This Demonstration shows several examples of twisted surfaces of revolution (also known as generalized helicoids). The surfaces are generated by taking a profile curve in the - plane, rotating it around the axis and, at the same time, translating it parallel to the same axis. Thus, the profile traces a circular helix.

Assuming that the parametrization of the profile curve is

,

the corresponding twisted surface of revolution has parametrization

,

where is the amount of twisting ( yields an ordinary surface of revolution).

The first three of the six examples in this Demonstration are particularly noteworthy:

"semicircle" has its center on the axis and gives a corkscrew surface (twisted sphere);

"tractrix" is a Dini's surface (twisted pseudosphere) that is a well-known example of a surface with constant negative Gaussian curvature;

"zero-curvature" is the curve with

,

,

that produces a surface with constant zero Gaussian curvature.

See, for example, [1, Chapter 15] and [2, Exercise 4.46].

References

[1] A. Gray, E. Abbena and S. Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed., Boca Raton, FL: Chapman & Hall/CRC, 2006.

[2] K. Tapp, Differential Geometry of Curves and Surfaces, New York: Springer Science+Business Media, 2016.