A question about a surface This is a question in "baby do Carmo" 2.2.14. The question says

A half-line $[0, \infty)$ is perpendicular to a line $E$ and rotates about $E$ from a given initial position while its origin $0$ moves along $E$. The movement is such that when $[0, \infty)$ has rotated through an angle $\theta$, the origin is at a distance $\displaystyle d=\sin^2\bigg(\frac{\theta}{2}\bigg)$ from its initial position on $E$. Verify that by removing the line $E$ from the image of the rotating line we obtain a regular surface. If the movement were such that $\displaystyle d=\sin\bigg(\frac{\theta}{2}\bigg)$, what else would need to be excluded to have a regular surface?

My idea is the following, which could be wrong.
Let say the half line is the $x$ axis and the line $E$ is the $z$-axis. I think this surface is quite similar to helicoid, except it's finite because when rotating the $x$-axis about the $z$-axis once, the height varies from 0 to 1 and back to 0.

My doubt:

I think one parametrization $x:\mathbb{R}^+\times(0, 2\pi)\to\mathbb{R}^3$ can be given by
$$x(u, \theta)=\bigg(u\cos\theta, u\sin\theta, \sin^2\bigg(\frac{\theta}{2}
\bigg)\bigg).$$
The surface we want should be the image of this map. I am not sure whether this is correct or not.

My question:

The latter part of the question says when $\displaystyle d=\sin\bigg(\frac{\theta}{2}\bigg)$, what else has to be excluded. Besides the line $E$ is there any other thing should be excluded?
The hint is to think about the parametrized curve
$$\alpha(t)=(\sin t,\sin 2t)$$
in the plane. If you restrict the domain to $(-\pi+\epsilon,\pi-\epsilon)$, it is a fine curve. But what happens when we allow either of $\pm \pi$? We still have a regular parametrization (i.e., the derivative is everywhere non-zero), but the curve now intersects itself.

A similar phenomenon is occuring with doCarmo's surface. With the latter parametrization, when $\theta = \pm 2\pi$ the surface now intersects itself along the curve coming from $\theta = 0$. It is an example of an immersed surface, just as the curve $\alpha$ is an immersed curve.
The hint is to think about the parametrized curve
$$\alpha(t)=(\sin t,\sin 2t)$$
in the plane. If you restrict the domain to $(-\pi+\epsilon,\pi-\epsilon)$, it is a fine curve. But what happens when we allow either of $\pm \pi$? We still have a regular parametrization (i.e., the derivative is everywhere non-zero), but the curve now intersects itself.

A similar phenomenon is occuring with doCarmo's surface. With the latter parametrization, when $\theta = \pm 2\pi$ the surface now intersects itself along the curve coming from $\theta = 0$. It is an example of an immersed surface, just as the curve $\alpha$ is an immersed curve.