Thinking in polar coordinates: #weareallcartesiansnow

You remember those polar coordinates we did in pre-calc/calc/trig classes? Yeah, it is clearly my desire to revive memories of those days on which we wrote equations as f (r,θ) instead of f (x,y), tried to plug-in values of -2π to 0 to 2π and figured out the direction of a graph, when we moaned and groaned about who would actually use something as pretentious as polar coordinates.

Turns out dart boards are a very straight-forward real-life application of polar coordinates. Each wedge extends from θ to θ+18 and from s to S where s is the radius of the outer bulls eye rings. The double and triple rings also span the same angles, but are characterized by different bounds from the center.

Why am I talking about polar coordinates and dart boards today? Well, due to sheer boredom I started throwing darts recently. It took me a while to get a majority of my throws on the board. I’m doing well now, and I’ve been trying to hit one number consistently. It’s very easy to get carried away throwing darts… the target is well-defined, it’s easy to experiment and improve your throw, and every so often you can hit the target, which is a major motivational factor.

I was trying to hit the double ring on the 17 today. It was way more difficult than I thought it would be. Initially it was difficult to pinpoint why it was so difficult, but I think it all has to do with the fact that we’re not good at dealing with polar coordinates.

Throw a dart randomly and hope it hits a spot in the scoring area. Somewhere along the radius passing through that point lie a few points which are included in the triple ring. If you haven’t already hit your goal of triple ring, you can say, “Ah, I need to throw at the same θ, but just need to change my r up a bit.”

Of course, our first instinct is to not say that… we say “I need to move it x units to the right [or left] and y units down [or up]” and try all kinds of stuff (gravity assist, monocular vision, offering of red jasmines hibiscus flowers and modaks to Ganesh) etc., with the hope that the dart will behave the way we want it to behave. This is perfectly fine. Gravity and initial velocity have an enormous influence on the trajectory of the dart, so we kind of have to isolate the vertical motion of the dart from the horizontal.

Yeah,  dart boards are a straightforward example of polar coordinates, but thinking in polar coordinates is decidedly not straightforward. Polar coordinates are a pretty elegant method to describe some stuff mathematically, but in everyday life it’s going to be really difficult to think of objects as being r units away and θ degrees clockwise/counterclockwise from some other point. To re-iterate the point in my title #weareallcartesiansnow.

So, yeah, it seems way easier to control the horizontal motion of a dart than it is to control the vertical motion. As noted above, there are 3 major factors affecting the trajectory (gravity, initial velocity, initial height; air resistance notwithstanding) whereas the horizontal coordinates are only affected by the initial position and velocity.

Phew, enough intellectualizing about darts. Time to sleep.

–Gautam

EDIT: h and j are very close to each in the English alphabet.

Pictures in the snow

I went out on an hour-long walk outside. It was amazing. I think I might go running in the snow in the future as well.

BIOL 3007 W: 17 Nov 2010

I took a nap in class today. It was amazing. We talked about ascomycota or something.

The greatest advertisement I have seen in the last few days

Speed is a function of stride length and stride rate

Lesson learnt: When performing an activity, decompose it into the various steps and actions necessary and understand the meaning of all the variables that affect the activity.

I ran today. The act wasn’t unusual. I have been running about 4 days a week, 14-15 miles a week. However, for whatever reason, my run felt a little weird. It felt much better to be running today the loop today than it had felt on most previous days. To be sure, I was breathing a little harder and going faster  than I usually do. The cause was confounding me. I had just consumed about 350 mLs of water, so if anything, I should have felt more uncomfortable running faster. I also have a mental beat (a usual 1-2-3-4-5-6-7-8 rhythm) that I run to, and it was pretty much invariant. Identifying changes is rhythm comes easy to me thanks to my Tabla background, and my feet were following this beat, eliminating the possibility that I was taking more steps in the same amount of time.

Then it occurred to me: I was taking longer strides. As simple as that. Speed definitely varies directly with stride length and stride rate. Fixing the stride rate (which I was doing today) and increasing the stride length (which I was also doing) definitely increases speed.

My new running goal should be improving stride length. It takes a lot more effort to swing the legs wider, and the impact of the legs hitting the ground is also larger. Finding the optimal stride length will be the key priority. Running on grass might be the best option, since it seems to me that the impact is relatively low. However, papers such as this one suggest that running on grass might actually have a higher impact since one cannot predict the level of impact of the next step while running on grass.

Anyway, decision will be made, performance will be (mentally) evaluated by moi, and results will be presented in the form of another long blog post.

Enjoy.

–Gautam

BBC Hindi – Crazy bikers from Chennai

http://www.bbc.co.uk/hindi/multimedia/2010/08/100803_roads_video_vv.shtml

The above is a video of some bikers (the motorbike kind) from Chennai who indulge in the road. Some have made trips to Kashmir and other such cool places. They talk about how the advent of 4-lane highways is changing their experiences, and they reminisce on their experiences in the past.

The video is in Hindi.

Robert Solow’s Testimony to Congress

It was given to the House Committee on Science and Technology and in it, Solow attempts to describe why modern macroeconomics in general and Dynamic Stochastic General Equilibrium models have “failed and [are] bound to fail.”

Solow_Testimony. (pdf)

“Economic theory is always and inevitably too simple; that can not be helped. But it is all the more important to keep pointing out foolishness wherever it appears. Especially when it comes to matters as important as macroeconomics, a mainstream economist like me insists that every proposition must pass the smell test: does this really make sense? I do not think that the currently popular DSGE models pass the smell test. They take it for granted that the whole economy can be thought about as if it were a single, consistent person or dynasty carrying out a rationally designed, long-term plan, occasionally disturbed by unexpected shocks, but adapting to them in a rational, consistent way. I do not think that this picture passes the smell test. The protagonists of this idea make a claim to respectability by asserting that it is founded on what we know about microeconomic behavior, but I think that this claim is generally phony. The advocates no doubt believe what they say, but they seem to have stopped sniffing or to have lost their sense of smell altogether.

[...]

The DSGE school populates its simplified economy – remember that all economics is about simplified economies just as biology is about simplified cells – with exactly one single combination worker-owner-consumer-everything-else who plans ahead carefully and lives forever. One important consequence of this “representative agent” assumption is that there are no conflicts of interest, no incompatible expectations, no deceptions.

[...]

An obvious example is that the DSGE story has no real room for unemployment of the kind we see most of the time, and especially now: unemployment that is pure waste. There are competent workers, willing to work at the prevailing wage or even a bit less, but the potential job is stymied by a market failure. The economy is unable to organize a win-win situation that is apparently there for the taking. This sort of outcome is incompatible with the notion that the economy is in rational pursuit of an intelligible goal. The only way that DSGE and related models can cope with unemployment is to make it somehow voluntary, a choice of current leisure or a desire to retain some kind of flexibility for the future or something like that. But this is exactly the sort of explanation that does not pass the smell test.

[...]

The point I am making is that the DSGE model has nothing useful to say about anti-recession policy because it has built into its essentially implausible assumptions the “conclusion” that there is nothing for macroeconomic policy to do. I think we have just seen how untrue this is for an economy attached to a highly-leveraged, weakly-regulated financial system. But I think it was just as visibly false in earlier recessions (and in episodes of inflationary overheating) that followed quite different patterns. There are other traditions with better ways to do macroeconomics.”

A good read. A good reason why I’ll continue to be taking my econ course material with a grain of salt.

–Gautam

Sitting in a cafe

What with this new found joy of a million hour battery life (okay, it’s closer to 8, which is about 6 times higher than what I’ve been used to previously) I’ve begun to actually spend time outside of my room working on things. I hated lugging around the tablet since its battery life was shit and any odd movements set off a non-stop ghost clicking run that was only solved by putting the computer to sleep for a few minutes and then reusing it. I’ve pretty much been using this laptop off the charger continuously since 2:30 PM today, and the battery info tells me I still have about 32% of battery life remaining. I’ve been using the internet for 4.5 of the 6 hours (Firefox with ~10 tabs open + Yoono), Adobe Reader (4 10 page pdfs), and MS Word (2 3-page Doc files.)

This is simply amazing. I wonder why I ever got the stupid tablet instead of a netbook in the first place. I guess I’ve learned my lesson. Its a great thing to be not sitting in my room, bored, waiting for pdfs to load on the glacially slow Argyle internet.

Everything comes with its costs though… I’ve spent $5 on coffee for two days now. At this rate, it would be nearly $1000 over 1 year, which I cannot afford! Maybe if I got more of the cheapest coffee on the menu (which is ~2.00 with tax) I can cut down the spending to $730. Still… $730 is 73% as bad as $1000. Or, maybe it’s 27% better than $1000. Who knows… Either way, it’s a huge hole burner in the wallet.

I hereby resolve that no more that no more than 1% of my annual discretionary spending will occur in coffee shops. i.e. I now need to spend 495 dollars on other things before I spend a penny in Expresso Expose. Sigh.

–Gautam

DIG marigold!

The baby has started germinating! I hope the photo is not too bad.

NPR : Ultra-Cheap ‘Nano’ Car Debuts in India

NPR : Ultra-Cheap ‘Nano’ Car Debuts in India

India’s Tata Motors has unveiled what is being billed as the world’s cheapest car — the $2,500 Nano.

At 9th Auto Expo in New Delhi on Thursday, Company Chairman Ratan N. Tata said the four-person sedan, also called the People’s Car, is an all-weather alternative to the motor scooter. He said he found inspiration for the vehicle on the streets.
——————-
Man. Rs. 1 Lakh = $2500 = pwn. They should star hitting the streets here

–Gautam