# How to find the formula of a spiral using least-squares(regression)?

Assume data from a plane which are roughly showing a spiral. I want employ the rationale of regression to find the parameters for the best fit by some spiral.

That means, I have to estimate the (optimal) center first.
Then I can convert the rectangular data-coordinates into polar coordinates, and then the radial distances from that center should be linearly (or polynomially) dependend on the angle by which some point is displaced by some point z0. (Here multiples of 2*Pi are significant).

My example data are in fact complex values, steming from iterations of $z_{k+1} = b^{z_k}$ where $z_0$ is some complex value in the near of a complex fixpoint and b is the base, say $b = \sqrt{2}$ , and I've already done some work with this.
But what I'm primarily looking for is an idea for the general ansatz: for the case where b is given I have the fixpoint and thus I need not estimate the center and the problem is not so difficult. But if I even do not have the center, only the data, how could I begin? I couldn't translate the rectangular coordinates into polar coordinates if the center is unknown in the beginning...

Q: How could I approach this with the method of least-squares-approximation?

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Note that if your objective function is the distance to the spiral, this can be driven arbitrarily close to $0$ by making the pitch of the spiral arbitrarily small -- so you would need either a fixed pitch or some penalty for small pitch. – joriki Nov 3 '12 at 15:29
I do not definitely know what "pitch" is here, but I think you say, that we can assume arbitrarily many windings between two consecutive data points - and thus make the radial expansion by one winding arbitrarily small... well, I've had this idea vaguely earlier, but also didn't know how I would deal with that properly evan after I had a general idea for some ansatz for regression... – Gottfried Helms Nov 3 '12 at 16:16