Hello there!

I am the author of EVE Mathematics article linked by Darinas. My article (Chapter 1) has some data specific to the solution of turning and the necessary mathematics is now posted there. I will reprise the method of solution here, and I have posted it along with some relevant graphs in my chapter here. The pursuit curve approach is also relevant, but only to the more complex piloting behavior that happens when using 'Approach' or other built-in motion programs. I have done plenty of math for these circumstances in EVE, however, I have not made time to write them up directly. Perhaps the day will come when there is sufficient interest for a complete treatise on more of the implications of EVE's mechanics...

For now lets talk about the 'double-clicking' problem that Werner Lucifer has outlined for us. To start, lets narrow the situation to motion in a 2-D plane. Indeed, any two-vector command can be represented in a plane chosen to contain both vectors.

What we need is a solution to Equation 1-5 that takes into account an initial ship vector state, and then a change in that state which represents our new motion direction vector and magnitude. Lets say that you are moving at V_{MAX} in the x direction, and you double-click to accelerate in the y direction. For right angle movement like this, it is particularly simple, because the dimensions are completely independent and a method of solution does not require a particular solution for the x direction. Writing the vectors, our initial motion is, V_{init} = [V_{MAX} 0], and the new direction vector would be, V_{comm} = [0 V_{MAX}].

Using Equation 1-5 I simply write the solution in the independent dimensions by inspection. So, in the x direction:

v_x(t) = V_{MAX} * exp(-t/tau)

and in the y direction:

v_y(t) = V_{MAX} * (1 - exp(-t/tau))

Now, you wanted to know what was the total forward velocity of your ship. I can find this by computing the total velocity vector length, or,

v_total(t) = V_{MAX} * sqrt( 2*exp(-t/tau)^2 + 1 - 2*exp(-t/tau))

As an exersize for the reader, see if you can compute a closed form solution for the minimum velocity obtained during motion for piloting changes (Hint: this simplifies nicely for the right-angle case). This has implications for piloting when taking damage from missiles because damage from them depends on total velocity, not angular velocity.

You can compute the velocity profile for an arbitrary angle, theta, by repeating the above process for velocity vectors [V_{MAX} 0] => [V_{MAX}*cos(theta) V_{MAX}*sin(theta)]. Just be careful to compute both homogeneous and particular solutions for the x direction.

Your other questions are, in part, answered in my notes, although I could certainly write more about it.

I thoroughly enjoyed finding this question on the front-page of skilltrainingcompleted. I will be sure to return to this site more often. With any luck we can build a community of people interested in both practical and theoretical understanding of EVE mechanics. And now, back to my tasty lunch of fried chicken...

Good luck, and fly smart.

upsideyourhead

-- EDIT -- I added data to my chapter in Figure I-10. I took data for 45, 90, and 135 degree turns. Enjoy.