Let $T$ be union of tangent lines to helix $C=(\cos x, \sin x,x)$.
1) I want to prove that $T - C$ is a smooth manifold and find equation for $T$.
2) I want to find how many times a line can intersect $T$.
T is parameterized by $\phi:(t,s) \mapsto (\cos t -s \sin t, \sin t +s \cos t, t+s).$ To prove that $T-C$ is a smooth manifold I need to show that $\phi$ is embedding and $\phi_s(t,s)$ and $\phi_t(t,s)$ are linearly independent for $s>0$. I can't prove that $\phi$ is embedding. And I don't know how can I find equation for $T$.
Parametrization $\phi$ is continuous and smooth in 3-space embeddable.
The developable helicoid has zero Gaussian curvature when the $ z = s+t \ne 0 $ in parametrization. So is the case when it is flat with $s+t$ zero. They are both two dimensional surfaces embeddable in 3D.
Both the surfaces are built on two parameters $t,s$.
They are isometrically equivalent, mappable bijectively, ie., both ways.
For tangents consider between two $s$ values, infinitely many in the $s$ interval at any $ (t=t_1)$
For helices, any $ (s = s_1)$ value with two limiting $t$ values of rotation infinitely many in the $t$ interval
Do you have these questions with respect to the flat spiral?