Initially I was a year advanced in math back in school, but by the time I reached grade 10, Algebra II, my interest in all school subjects was so low that I simply did no homework, failed all classes (particularly, I failed Algebra II 3 times), and finally dropped out of high school in the middle of grade 12.

It was not until I started making video games seriously (and professionally) that I found a reason for any of the math they were trying to put into my head.
Luckily, due to the fact that I am not an idiot (in grade 9 when I actually did do my homework, I got a 100% in Geometry, having never missed a single answer on any homework, test or final; something never done before in my school’s history), I was able to catch up on most of all that math on my own, but finally I have created an equation that suits a physical model I also designed, but I am still not at the mathematical level needed to actually solve it.

Now I am the CTO of a game company in Tokyo and I need this solved (that is, better than my guess-and-check iterative method, which does work, but is too slow).
Without further ado:

Tp = target position.Tv = target velocity.Ta = target acceleration.Bp = bullet position (when originally fired).Bs = bullet speed.length² = A²+B²+C² in 3D.length²((Tp+Tvt+½×Tat²)-Bp) = (Bs*Bs)or:length²((Tp+Tvt+½×Tat²)-Bp)-(Bs*Bs) = 0Solve for t (time)

The keen observer can see a target moving along a trajectory and a bullet moving as well.
But this is not how to intercept the bullet with the target. As you can see, the direction of the bullet does not matter.
Bullet velocity is not a factor, it is the bullet speed.

This will ultimately be used in the game engine being developed in my company to allow bots to aim at targets.
Here, however, I am not interested in aiming a bot at a target; this equation requires that you solve for t, while the actual direction of the bullet to be fired is unknown.

I will also include this in an upcoming article on game AI and I will give credits to whoever can solve this (please show your work so that I can include an explanation of each step in the article).

Note: I already know how to solve this using iterative methods in C++. I already have bots that are devastatingly accurate (you have a 100% chance of being hit unless you manually change your own trajectory while the bullet is traveling). But when there are too many bots, the engine slows unacceptably. This equation correctly describes the same thing I am performing iteratively, so by solving for t in the equations above I hope I can increase the efficiency of the bot AI dramatically.


Thank you,
L. Spiro

I believe the dimensions do match.

length²() takes a 3D vector and returns a scalar value.
Bs is a scalar value, so Bs² is as well.


Think of it this way:

Bv = bullet velocity (3D vector).length²((Tp+Tvt+½×Tat²)-Bp)-length²(Bv) = 0

Now Bv is a 3D vector just like all the other variables, and like the rest it goes through the length²() function, which results in a single scalar.

Since length²(Bv) basically throws away the directional component of the velocity, it is equivalent to Bs*Bs (where Bs—speed—is a single scalar).
So the simplified and faster form of the equation is what I gave above.

However, since length²() is a function that changes the dimensions, perhaps this equation cannot be solved in this form?


I can see how to rewrite it such that we can remove the length²() function and instead repeat the equation 3 times, once for each of the X, Y, and Z scalars on a 3D vector, but solving for that would give me 3 different t (time) values and I don’t see how that can be useful.

Here is the expanded form:

(((Tpx+Tvxt+½×Taxt²)-Bpx)² + ((Tpy+Tvyt+½×Tayt²)-Bpy)² + ((Tpz+Tvzt+½×Tazt²)-Bpz)²) - (Bs*Bs) = 0

To me that looks even harder to solve (though it is exactly the same thing) but to a mathematical person perhaps it is easier.


I hope to include a solution to this equation in an upcoming article I will be writing for this site, and I will give credit in the article to whoever can solve this (showing each step for how it is solved). I will also provide code for solving it.


Thank you,
L. Spiro

Whoops, I made a mistake when I rewrote my equation (before I posted it here it was in another format, which was correct).

It was supposed to be:

length²((Tp+Tvt+½×Tat²)-Bp)-(Bst*Bst) = 0

Or:

(((Tpx+Tvxt+½×Taxt²)-Bpx)² + ((Tpy+Tvyt+½×Tayt²)-Bpy)² + ((Tpz+Tvzt+½×Tazt²)-Bpz)²) - Bst² = 0

Naturally, speed multiplied by time (then squared) was my intent. Does this fix the dimension problem?
I am pretty sure this accurately reflects my concept, and I know that my concept is correct and works.

Considering my explained self-taught mathematical background, it is reasonable for me not to understand his terminology (how “dimensions” apply to math are obviously not how they apply to vectors), and I felt I responded cordially, taking into account “I might not know exactly what he means so I should explain how I think it is correct directly and without sounding as though I am arguing,” so was a drop to my “user rating” really necessary?
Especially when this is for a GameDev article (in addition to my work), meaning something I plan to give back to the community…


With my hotfix, is the dimensionality problem solved, and can this equation be solved for t?


L. Spiro