Back On Track, Part II

Let me introduce you to the concept of a transformation. In our limited definition, a transformation takes a set of coordinate spaces and maps it to a new set of coordinates. For example, a translation, something we should all be familiar with, takes a coordinate and 'slides' it across. In mathematical parlance, T_{(u, v)}(x, y) = (x', y'). But there's slightly more to this story. A transformation is a function, and adds a new set of equalities. In this case, x' = x + u and y' = y + v.

I said in the previous post that the point (X, Y) was the center of the turning circle, and that we would prove that here. We actually don't have to; these are defined to be the center of the turning circle by definition of a circle using parametric equations. However, to find this coordinate is slightly more difficult.

Consider in a real world situation, we would only have x(0), y(0), and θ(0), the starting conditions of the vehicle. But the equations we have derived demand (X, Y) and thus we have to solve for them. But there's actually a slight caveat that I did not mention earlier: notice the specific definition for s(t). We defined that at t = 0, the vehicle starts with a 'clean slate'. That the initial position is zero, the initial distance traveled is 0, and the initial angle is 0. This is not necessarily a bad thing, because we can finagle reality into bending to our wishes. (By the way, the value for (X, Y) is (0, r(t)), which is fairly trivial to calculate under our premises; just simply plug in appropriate values, i.e. t = 0, x(t) = 0, etc... to find these values).

Let's define four new variables, t_r, x_r, y_r, and θ_r, which represent the 'real' initial x and y coordinates and the starting angle, respectively. These might be CPU clock time, GPS coordinates and angle relative to the equator; or unix timestamp, coordinates relative to a nearby building and angle relative to said building; the actual values of these numbers can vary wildly. In order to deal with this, we will first define a transformation that maps t' = t - t_r, another transformation that maps x' = x - x_r and y' = y - y_r, and finally a transformation that maps θ' = θ - θ_r.

There is one special condition. We can translate time and spatial coordinates separately from each other because they are separate dimensions, i.e. for the purposes of translations, they act independently of one another. However, θ is a different story. When we do such a transformation, we aren't actually doing a translation: we are doing a rotation. Sparing you the details, just suffice to know that using a rotational matrix, you find new equations x" = x' cos(-θ_r) - y' sin(-θ_r) and y" = x' sin(-θ_r) + y' cos(-θ_r).

Now, there are no caveats to our equations. Everything works out beautifully, now that our equations are done in terms of t', x", y", and θ' instead of t_r, x_r, y_r, and θ_r. To convert between the numbers our formulas output and numbers in the 'real' world, simply apply the inverse of these transformations backwards. That is an exercise left up to the reader; have some fun trying to figure it out.

The important thing to remember is that you can always convert from one system of coordinates to another by replacing the variables by their original mapping. In our various calculations, we could have replaced every instance of t by it's actual value, t - t_r, but you can see how quickly that would get even more confusing. Mathematicians strive to put constraints on problems so that they are more manageable; it is the same reason that up to this point, we have been assuming speed and angular velocity are constant.

So, what if we stop assuming speed and angular velocity are constant? Well, I don't know. The farthest I've gotten in the formula is actually only one step away from our formal definitions in the previous part:

In these equations, I have removed the superfluous (t)'s from all of the functions, mostly just for aesthetic concerns, and I have reexpressed the integral in terms of θ. The chain of logic is actually pretty simple: rearranging dθ/dt = ω, you get dt = ω^-1dθ. Replacing dt in the original integral gets you the formulas above. But it isn't actually any closer to solving the problem. Only when you are given more constraints, such as the value of ω, or dv/dt, also known as the acceleration, can you begin to work somewhere else. Or not.

Some might paint me as dejected for not getting anywhere. But quite the contrary; I could not be happier. Mathematics is not always in the answer; sure, the rush of epiphany is unparalleled. But the true object of mathematics is to push one's own boundaries. I learned a lot attacking this problem, tracking down false leads, making mistakes, referencing material. But even though I have not succeeded, I have learned a lot, and I have pushed my boundaries. Isn't this all that we can do?