TITLE: "Self-sliding" surfaces
QUESTION [23 upvotes]: There are surfaces which you can "slide" onto themselves so that you see no change.
In particular, the plane, the cylindre, the surfaces of revolution (circular cone, sphere, torus...), the helicoid. The sliding motion can be a combination of translation and rotation.
I want to know if there is some theory about these, and in particular if they have a name, if there are others than those I listed and if they can be characterized by some property of their principal curvatures. (I guess that at least one of them must be constant.)
Update:
If I am right, those surfaces are obtained by sweeping an arbitrary curve with a constant rigid motion, one of translation, rotation about an axis or "screw displacement". Then during surface motion, every point follows a straight line, a circular arc or an helix.

Using cylindrical coordinates $(r,\theta,z)$, where the $z$ axis is that of the helix, and using the parametric equation of the section curve of the surface by the plane $xz$, $(r(u),0,z(u))$, the equation of the swept surface is
$$(r(u),v,z(u)+\sigma v)$$ where $\sigma$ is the pitch.
In Cartesian coordinates,
$$(r(u)\cos v,r(u)\sin v,z(u)+\sigma v).$$
Then the first fundamental form is
$$\begin{align}E&=r'^2(u)+z'^2(u),\\F&=\sigma z'(u),\\G&=r^2(u)+\sigma^2.\end{align}$$
The normal vector,
$$\frac1{\sqrt{\sigma^2r'^2(u)+z^2(u)r^2(u)+r^2(u)r'^2(u)}}\\(\sigma r'(u)\sin v-z'(u)r(u)\cos v,-\sigma r'(u)\cos v-z'(u)r(u)\sin v,r(u)r'(u)).$$
And the second fundamental form,
$$\begin{align}
L&=-z'(u)r(u)r''(u)-\sigma r'(u)r''(u)+r(u)r'(u)z'(u),\\
M&=-\sigma r'(u),\\N&=-z'(u)r^2(u),\end{align}$$ scaled by $\dfrac1{\sqrt{\sigma^2r'^2(u)+z^2(u)r^2(u)+r^2(u)r'^2(u)}}$.

I guess that a line of attack is to determine the position of the helix axis and the value of the step from the local differential forms, and to show that they are constant. Given the complexity of the expressions, this looks like a serious endeavor.

REPLY [5 votes]: of quite an aside comment relevance (no answer) .. such dry bean pod examples are plenty in Nature. Similar to our geometry here  when toroidal arching up is disregarded. Posted it elsewhere before.