Thinking in polar coordinates: #weareallcartesiansnow

by Gilbert Keith

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.

Dartboard.svg

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.

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