Surface of screw motion

helical surface

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

and its line element is

A surface of screw motion can be deformed into a rotation surface so that the generating helical lines are parallel (Boor's theorem). If , one has a helicoid; if , one has a rotation surface, or surface of revolution.

References

[a1]	M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)
[a2]	H.S.M. Coxeter, "Introduction to geometry" , Wiley (1961)
[a3]	M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976) pp. 145