feat: Schwarzschild RK4 geodesic integrator + black shadow

assets/shaders/black_hole.wgsl

@@ -1,5 +1,6 @@
 #import bevy_sprite::mesh2d_vertex_output::VertexOutput
 #import "shaders/ray_gen.wgsl"
+#import "shaders/geodesic_schwarzschild.wgsl"
 
 struct BlackHoleUniforms {
     eye: vec4<f32>,
@@ -31,11 +32,17 @@ struct BlackHoleUniforms {
 
 @fragment
 fn fragment(in: VertexOutput) -> @location(0) vec4<f32> {
-    // in.uv is [0,1] across the quad. Center and flip y, apply aspect.
     let aspect = uniforms.resolution.x / uniforms.resolution.y;
     var uv = (in.uv * 2.0 - 1.0);
     uv.x *= aspect;
     let dir = ray_direction(uv);
-    // Visualize ray direction as a color (sanity check).
-    return vec4<f32>(abs(dir), 1.0);
+    // Step size scales with how far we need to travel (~ distance/ steps).
+    let dt = max(length(uniforms.eye.xyz), 20.0) / f32(uniforms.steps);
+    let res = classify_ray(uniforms.eye.xyz, dir, uniforms.steps, dt);
+    if (res.status == 1u) {
+        // Captured = shadow.
+        return vec4<f32>(0.0, 0.0, 0.0, 1.0);
+    }
+    // Escaped: dim grey for now (stars come in Task 11).
+    return vec4<f32>(0.02, 0.02, 0.02, 1.0);
 }

assets/shaders/geodesic_schwarzschild.wgsl

@@ -0,0 +1,55 @@
+const R_ESCAPE: f32 = 1000.0;
+
+// One RK4 sub-step derivative of (pos, dir) under the Schwarzschild
+// bending acceleration. Rs is uniforms.rs.
+fn deriv(pos: vec3<f32>, dir: vec3<f32>) -> (vec3<f32>, vec3<f32>) {
+    let r = length(pos);
+    let rs = uniforms.rs;
+    // Angular momentum squared: |cross(pos, dir)|^2
+    let h = cross(pos, dir);
+    let h2 = dot(h, h);
+    // Avoid division by zero.
+    let r5 = max(r * r * r * r * r, 1e-6);
+    // d(pos)/dt = dir
+    let dpos = dir;
+    // d(dir)/dt = bending acceleration (re-normalized each step in integrate)
+    let accel = -1.5 * rs * h2 / r5 * pos;
+    return (dpos, accel);
+}
+
+// Integrate a ray from `pos` along `dir`. Returns:
+//   .status: 0 = escaped, 1 = captured (shadow)
+//   .final_pos, .final_dir: end state (used for sky sampling on escape)
+struct RayResult {
+    status: u32,
+    final_pos: vec3<f32>,
+    final_dir: vec3<f32>,
+}
+
+// Accumulator callback pattern: the caller passes a function-style body via
+// a per-step check. Because WGSL has no first-class closures, we inline the
+// per-step intersection tests in black_hole.wgsl's integrate_and_trace().
+// This function returns ONLY the escape/capture classification, used as a
+// fallback when no scene object is hit.
+fn classify_ray(start_pos: vec3<f32>, start_dir: vec3<f32>, steps: u32, dt: f32) -> RayResult {
+    var pos = start_pos;
+    var dir = start_dir;
+    for (var i: u32 = 0u; i < steps; i = i + 1u) {
+        let r = length(pos);
+        if (r < uniforms.rs) {
+            return RayResult(1u, pos, dir);
+        }
+        if (r > R_ESCAPE) {
+            return RayResult(0u, pos, dir);
+        }
+        // RK4
+        let (k1p, k1d) = deriv(pos, dir);
+        let (k2p, k2d) = deriv(pos + k1p * dt * 0.5, normalize(dir + k1d * dt * 0.5));
+        let (k3p, k3d) = deriv(pos + k2p * dt * 0.5, normalize(dir + k2d * dt * 0.5));
+        let (k4p, k4d) = deriv(pos + k3p * dt,     normalize(dir + k3d * dt));
+        pos = pos + (k1p + 2.0 * k2p + 2.0 * k3p + k4p) * dt / 6.0;
+        dir = normalize(dir + (k1d + 2.0 * k2d + 2.0 * k3d + k4d) * dt / 6.0);
+    }
+    // Ran out of steps without a clear verdict: treat as escaped.
+    return RayResult(0u, pos, dir);
+}