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Author: Tspk91

ProjectileUtil.cs

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+//All in world space! Gets point you have to aim to
+//NOTE: this will break with infinite speed projectiles!
+//https://gamedev.stackexchange.com/questions/149327/projectile-aim-prediction-with-acceleration
+public static Vector3 GetTargetLeadingPositionQuadratic(Vector3 launchPosition, Vector3 launcherVelocity, Vector3 projectileAcceleration, Vector3 targetPosition, Vector3 targetVelocity, Vector3 targetAcceleration, float distance, float projectileSpeed, int iterations)
+{
+    Vector3 pT = targetPosition - launchPosition;
+    Vector3 vT = targetVelocity - launcherVelocity;
+    Vector3 aT = targetAcceleration;
+    float s = projectileSpeed;
+    Vector3 aP = projectileAcceleration;
+
+    Vector3 accel = aT - aP;
+
+    //time to target guess
+    float guess = distance / s;
+
+    if (iterations > 0)
+    {
+        //quartic coefficients
+        float a = Vector3.Dot(accel, accel) * 0.25f;
+        float b = Vector3.Dot(accel, vT);
+        float c = Vector3.Dot(accel, pT) + Vector3.Dot(vT, vT) - s * s;
+        float d = 2f * Vector3.Dot(vT, pT);
+        float e = Vector3.Dot(pT, pT);
+
+        //solve with newton
+        float finalGuess = SolveQuarticNewton(guess, iterations, a, b, c, d, e);
+
+        //use first guess if negative or zero
+        if (finalGuess > 0f)
+        {
+            guess = finalGuess;
+        }
+    }
+
+    Vector3 travel = pT + vT * guess + 0.5f * aT * guess * guess;
+
+    return launchPosition + travel;
+}
+
+//All in world space! Gets launch velocity you have to aim to
+//NOTE: this will break with infinite speed projectiles!
+//https://gamedev.stackexchange.com/questions/149327/projectile-aim-prediction-with-acceleration
+public static Vector3 GetTargetLeadingVelocityQuadratic(Vector3 launchPosition, Vector3 launcherVelocity, Vector3 projectileAcceleration, Vector3 targetPosition, Vector3 targetVelocity, Vector3 targetAcceleration, float distance, float projectileSpeed, int iterations)
+{
+    Vector3 pT = targetPosition - launchPosition;
+    Vector3 vT = targetVelocity - launcherVelocity;
+    Vector3 aT = targetAcceleration;
+    float s = projectileSpeed;
+    Vector3 aP = projectileAcceleration;
+
+    Vector3 accel = aT - aP;
+
+    //time to target guess
+    float guess = distance / s;
+
+    if (iterations > 0)
+    {
+        //quartic coefficients
+        float a = Vector3.Dot(accel, accel) * 0.25f;
+        float b = Vector3.Dot(accel, vT);
+        float c = Vector3.Dot(accel, pT) + Vector3.Dot(vT, vT) - s * s;
+        float d = 2f * Vector3.Dot(vT, pT);
+        float e = Vector3.Dot(pT, pT);
+
+        //solve with newton
+        float finalGuess = SolveQuarticNewton(guess, iterations, a, b, c, d, e);
+
+        //use first guess if negative or zero
+        if (finalGuess > 0f)
+        {
+            guess = finalGuess;
+        }
+    }
+
+    Vector3 travel = pT + vT * guess + 0.5f * aT * guess * guess;
+
+    Vector3 launchVelocity = travel / guess - 0.5f * aP * guess;
+
+    return launchVelocity;
+}
+
+static float SolveQuarticNewton(float guess, int iterations, float a, float b, float c, float d, float e)
+{
+    for (int i = 0; i < iterations; i++)
+    {
+        guess = guess - EvalQuartic(guess, a, b, c, d, e) / EvalQuarticDerivative(guess, a, b, c, d);
+    }
+    return guess;
+}
+
+static float EvalQuartic(float t, float a, float b, float c, float d, float e)
+{
+    return a * t * t * t * t + b * t * t * t + c * t * t + d * t + e;
+}
+
+static float EvalQuarticDerivative(float t, float a, float b, float c, float d)
+{
+    return 4f * a * t * t * t + 3f * b * t * t + 2f * c * t + d;
+}