Add parabolic orbit rendering function

- Add render_parabolic_orbit() with true anomaly range -π*0.95 to π*0.95
- Use 80 segments for smooth parabolic curve visualization
- Update render_orbit() branching to handle parabolic (0.98 <= e <= 1.02)
- Calculate semi-latus rectum p = h²/μ for parabolic orbits
- Maintain existing elliptical/hyperbolic orbit rendering

Claude

src/renderer.cpp

@@ -174,7 +174,7 @@ static void render_elliptical_orbit(double a, double e, OrbitalBasis basis,
 }
 
 static void render_hyperbolic_orbit(double p, double e, OrbitalBasis basis,
-                                    Vec3 parent_pos, RenderState* render_state, Color color) {
+                                     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;
@@ -194,6 +194,27 @@ static void render_hyperbolic_orbit(double p, double e, OrbitalBasis basis,
     }
 }
 
+static void render_parabolic_orbit(double p, OrbitalBasis basis,
+                                   Vec3 parent_pos, RenderState* render_state, Color color) {
+    double max_true_anomaly = PI * 0.95;
+
+    int segments = 80;
+    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 + cos(theta1));
+        double r2 = p / (1.0 + 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) {
@@ -232,13 +253,25 @@ void render_orbit(CelestialBody* body, CelestialBody* parent, RenderState* rende
         if (a <= 0.0) return;
 
         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);
+    } else if (e > 1.02) {
+        double a = mu / (2.0 * (-specific_energy));
         double p = a * (1.0 - e * e);
 
         if (p <= 0.0) return;
 
         render_hyperbolic_orbit(p, e, basis, parent->position, render_state, orbit_color);
+    } else {
+        Vec3 h_vec = {
+            r_vec.y * body->velocity.z - r_vec.z * body->velocity.y,
+            r_vec.z * body->velocity.x - r_vec.x * body->velocity.z,
+            r_vec.x * body->velocity.y - r_vec.y * body->velocity.x
+        };
+        double h_squared = h_vec.x * h_vec.x + h_vec.y * h_vec.y + h_vec.z * h_vec.z;
+        double p = h_squared / mu;
+
+        if (p <= 0.0) return;
+
+        render_parabolic_orbit(p, basis, parent->position, render_state, orbit_color);
     }
 }