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.
[Math] Archimedean spiral – rotation of polar axis
analytic geometrycalculuspolar coordinates
Best Answer
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.