# Trying to reverse engineer this pattern…

this is my first post on the mathematics node of stack exchange, so please forgive me if I'm not posting an appropriate question, but I'm not sure where else to address this. I'm trying to figure out what equation might generate the following pattern. (keep in mind the numbers might be slightly off, not exact)

 0  :: 0, 0
10 :: 19, -16 (0.17453)
20 :: 41, -28 (0.34906)
30 :: 64, -37 (0.52359)
40 :: 89, -41 (0.69813)
50 :: 113,-41 (0.87266)
60 :: 138,-37 (1.04719)
70 :: 162,-28 (1.22)
80 :: 182,-16 (1.392)
90 :: 201, 0  (1.570)
100:: 217,19 (1.745)
110:: 229,41 (1.919)
120:: 238,64 (2.094)
130:: 241,89 (2.268)
140:: 241,113 (2.443)
150:: 238,138 (2.617)
160:: 229,162 (2.792)
170:: 217,182 (2.792)
180:: 201,201 (2.967)
190:: 182,217 (3.316)
200:: 162,229 (3.490)
210:: 138,238 (3.665)
220:: 113,241 (3.839)
230:: 89,241 (4.0142)
240:: 64,238 (4.188)
250:: 41,229
260:: 19,217
270:: 0,201
280:: -16,182
290:: -28,160
300:: -37,137
310:: -41,113
320:: -41,89
330:: -37,64
340:: -28,41
350:: -16,19
360:: 0,0


as you math wizez may have guessed it has to do with adjusting coordinates for rotation around an axis. Any help would be great!

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This kind of thing has been tackled several times before on the site: here, here, here, and here. The upshot is that there are infinitely many functions that would produce any given finite collection of data, and so there is no single "answer"; but if you have information or assumptions about the function, then there are methods that can produce interpolating functions with those properties. – Zev Chonoles Jun 23 '12 at 6:09
It would be rather helpful to know how you came across this set of data. Also, Zev, I'm not sure the point you're making with those examples. – Eric Stucky Jun 23 '12 at 6:15
@Eric: I think this question qualifies as an abstract duplicate (this doesn't necessarily mean the question will be closed though). But providing links to relevant earlier questions seems helpful regardless. – Zev Chonoles Jun 23 '12 at 6:21
@Zev: I think there's a qualitative difference from the examples you link to. In the typical questions that your criticism applies to there is only a few (up to 7 in your examples) data points, which isn't enough to decide the general shape of a fit with any degree of accuracy. But with 37 data points, as here, there is a good chance that someone with mathematical experience can look at a plot and see a solution type that can fit the data better than other solutions of comparable simplicity, whereas an OP with less experience perhaps can't. That makes it a good question in my opinion. – Henning Makholm Jun 23 '12 at 17:42

Let $f(x) = 142 \sin \frac{\pi (t -45)}{180}+100$. Then your data is reasonably well approximated by $\{t, f(t), f(360-t)\}_{t\in \{0,\cdots,360\}}$.
Here is a plot of $f$ and the first two columns of the data above:
I dare guess that your $142$ might actually be $100\sqrt2$. – Jyrki Lahtonen Jun 23 '12 at 9:24
You might be right. All the data (at least the first 3 columns) are integers, so I would guess there was some rounding. The $2$-norm with $100 \sqrt{2}$ is slightly larger than with 142, but this may be due to rounding (I'm treating the data as a vector in $\mathbb{R}^{360}$). – copper.hat Jun 23 '12 at 14:53