What is the surface equation of a spiral strip? I have a strip in the $xy$ plane, whose inner side is a spiral. The strip has a finite width. The image of the strip is shown below. Since, the inner side is spiral (shown in red), the polar equation of it is:  $r_1=a.e^{b\theta}$ and the other side's equation is, $r_2=r_1+d$  ,where $d$ is the width of the strip. for both $r_1$ and $r_2$ , $0\leq\theta\leq2\pi$. Now my question is how can we express the surface equation? My attempt is simple, the equation (in cylindrical co-ordinate) is: $$z=0, r_2\leq \rho \leq r_1 , 0\leq\theta\leq2\pi$$
Is there any better way to write down the strip equation?

 A: If $d$ is intended to represent the minimum distance between the two edges of the spiral strip, then your parametrization $$r_1 = a e^{b \theta}, \quad r_2 = r_1 + d,$$ is not correct.  However, it is true that the curves $r_1$, $r_2$ are separated by a constant width; it just isn't $d$, as your own figure shows.  In your figure, $d$ is the length of the black line segments.  However, the width of the strip is measured along a line perpendicular to a given point on either curve.
You have in essence stumbled upon a unique property of the logarithmic spiral $$r = a e^{b \theta},$$  namely any ray from the origin intersecting the spiral does so at a constant angle.  This is a consequence of the spiral's self-similarity.  The composite transformation $$(r, \theta) \mapsto (r e^{b \varphi}, \theta + \varphi)$$comprising an anticlockwise rotation of $\varphi$ and dilation $e^{b \varphi}$ will map the spiral to itself for any angle $\varphi$.
Unfortunately, the relationship between the strip width and the value of $d$ does not have an elementary closed form.
