Polar Coordinates Intuition

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kcaze
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Polar Coordinates Intuition

Postby kcaze » Sun Dec 05, 2010 1:16 pm UTC

So I've just started learning polar coordinates now and I don't really have much intuition for the graphs of polar functions.

With rectangular coordinates, I can picture the approximate shape of a function if it's a polynomial or a simple trigonometric function. But with polar coordinates, even what should be simple exercises such as graphing [imath]r = cot\theta csc\theta[/imath] tend to trip me up (I'd have to convert it into rectangular coordinates and then realize that it's just a parabola).

So how do I develop the intuition for visualizing the graph of a polar function in my mind? Are there any tricks to it or any general forms to memorize?

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Re: Polar Coordinates Intuition

Postby the tree » Sun Dec 05, 2010 10:03 pm UTC

kcaze wrote:So how do I develop the intuition for visualizing the graph of a polar function in my mind? Are there any tricks to it or any general forms to memorize?
I can't really think of any. I just think, what is r when theta is 0 and when you deviate from zero the left or the right - how does r change. Similar things for when theta is pi/2, pi, 3pi/2. I suppose it's similar to plotting rectangular functions - you just look for zeros and turning points and asymptotes and then you piece those together.

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Re: Polar Coordinates Intuition

Postby cba » Mon Dec 06, 2010 12:46 am UTC

it annoyed me when we were taught how to graph polar functions that the best way really was to just learn what a few simple ones looked like, but mainly just do a table of values and plot it.
That annoyed me becasue i quite like graphing in a normal x-y co-ordinate system, where doing a table of values really doesnt help you at all...
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skullturf
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Re: Polar Coordinates Intuition

Postby skullturf » Mon Dec 06, 2010 2:04 am UTC

This is partly a subjective matter and thus opinions may differ, but I tend to think that plotting in any coordinate system whatsoever is, fundamentally, "just" plugging points in.

Of course, there are various useful shortcuts: look for horizontal and vertical tangents, consider concavity, and so forth. These apply in either rectangular or polar coordinates.

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Re: Polar Coordinates Intuition

Postby xepher » Mon Dec 06, 2010 1:55 pm UTC

The argument is theta, and the radius is the value. In normal rectangular geometry, the horizontal (x) is the argument, and the vertical (y) is the value. So, when it came to polar graphs,

Imagine a sheet of paper as the plane. One edge can be the x, the other y. The only thing to do is curve the paper into a cone, and look at it from the top down.

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Re: Polar Coordinates Intuition

Postby the tree » Mon Dec 06, 2010 3:19 pm UTC

cba wrote:That annoyed me becasue i quite like graphing in a normal x-y co-ordinate system, where doing a table of values really doesnt help you at all...
If you're plotting an explicit function, and I say plotting rather than sketching, a table of values is really all you're aiming for.

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kcaze
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Re: Polar Coordinates Intuition

Postby kcaze » Tue Dec 07, 2010 12:36 am UTC

xepher wrote:Imagine a sheet of paper as the plane. One edge can be the x, the other y. The only thing to do is curve the paper into a cone, and look at it from the top down.


Except you can't curve an infinite plane into a cone can you?

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Re: Polar Coordinates Intuition

Postby xepher » Tue Dec 07, 2010 1:17 pm UTC

kcaze wrote:
xepher wrote:Imagine a sheet of paper as the plane. One edge can be the x, the other y. The only thing to do is curve the paper into a cone, and look at it from the top down.


Except you can't curve an infinite plane into a cone can you?

Not unless you wrap it around multiple times. Infinite, actually. But the main thing to get is that the argument (x) just keeps curving around the origin. That's how I thought of it; I'm not sure about others.


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