3D Target Leading formula for a ballistic trajecto

Question

I was wondering if there was a target leading formula for target leading in 3d with a ballistic trajectory and if not i was wondering if it is possible to convert a 2d target leading formula to work as a 3d target leading formula by aligning the plane it is on along the motion of the target?

Answer 1

Leading the target can be easily calculated at the time the projectile is fired, however there is no guarantee the target won't change course, accelerate, or decelerate while the projectile is in the air. In any case, when you're leading the target you can either assume that the target's velocity will remain the same, or if it is accelerating, that its acceleration will remain the same.

Here's a blog post I wrote about predicting a traveled distance over time, accounting for acceleration. And here is the code I use.

This will calculate a distance traveled, over time, given a constant acceleration. In this case constantAccel is a speed, and if your target isn't accelerating then you would just use 0.0f for that parameter.

float CalcDistanceOverTime(const float initVelocity, 
                           const float constantAccel,
                           const float timeDelta)
{
    return  (initVelocity * timeDelta)
          + (0.5f * constantAccel * (timeDelta * timeDelta);
}

Here's an example:

Vector3 targetsVelocity(3.0f, 0.0f, 3.0f);
float targetsAcceleration = 1.0f;

float distanceTraveled = CalcDistanceOverTime(targetsVelocity, targetsAcceleration, timeDelta)

Vector3 finalPosition = targetsVelocity * distanceTraveled;

You may notice that you'll need a timeDelta to pass to this formula. This means, based on your projectile's trajectory and speed, you'll need to know about how long it will take to reach the target, however it is made more difficult by the fact that you don't know exactly how long that may take until you know where it will be. I'm not sure of the exact formula for this, but I believe using Calculus you could calculate, based on the speed and velocity of your projectile, and the speed and velocity of your target, accounting for gravity with both, that you should be able to find a point in space where these two can collide.

If the above method isn't feasible then you may be able to choose a fixed timeDelta, if you can guarantee your projectile can go at whatever angle and speed that you would like. For example, pick 3 seconds as your timeDelta, you know where the target will be in 3 seconds, and immediately fire a projectile that you know will reach that point in space within 3 seconds.

And just in case, here's a blog post about calculating ballistic trajectory in 3D. Calculating the time to target with this method should be simple, based on outgoing vertical velocity and position, just use gravity to calculate how many seconds until that position reaches the elevation at the target position.