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Let's say we have an Archimedean spiral in Cartesian coordinates. This corresponds to a line in polar system (i.e. $r=a\theta+b$).

Now if I move the origin of the Cartesian coordinates system to $\begin{bmatrix} x_1 \\ y_1 \end{bmatrix}$ and rotate its axes by $\alpha$ counter-clockwise, the spiral is still a spiral in Cartesian system (the shape does not change).

Therefore, it should still be a line in polar system too (i.e. $r_1 = a_1\theta_1+b_1$).

So what is the equation for the second spiral in polar system (in terms of $x_1$ and $y_1$ and $\alpha$)?

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Your reasoning is not quite right. An Archimedean spiral will only have a nice simple equation of the form $r = a\theta + b$ if you use the center of the spiral as your origin. If you use some other origin, the equation will be a mess.

The same is true of most (maybe all?) types of curves. Circles are the simplest example. A circle of radius $k$ has the polar equation $r=k$ if you place the coordinate origin at its center. If you place the origin somewhere else, you'll get a much more complex equation.

It's generally true (either in polar or rectangular coordinates) that a judicious choice of coordinate system might dramatically simplify the equation of a curve. And, in fact, there are many mathematical techniques that are essentially just clever ways of choosing coordinate systems. Eigenvectors/eigenvalues, for example, and diagonalization of matrices -- these are really just tricks for picking good coordinate systems (or, that was one of their original purposes, anyway).

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Thank you, It was a good explanation. You were right about my reasoning. Because I reached an equation for the second spiral that contains a bunch of sin and cos functions and I thought maybe it is not correct. Where in fact the equation is correct and my reasoning is wrong. So there is no such thing as the transformation matrix as it is in Cartesian space? I mean is there any unique transformation function when the origin is moved in Cartesian space? –  Hesam Mar 9 '13 at 20:15
    
In cartesian coordinates, translation of the origin is easy, and rotation is a bit more difficult. In polar coordinates, rotation and scaling are easy, but translation is a mess. So, it depends what kind of transformation you want to do. –  bubba Mar 10 '13 at 1:32

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