# Surface of screw motion

helical surface

A surface described by a plane curve $L$ which, while rotating around an axis at a uniform rate, also advances along that axis at a uniform rate. If $L$ is located in the plane of the axis of rotation $z$ and is defined by the equation $z=f(u)$, the position vector of the surface of screw motion is

$$r=\{u\cos v,u\sin v,f(u)+hv\},\quad h=\text{const},$$

and its line element is

$$ds^2=(1+f'^2)du^2+2hf'dudv+(u^2+h^2)dv^2.$$

A surface of screw motion can be deformed into a rotation surface so that the generating helical lines are parallel (Boor's theorem). If $f=\text{const}$, one has a helicoid; if $h=0$, one has a rotation surface, or surface of revolution.