Time Spent Programming

Life

Well, writing argumentative essays and doing math and being intellectual is fun and all, sometimes it's nice to just kick back and relax a bit, eh?

Working on Artificial Intelligence (actually "Pseudointelligence" allegedly; but AI sounds better), and having a blast not getting it done in code. These things get really complicated really fast when trying to transition between mathematical language and computer language; even more so when trying to get it into plain language, so I will not bother explaining it here.

Wasting time on purpose is a great way to relax. Clearing your mind out of all the loose little ends every now and then does wonders for stress. Have to steel yourself for senior year, eh? Between independent research, challenging AP exams, leadership positions, college applications, and social life, I might as well screw around while I can.

Speaking of screwing around, I might as well get something done for the Annual Roguelike Release Party. Expect to see something out of me September 19th, 2010. Hold me to this, please.

Meh, anyways, have a good last few weeks before Labor Day. I have to get some work done.

A Time and a Place

RescueTime Efficiency Summary:

-0.58 over 17h 8m vs. Avg. User 0.18 over 11h 10m

Sorry about that; bbl.

Writer's Block

What is writer's block?

Lack of skill, of concentration?

What lies between our conception and our paper?

... A writer's block?

Was there anything there to write in the first place?

How would I know?

Why can't I overcome this hurdle?

Am I good enough for it?

What do I have to do to see?

... Adequate enough?

When will my idea become clear?

The idea is mine?

Where is the inspiration, the light?

Is this the inspiration, the light?

Who is writer's block?

Me.

Web 2.0

Web 2.0; in an era where bandwidth is cheap, and storage space even cheaper, there has been a proliferation of blogs, news websites, and public forums where once there were none. Electronic development has given the many a voice that can be heard around the world; but at what price? When information is cheap, the emphasis is no longer on the information itself, but on those who present it. This is the problem with Web 2.0.

Web 2.0 has been, and will continue being, an amazing step forward in humanity's capacity to communicate. The fact that I can look up one of over 3 million articles on the Wikipedia and have the luxury of demanding that each point that I read be clearly cited with a respectable source is an advantage that I will not deny of Web 2.0. Or the fact that I can catch up on my friends with the click of a single button; no more rummaging through lost and broken contacts, scouring social networks for someone who might still be in contact.

Back in the "ye olde days" of the late 1990s, Yahoo had a single news portal, and that was all that was deemed necessary. Today, Google has a separate search algorithm for news stories, designed specifically to sort through the hundreds, if not thousands of mainstream blogs catering to news ranging from the intimately local to the expansively global. But, when attempting to do research for a political piece I was writing, it struck me how useless this search algorithm was.

On the sidebar of common options, Google News allows you to choose between three different categories: "All News", "Images", and "Blogs". Notice that a "News" category is missing? In an era when information is cheap, the emphasis is no longer on the information. Although I lack hard numbers to cite here, a Google News search will bring up around 9 editorials or other opinion pieces out of every 10 stories returned. When did editorials become so popular? My local newspaper runs around 2 or 3 pages of editorials out of a 20 page newspaper (not counting sports, lifestyle, advertisements, etc...). On the Internet, information is cheap; and if writers want to make a living peddling something cheap, they have to either dress it up as something more expensive, like an "opinion article", as if that added any intellectual value, or deliver it en masse; sometimes both.

Looking up information on the proposed Cordoba House to write an unbiased summary of its political effects on American politics, I came across the fact that the Anti-Defamation League had asked the Cordoba House to reconsider its location. This seemed out of the ordinary, so I searched this story up; lo and behold, scores of articles proclaiming the ADL's incredible hypocrisy, or how the ADL was ardently against the Cordoba House, or the reactions of the Jewish community on the ADL's decision. About two or three of these articles were polite enough to actually quote more than a line or two from the ADL's own words. About zero of them were polite enough to actually link to the ADL's press release, which I had to find myself.

Some pundits of the Internet claim the problem is that "we're drowning in information". To the contrary, I'd propose that we're drowning the information.

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?

Back On Track, Part I

Sorry about that. Back to normal, scheduled updates.

Let me introduce you to one of my favorite subjects, math. I've been working on a problem interesting to me these past few days:

Say you have a vehicle traveling forward at a speed given by v(t), that can only change its direction by turning with an angular velocity ω(t). What is the equation describing the position of the object, , at any given time?

The reason this is so interesting a question is that it involves combining both linear and angular motion; a complex and non-trivial task. First, let us set some constraints on the problem to give us an idea of where we have to go: let us set a constant speed of V and constant angular velocity of T, to simplify our equations vastly. We can easily see that the total distance traveled is given by the following equation:

By similar math, the vehicle will have traveled a total arc of θ(t) = Tt. But that is not enough to fully describe the motions of the vehicle. One more piece is needed, the turning radius.

Consider for a moment the formula s=θr, the formula for the arc length of a circle. But what does this have to do with velocity? Assume with me for a moment that the vehicle is moving along a circle (a fact we will prove in a little bit). Using the formulas we have defined earlier, we recognize that s(t) = θ(t)r. We have left the radius in boldface just to illustrate how the parts fit together. Defining some equation r(t) as the turning radius, and solving for r, we find:

It is a reasonable to assume the vehicle is moving in a circle, but assumptions are not good enough for mathematics. We will show that the vehicle is actually traveling in a circle. More rigorously we define x(t) and y(t).

Why? The actual velocity of the vehicle at any time t is a vector with an angle of θ(t) and a magnitude of v(t). Notice an important distinction; speed is a scalar quantity, and hence there is no arrow over our speed function. Velocity is separate from speed. The same goes for the position  and the x-coordinate x(t). The appropriate x and y is obtained by integrating the respective components of the velocity vector, namely the formulas you see in the integrals. Once again, we will assume a constant speed of V and a constant angular velocity of T. Integrating these formulas out using previously discussed, we find that:

This is actually a tricky answer. The naive interpretation of "moving in a circle" would be to assume that the x component is found by multiplying the radius by the cosine, and the y component by the sine, but one of the harshest lessons mathematicians learn is to never place too much trust intuition. It can point you in the right direction, but it is no substitute for proof. I actually fell victim to this. There is one point of interest in these two equations, the two constants of integration. The point (X, Y) is the center of the turning circle. To discuss proof of this is another non-trivial problem which will lead us into tomorrow's discussion.

PS: Images are kind of mangled in this post. I had to tinker around with blogger for a little bit before fixing them. Next post will have functional images.