 }
 
+#[cfg(test)]
+mod spawn_contract_tests {
+    use super::*;
+    use crate::components::ScriptedPath;
+    use crate::flight_paths::PathType;
+
+    #[test]
+    fn is_special_stage_0_returns_false() {
+        assert!(!is_special_stage(0));
+    }
+
+    #[test]
+    fn is_special_stage_1_returns_false() {
+        assert!(!is_special_stage(1));
+    }
+
+    #[test]
+    fn is_special_stage_2_returns_false() {
+        assert!(!is_special_stage(2));
+    }
+
+    #[test]
+    fn is_special_stage_3_returns_true() {
+        assert!(is_special_stage(3));
+    }
+
+    #[test]
+    fn is_special_stage_6_returns_true() {
+        assert!(is_special_stage(6));
+    }
+
+    #[test]
+    fn default_flight_patterns_has_two_entries() {
+        let cfg = StageConfigurations::default();
+        assert_eq!(
+            cfg.flight_patterns.len(),
+            2,
+            "default StageConfigurations should have exactly 2 flight patterns"
+        );
+    }
+
+    #[test]
+    fn flight_patterns_not_empty() {
+        let cfg = StageConfigurations::default();
+        assert!(
+            !cfg.flight_patterns.is_empty(),
+            "flight_patterns must not be empty to avoid panic in spawn logic"
+        );
+    }
+
+    #[test]
+    fn scripted_path_component_constructs() {
+        let _sp = ScriptedPath;
+    }
+
+    #[test]
+    fn following_path_variant_has_correct_fields() {
+        let path = PathType::Lissajous {
+            center_x: 0.0,
+            center_y: 0.0,
+            amplitude_x: 100.0,
+            amplitude_y: 200.0,
+            ax: 1.0,
+            ay: 1.0,
+            delta: 0.0,
+        };
+        let _state = EnemyState::FollowingPath {
+            path,
+            t: 0.0,
+            delay: 0.0,
+            elapsed: 0.0,
+        };
+    }
+}
+
 #[cfg(test)]
 mod beam_tests {
     use super::*;
@@ -565,3 +743,80 @@ mod beam_tests {
         assert_eq!(result, 1.0);
     }
 }
+
+#[cfg(test)]
+mod movement_contract_tests {
+    use super::*;
+    use crate::flight_paths::PathType;
+
+    #[test]
+    fn bezier_evaluate_at_t_half_advances_position() {
+        // A Bezier path from origin to off-screen bottom;
+        // position at t=0.5 must differ from t=0.0, proving movement.
+        let path = PathType::CubicBezier {
+            p0: Vec3::ZERO,
+            p1: Vec3::ZERO,
+            p2: Vec3::ZERO,
+            p3: Vec3::new(0.0, -450.0, 0.0),
+        };
+        let pos_start = flight_paths::evaluate(&path, 0.0);
+        let pos_mid = flight_paths::evaluate(&path, 0.5);
+        assert_ne!(
+            pos_mid, pos_start,
+            "Bezier evaluate at t=0.5 ({:?}) should differ from t=0.0 ({:?}), proving movement",
+            pos_mid, pos_start
+        );
+    }
+
+    #[test]
+    fn bezier_evaluate_at_t1_is_off_screen() {
+        // p3.y = -450, and the off-screen threshold is
+        // -WINDOW_HEIGHT/2.0 - ENEMY_SIZE.y = -400 - 40 = -440.
+        // At t=1.0 the Bezier lands exactly on p3, so y must be < -440.
+        let path = PathType::CubicBezier {
+            p0: Vec3::ZERO,
+            p1: Vec3::ZERO,
+            p2: Vec3::ZERO,
+            p3: Vec3::new(0.0, -450.0, 0.0),
+        };
+        let pos = flight_paths::evaluate(&path, 1.0);
+        assert!(
+            pos.y < -440.0,
+            "Bezier y at t=1.0 ({}) should be off-screen (< -440)",
+            pos.y
+        );
+    }
+
+    #[test]
+    fn lissajous_duration_is_4_seconds() {
+        let path = PathType::Lissajous {
+            center_x: 0.0,
+            center_y: 0.0,
+            amplitude_x: 100.0,
+            amplitude_y: 200.0,
+            ax: 1.0,
+            ay: 1.0,
+            delta: 0.0,
+        };
+        assert_eq!(
+            path.duration(),
+            4.0,
+            "Lissajous duration should be 4.0 seconds"
+        );
+    }
+
+    #[test]
+    fn cubic_bezier_duration_is_5_seconds() {
+        let path = PathType::CubicBezier {
+            p0: Vec3::ZERO,
+            p1: Vec3::ZERO,
+            p2: Vec3::ZERO,
+            p3: Vec3::ZERO,
+        };
+        assert_eq!(
+            path.duration(),
+            5.0,
+            "CubicBezier duration should be 5.0 seconds"
+        );
+    }
+}
diff --git a/src/flight_paths.rs b/src/flight_paths.rs
new file mode 100644
index 0000000..9986cd0
--- /dev/null
+++ b/src/flight_paths.rs
@@ -0,0 +1,141 @@
+use bevy::prelude::*;
+
+#[derive(Clone, Debug, PartialEq)]
+pub enum PathType {
+    Lissajous {
+        center_x: f32,
+        center_y: f32,
+        amplitude_x: f32,
+        amplitude_y: f32,
+        ax: f32,
+        ay: f32,
+        delta: f32,
+    },
+    CubicBezier {
+        p0: Vec3,
+        p1: Vec3,
+        p2: Vec3,
+        p3: Vec3,
+    },
+}
+
+impl PathType {
+    pub fn duration(&self) -> f32 {
+        match self {
+            PathType::Lissajous { .. } => 4.0,
+            PathType::CubicBezier { .. } => 5.0,
+        }
+    }
+}
+
+pub fn evaluate(path_type: &PathType, t: f32) -> Vec3 {
+    match path_type {
+        PathType::Lissajous {
+            center_x,
+            center_y,
+            amplitude_x,
+            amplitude_y,
+            ax,
+            ay,
+            delta,
+        } => {
+            let x = *center_x + *amplitude_x * (2.0 * std::f32::consts::PI * ax * t + delta).sin();
+            let y = *center_y + *amplitude_y * (2.0 * std::f32::consts::PI * ay * t).sin();
+            Vec3::new(x, y, 0.0)
+        }
+        PathType::CubicBezier { p0, p1, p2, p3 } => {
+            let u = 1.0 - t;
+            let u3 = u * u * u;
+            let u2t = u * u * t;
+            let ut2 = u * t * t;
+            let t3 = t * t * t;
+            u3 * *p0 + 3.0 * u2t * *p1 + 3.0 * ut2 * *p2 + t3 * *p3
+        }
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+    use std::f32::consts::PI;
+
+    #[test]
+    fn lissajous_y_is_non_monotonic() {
+        let path = PathType::Lissajous {
+            center_x: 0.0,
+            center_y: 100.0,
+            amplitude_x: 120.0,
+            amplitude_y: 300.0,
+            ax: 2.0,
+            ay: 1.0,
+            delta: PI / 2.0,
+        };
+        let y0 = evaluate(&path, 0.0).y;
+        let y_mid = evaluate(&path, 0.25).y;
+        let y_end = evaluate(&path, 0.5).y;
+        // y rises then falls, proving the curve is not a straight line
+        assert!(
+            y_mid > y0,
+            "Lissajous y at t=0.25 ({}) should be above y at t=0 ({})",
+            y_mid,
+            y0
+        );
+        assert!(
+            y_end < y_mid,
+            "Lissajous y at t=0.5 ({}) should be below y at t=0.25 ({})",
+            y_end,
+            y_mid
+        );
+    }
+
+    #[test]
+    fn bezier_control_points_deviate_from_linear() {
+        let path = PathType::CubicBezier {
+            p0: Vec3::new(0.0, 450.0, 0.0),
+            p1: Vec3::new(0.0, 200.0, 0.0),
+            p2: Vec3::new(0.0, 100.0, 0.0),
+            p3: Vec3::new(0.0, -450.0, 0.0),
+        };
+        let y_bezier = evaluate(&path, 0.25).y;
+        // Linear interpolation between p0.y=450 and p3.y=-450 at t=0.25
+        let y_linear = 450.0 + (-450.0 - 450.0) * 0.25; // = 225
+        assert!(
+            y_bezier > y_linear,
+            "Bezier y ({}) should be strictly greater than linear interpolation y ({})",
+            y_bezier,
+            y_linear
+        );
+    }
+
+    #[test]
+    fn bezier_at_t1_equals_p3() {
+        let path = PathType::CubicBezier {
+            p0: Vec3::ZERO,
+            p1: Vec3::ZERO,
+            p2: Vec3::ZERO,
+            p3: Vec3::new(0.0, -450.0, 0.0),
+        };
+        let result = evaluate(&path, 1.0);
+        assert!(
+            (result.y - (-450.0)).abs() < 1e-4,
+            "Bezier y at t=1.0 ({}) should equal p3.y (-450)",
+            result.y
+        );
+    }
+
+    #[test]
+    fn bezier_at_t0_equals_p0() {
+        let path = PathType::CubicBezier {
+            p0: Vec3::new(0.0, 450.0, 0.0),
+            p1: Vec3::ZERO,
+            p2: Vec3::ZERO,
+            p3: Vec3::ZERO,
+        };
+        let result = evaluate(&path, 0.0);
+        assert!(
+            (result.y - 450.0).abs() < 1e-4,
+            "Bezier y at t=0.0 ({}) should equal p0.y (450)",
+            result.y
+        );
+    }
+}
diff --git a/src/game_state.rs b/src/game_state.rs
index 35c6f11..5175011 100644
--- a/src/game_state.rs
+++ b/src/game_state.rs
@@ -1,6 +1,8 @@
 use bevy::prelude::*;
 
-use crate::components::{Bullet, Enemy, EnemyBullet, GameOverUI, RestartMessage, StartButton, StartMenuUI};
+use crate::components::{
+    Bullet, Enemy, EnemyBullet, GameOverUI, RestartMessage, StartButton, StartMenuUI,
+};
 use crate::resources::RestartPressed;
 
 #[derive(Clone, Eq, PartialEq, Debug, Hash, Default, States)]
diff --git a/src/lib.rs b/src/lib.rs
index 9c16d49..6ca9739 100644
--- a/src/lib.rs
+++ b/src/lib.rs
@@ -7,6 +7,7 @@ pub mod bullet;
 pub mod components;
 pub mod constants;
 pub mod enemy;
+pub mod flight_paths;
 pub mod game_state;
 pub mod player;
 pub mod resources;
@@ -15,8 +16,7 @@ pub mod starfield;
 pub mod systems;
 
 use bullet::{
-    check_bullet_collisions, check_enemy_bullet_player_collisions, move_bullets,
-    move_enemy_bullets,
+    check_bullet_collisions, check_enemy_bullet_player_collisions, move_bullets, move_enemy_bullets,
 };
 use components::TractorBeam;
 use constants::{PLAYER_RESPAWN_DELAY, STARTING_LIVES, WINDOW_HEIGHT, WINDOW_WIDTH};
diff --git a/src/player.rs b/src/player.rs
index a552694..67cfb31 100644
--- a/src/player.rs
+++ b/src/player.rs
@@ -58,10 +58,10 @@ pub fn move_player(
 
     transform.translation.x += direction * player.speed * time.delta_secs();
     let half_width = PLAYER_SIZE.x / 2.0;
-    transform.translation.x = transform
-        .translation
-        .x
-        .clamp(-WINDOW_WIDTH / 2.0 + half_width, WINDOW_WIDTH / 2.0 - half_width);
+    transform.translation.x = transform.translation.x.clamp(
+        -WINDOW_WIDTH / 2.0 + half_width,
+        WINDOW_WIDTH / 2.0 - half_width,
+    );
 }
 
 pub fn handle_captured_player(
diff --git a/src/resources.rs b/src/resources.rs
index a2a0a9e..48effc5 100644
--- a/src/resources.rs
+++ b/src/resources.rs
@@ -5,6 +5,7 @@ use crate::constants::{
     ENEMY_SHOOT_INTERVAL, FORMATION_BASE_Y, FORMATION_COLS, FORMATION_ENEMY_COUNT,
     FORMATION_X_SPACING, FORMATION_Y_SPACING, SPECIAL_STAGE_INTERVAL, WINDOW_WIDTH,
 };
+use crate::flight_paths::PathType;
 
 /// Returns `true` when `stage_number` is a special stage (every 3rd stage).
 pub fn is_special_stage(stage_number: u32) -> bool {
@@ -64,6 +65,7 @@ pub struct StageConfig {
 pub struct StageConfigurations {
     pub stages: Vec<StageConfig>,
     pub special_stage: StageConfig,
+    pub flight_patterns: Vec<PathType>,
 }
 
 impl StageConfigurations {
@@ -111,9 +113,28 @@ impl Default for StageConfigurations {
             enemy_shoot_interval: ENEMY_SHOOT_INTERVAL,
         };
 
+        let flight_patterns = vec![
+            PathType::Lissajous {
+                ax: 2.0,
+                ay: 1.0,
+                delta: std::f32::consts::FRAC_PI_2,
+                amplitude_x: 120.0,
+                amplitude_y: 300.0,
+                center_x: 0.0,
+                center_y: 100.0,
+            },
+            PathType::CubicBezier {
+                p0: Vec3::new(-250.0, 450.0, 0.0),
+                p1: Vec3::new(-100.0, 200.0, 0.0),
+                p2: Vec3::new(100.0, 100.0, 0.0),
+                p3: Vec3::new(250.0, -450.0, 0.0),
+            },
+        ];
+
         Self {
             stages: vec![stage1, stage2],
             special_stage,
+            flight_patterns,
         }
     }
 }
@@ -167,3 +188,77 @@ pub struct AttackDiveTimer {
 pub struct RestartPressed {
     pub pressed: bool,
 }
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+    use std::f32::consts::FRAC_PI_2;
+
+    #[test]
+    fn default_has_at_least_two_flight_patterns() {
+        let cfg = StageConfigurations::default();
+        assert!(
+            cfg.flight_patterns.len() >= 2,
+            "default StageConfigurations should have at least 2 flight patterns, found {}",
+            cfg.flight_patterns.len()
+        );
+    }
+
+    #[test]
+    fn default_flight_patterns_have_different_path_types() {
+        let cfg = StageConfigurations::default();
+        let first = &cfg.flight_patterns[0];
+        let second = &cfg.flight_patterns[1];
+        // The two entries must be different PathType variants
+        match (first, second) {
+            (PathType::Lissajous { .. }, PathType::CubicBezier { .. }) => {}
+            (PathType::CubicBezier { .. }, PathType::Lissajous { .. }) => {}
+            _ => panic!(
+                "first and second flight patterns must have different PathType variants: {:?} vs {:?}",
+                first, second
+            ),
+        }
+    }
+
+    #[test]
+    fn first_default_pattern_is_lissajous_with_correct_params() {
+        let cfg = StageConfigurations::default();
+        let PathType::Lissajous {
+            ax,
+            ay,
+            delta,
+            amplitude_x,
+            amplitude_y,
+            center_x,
+            center_y,
+        } = &cfg.flight_patterns[0]
+        else {
+            panic!(
+                "first flight pattern should be Lissajous, got {:?}",
+                cfg.flight_patterns[0]
+            );
+        };
+        assert_eq!(*ax, 2.0, "Lissajous ax");
+        assert_eq!(*ay, 1.0, "Lissajous ay");
+        assert_eq!(*delta, FRAC_PI_2, "Lissajous delta");
+        assert_eq!(*amplitude_x, 120.0, "Lissajous amplitude_x");
+        assert_eq!(*amplitude_y, 300.0, "Lissajous amplitude_y");
+        assert_eq!(*center_x, 0.0, "Lissajous center_x");
+        assert_eq!(*center_y, 100.0, "Lissajous center_y");
+    }
+
+    #[test]
+    fn second_default_pattern_is_cubic_bezier_with_correct_control_points() {
+        let cfg = StageConfigurations::default();
+        let PathType::CubicBezier { p0, p1, p2, p3 } = &cfg.flight_patterns[1] else {
+            panic!(
+                "second flight pattern should be CubicBezier, got {:?}",
+                cfg.flight_patterns[1]
+            );
+        };
+        assert_eq!(*p0, Vec3::new(-250.0, 450.0, 0.0), "CubicBezier p0");
+        assert_eq!(*p1, Vec3::new(-100.0, 200.0, 0.0), "CubicBezier p1");
+        assert_eq!(*p2, Vec3::new(100.0, 100.0, 0.0), "CubicBezier p2");
+        assert_eq!(*p3, Vec3::new(250.0, -450.0, 0.0), "CubicBezier p3");
+    }
+}
diff --git a/src/systems.rs b/src/systems.rs
index 563cf7d..bbb9993 100644
--- a/src/systems.rs
+++ b/src/systems.rs
@@ -2,9 +2,11 @@ use bevy::prelude::*;
 use std::time::Duration;
 
 use crate::components::{Explosion, Invincible, Player};
-use crate::constants::{EXPLOSION_BASE_SIZE, EXPLOSION_COLOR, EXPLOSION_DURATION, EXPLOSION_MAX_SIZE};
+use crate::constants::{
+    EXPLOSION_BASE_SIZE, EXPLOSION_COLOR, EXPLOSION_DURATION, EXPLOSION_MAX_SIZE,
+};
 use crate::player::spawn_player_ship;
-use crate::resources::{CurrentStage, PlayerLives, Score, is_special_stage};
+use crate::resources::{is_special_stage, CurrentStage, PlayerLives, Score};
 use crate::starfield::spawn_starfield;
 
 pub fn setup(mut commands: Commands) {
diff --git a/tests/special_stage.rs b/tests/special_stage.rs
index 2c744f3..fd8e983 100644
--- a/tests/special_stage.rs
+++ b/tests/special_stage.rs
@@ -110,7 +110,10 @@ fn for_stage_returns_normal_config_at_stage_4() {
 fn special_stage_has_full_formation() {
     let configs = StageConfigurations::default();
     let config = configs.for_stage(3);
-    assert_eq!(config.enemy_count, 32, "Special stage should have 32 enemies");
+    assert_eq!(
+        config.enemy_count, 32,
+        "Special stage should have 32 enemies"
+    );
     assert_eq!(
         config.formation_layout.positions.len(),
         32,