Determine the radius of the tubular surface for which the tubular surface is not regular I'm new to the frenet frame and I just found this exercise to be super helpful. < However, I'm wondering that how can I calculate the radius of the tubular surface for which the tubular surface is not regular? I know that I need to set $x'_s = 0$ and $x'_θ = 0$. But then I get a mess...Any idea will be appreciated, thanks!

Remember that the requirement for a regular parametrized surface is that the vectors $x_s$ and $x_\theta$ should be linearly independent (not merely nonzero). So you want to find for what value(s) of $r$ the equation $x_s\times x_\theta = 0$ can occur at some point $(s,\theta)$.