One of the examples of curves in polar coordinates in my book is an Archimedean spiral $$ r=a\theta $$ and the book says that the equation $$ r=a\theta + b $$ also represents and Archimedean spiral because if we would rotate the polar axis through an angle $\alpha = -\frac{b}{a}$ it would change to the previous one $r=a\theta$. Can anyone explain to me how is the rotation made? I don't think that I get it quite right.
2026-04-03 15:47:49.1775231269
Archimedean spiral - rotation of polar axis
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The spiral $ r=a\theta$ goes through origin. Rotate the full spiral as a rigid spiral by an angle $\beta. $
$ r= a(\theta + \beta) = a\theta + a \beta = a\theta + b$. The spiral has nonzero value where it cuts x-axis. Only after looking at $\beta$ in the opposite direction does the spiral goes through x-axis.