Add dynamic orbital rendering with hyperbolic/parabolic support

- Calculate orbital elements dynamically from current state and parent
- Orbits now update automatically when parent body changes
- Add support for parabolic trajectories (e ≈ 1)
- Add support for hyperbolic trajectories (e > 1) with limited extent
- Properly orient orbits in 3D space using eccentricity vector
- Limit hyperbolic/parabolic drawing to avoid infinite vertices

Refactored for code reuse and separation of concerns:
- calculate_orbital_basis(): Computes orbital reference frame
- orbital_to_cartesian(): Transforms orbital coords to 3D space
- draw_orbit_segment(): Handles rendering of individual segments
- render_elliptical_orbit(): Draws closed elliptical paths
- render_hyperbolic_orbit(): Draws open hyperbolic/parabolic paths

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-Authored-By: Claude Sonnet 4.5 <noreply@anthropic.com>

src/renderer.cpp

@@ -102,23 +102,115 @@ void render_body(CelestialBody* body, RenderState* render_state) {
     DrawSphereWires(position, radius, 16, 16, color);
 }
 
+struct OrbitalBasis {
+    Vec3 periapsis_dir;
+    Vec3 normal;
+    Vec3 q_vec;
+};
+
+static OrbitalBasis calculate_orbital_basis(Vec3 r_vec, Vec3 velocity, Vec3 e_vec) {
+    OrbitalBasis basis;
+    basis.periapsis_dir = vec3_normalize(e_vec);
+
+    Vec3 h_vec = {
+        r_vec.y * velocity.z - r_vec.z * velocity.y,
+        r_vec.z * velocity.x - r_vec.x * velocity.z,
+        r_vec.x * velocity.y - r_vec.y * velocity.x
+    };
+    basis.normal = vec3_normalize(h_vec);
+
+    basis.q_vec = {
+        basis.normal.y * basis.periapsis_dir.z - basis.normal.z * basis.periapsis_dir.y,
+        basis.normal.z * basis.periapsis_dir.x - basis.normal.x * basis.periapsis_dir.z,
+        basis.normal.x * basis.periapsis_dir.y - basis.normal.y * basis.periapsis_dir.x
+    };
+
+    return basis;
+}
+
+static Vec3 orbital_to_cartesian(double x, double y, OrbitalBasis basis, Vec3 parent_pos) {
+    return {
+        basis.periapsis_dir.x * x + basis.q_vec.x * y + parent_pos.x,
+        basis.periapsis_dir.y * x + basis.q_vec.y * y + parent_pos.y,
+        basis.periapsis_dir.z * x + basis.q_vec.z * y + parent_pos.z
+    };
+}
+
+static void draw_orbit_segment(double x1, double y1, double x2, double y2,
+                               OrbitalBasis basis, Vec3 parent_pos,
+                               RenderState* render_state, Color color) {
+    Vec3 p1_sim = orbital_to_cartesian(x1, y1, basis, parent_pos);
+    Vec3 p2_sim = orbital_to_cartesian(x2, y2, basis, parent_pos);
+
+    Vector3 p1 = sim_to_render(p1_sim, render_state->distance_scale);
+    Vector3 p2 = sim_to_render(p2_sim, render_state->distance_scale);
+
+    DrawLine3D(p1, p2, color);
+}
+
+static void render_elliptical_orbit(double a, double e, OrbitalBasis basis,
+                                   Vec3 parent_pos, RenderState* render_state, Color color) {
+    double b = a * sqrt(1.0 - e * e);
+    double c = a * e;
+
+    int segments = 100;
+    for (int i = 0; i < segments; i++) {
+        float theta1 = (float)i / segments * 2.0f * PI;
+        float theta2 = (float)(i + 1) / segments * 2.0f * PI;
+
+        double x1 = a * cos(theta1) - c;
+        double y1 = b * sin(theta1);
+        double x2 = a * cos(theta2) - c;
+        double y2 = b * sin(theta2);
+
+        draw_orbit_segment(x1, y1, x2, y2, basis, parent_pos, render_state, color);
+    }
+}
+
+static void render_hyperbolic_orbit(double p, double e, OrbitalBasis basis,
+                                    Vec3 parent_pos, RenderState* render_state, Color color) {
+    double max_true_anomaly = (e > 1.01) ? acos(-1.0 / e) * 0.95 : PI * 0.48;
+
+    int segments = 60;
+    for (int i = 0; i < segments; i++) {
+        float theta1 = -max_true_anomaly + (float)i / segments * 2.0f * max_true_anomaly;
+        float theta2 = -max_true_anomaly + (float)(i + 1) / segments * 2.0f * max_true_anomaly;
+
+        double r1 = p / (1.0 + e * cos(theta1));
+        double r2 = p / (1.0 + e * cos(theta2));
+
+        double x1 = r1 * cos(theta1);
+        double y1 = r1 * sin(theta1);
+        double x2 = r2 * cos(theta2);
+        double y2 = r2 * sin(theta2);
+
+        draw_orbit_segment(x1, y1, x2, y2, basis, parent_pos, render_state, color);
+    }
+}
+
 // Render orbit path for a body
 void render_orbit(CelestialBody* body, CelestialBody* parent, RenderState* render_state) {
     if (body->parent_index == -1 || parent == NULL) {
         return;
     }
 
-    double a = body->semi_major_axis;
-    double e = body->eccentricity;
+    Vec3 r_vec = vec3_sub(body->position, parent->position);
+    double r = vec3_magnitude(r_vec);
+    double v = vec3_magnitude(body->velocity);
 
-    if (a <= 0.0) {
-        return;
-    }
+    if (r < 1.0) return;
 
-    double b = a * sqrt(1.0 - e * e);
-    double c = a * e;
+    double mu = G * parent->mass;
+    double specific_energy = (v * v) / 2.0 - mu / r;
 
-    Vector3 parent_pos = sim_to_render(parent->position, render_state->distance_scale);
+    double v_squared = v * v;
+    double r_dot_v = r_vec.x * body->velocity.x + r_vec.y * body->velocity.y + r_vec.z * body->velocity.z;
+    Vec3 e_vec = {
+        (v_squared - mu / r) * r_vec.x - r_dot_v * body->velocity.x,
+        (v_squared - mu / r) * r_vec.y - r_dot_v * body->velocity.y,
+        (v_squared - mu / r) * r_vec.z - r_dot_v * body->velocity.z
+    };
+    double e = vec3_magnitude(e_vec) / mu;
 
     Color orbit_color = {
         (unsigned char)(body->color[0] * 128),
@@ -127,28 +219,20 @@ void render_orbit(CelestialBody* body, CelestialBody* parent, RenderState* rende
         128
     };
 
-    int segments = 100;
-    for (int i = 0; i < segments; i++) {
-        float theta1 = (float)i / segments * 2.0f * PI;
-        float theta2 = (float)(i + 1) / segments * 2.0f * PI;
+    OrbitalBasis basis = calculate_orbital_basis(r_vec, body->velocity, e_vec);
 
-        float x1 = (float)(a * cos(theta1) - c);
-        float y1 = (float)(b * sin(theta1));
-        float x2 = (float)(a * cos(theta2) - c);
-        float y2 = (float)(b * sin(theta2));
+    if (e < 0.98) {
+        double a = -mu / (2.0 * specific_energy);
+        if (a <= 0.0) return;
 
-        Vector3 p1 = {
-            parent_pos.x + x1 * (float)render_state->distance_scale,
-            parent_pos.y,
-            parent_pos.z + y1 * (float)render_state->distance_scale
-        };
-        Vector3 p2 = {
-            parent_pos.x + x2 * (float)render_state->distance_scale,
-            parent_pos.y,
-            parent_pos.z + y2 * (float)render_state->distance_scale
-        };
+        render_elliptical_orbit(a, e, basis, parent->position, render_state, orbit_color);
+    } else {
+        double a = (e > 1.01) ? mu / (2.0 * (-specific_energy)) : r / (1.0 + e);
+        double p = a * (1.0 - e * e);
+
+        if (p <= 0.0) return;
 
-        DrawLine3D(p1, p2, orbit_color);
+        render_hyperbolic_orbit(p, e, basis, parent->position, render_state, orbit_color);
     }
 }