# 4 question for HallsofIvy

1. Sep 16, 2006

### Duhoc

x = r1cos(omegat) + r2cos(2omegat)
y= r1 sin (omegat) + r2 sin (2omegat)

1) Is this the equation of the graph of the curve in parametric form?

2)Do you have any idea what this curve looks like?

3)How would I determine the area inscribed by the curve at time t?

4)I don't know what ITEX means. Can you clarify?

Thank you,
Duhoc

2. Sep 16, 2006

I know I'm not halls. But what is omegat?

might be nice to learn latex, you can turn omegat into $\omega t [/tex] 3. Sep 16, 2006 ### Data You already have a thread; I don't know why you made a new one. [ itex] allows you to use latex code inline. If you want things to look nicer when all you want on a line is math, then you can use [ tex]. Those equations certainly define a curve in parametric form. The "area inscribed by the curve" only makes sense for a closed curve, so you can't evaluate it as a function of t. I've attached a picture of the curve plotted from [itex]t=0$ to $t=2\pi$ for $\omega = r_1 = r_2 = 1$.

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Last edited: Sep 16, 2006
4. Sep 17, 2006

### HallsofIvy

Staff Emeritus
1) Since those equations define a specific (x,y) for each t, yes, those are parametric equations for some curve.

As far as (2) and (3) are concerned, FrogPad and Data did a very nice job of answering them. Thanks, guys!

5. Sep 18, 2006

### jpr0

I think this is how a solution should look. I'm really not sure of its correctness though, so if anyone would like to offer some feedback... I couldn't be bothered typing it into here, it would take too long, so I just attached a pdf.

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• ###### asdasd.pdf
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6. Sep 18, 2006

You thanked me for being a smart a_* to him :)
No problem. Anytime anyone addresses you specifically, I'll be sure to say something snide to them.

7. Sep 18, 2006

### Duhoc

Solution

JPRO
Thank you for this beautiful solution. I will be off from work tomorrow and study it. Perhaps HallsofIvy might view it as well. I would be most interested in his analysis. As for frogpad, I don't think he was referring to me. In any case I'm not bright enough to be snide. But there is a fine line between smart and smart aleck. But if you are really bright it doesn't make much difference to me even if you cross it.

Duhoc

8. Sep 18, 2006

oh no man... I wasn't really being nasty. I was just kidding around

I just thought it was funny how you addressed hallsofivy specifically. There are some BRILLIANT people on this board (hallsofivy is definitely one of them). Anyways... have a good one man!

9. Sep 19, 2006

### Duhoc

1-Is there a solution for a third compass with a wheel tracing out the curve inscribed by compass 2 at some multiple of omega?
2-For any number of compasses tracing out the contour of the prior curve?
3-Could this expression, if it existed, be integrated to describe a three dimensional curve in 4 dimensional space time?

Duhoc

10. Sep 19, 2006

### jpr0

Hi Duhoc,

I don't have much time at the moment, so I'll try and give you a fuller reply later. To visualize the curves you are describing there is a free tool for plotting graphs of functions, of raw data and so on. It's called gnuplot. If you're using win32 (windows XP say) then you can download it from here:

ftp://ftp.gnuplot.info/pub/gnuplot/gp400win32.zip

If you're using a different OS you can look here for the relevant version:

ftp://ftp.gnuplot.info/pub/gnuplot/

Just unzip the zip file, go to the "bin" directory, and run the binary wgnuplot.exe. This gives you a terminal. To plot the kind of curves you're describing you need a parametric plot, so type this into the terminal :

set parametric
pl cos(t),sin(t)

This will give you a circle (although it dosen't look like of the dimensions of the plot window, just readjust them). If you want to to plot the curve discussed above, try something like this

pl cos(t)+0.5*sin(2*t),sin(t)+0.5*sin(2*t)

Or if you want to see the plot of the first "compass trace" and then the second,

pl cos(t),sin(t) , cos(t)+0.5*cos(2*t),sin(t)+0.5*cos(2*t)

If you want to see how adding another compass ontop of the second looks try something like this:

pl cos(t)+0.3*cos(2*t) + 0.1*cos(3*t), sin(t)+0.3*sin(2*t) + 0.1*sin(3*t)

etc.. you can play around changing the radii here (the 0.1, 0.3 etc) and the angular frequencies.

1) I don't see any trouble with this approach adding a third compass, it just increases the size of the expression for z(t). And it would have to be an integer multiple of omega so that the curve you trace out is closed. There should probably be some restrictions on how fast the angular rotation of the 2nd, and 3rd compass can be. For instance if you have the third compass whizzing round at 300.omega then it will start producing crossing points in the curve. I'll have a closer look to see how to derive what values of R and omega are valid.

2) Again I don't see any problem with adding more and more compasses, if you just work out what restrictions on the radii and angular frequency are. Probably the radii will get progressively smaller, and probably their sum shouldn't be more than R1, or some expression involving R1....

3) I'm not sure what the question is here. I'll re-read it later to see if makes any more sense.

11. Sep 19, 2006