Additional geometric investigation of carved Monge surfaces

The large number of scientific papers on carved Monge surfaces give an idea of the dynamism of the research activity on them. Based on a definition of the concerned surfaces, several investigations have been carried out and many scientific works produced. The methods used by scientists to investigate the geometry of these surfaces are those of the differential geometry. Additional investigations, by mean of the kinematic method gave as a result, an alternative definition: a carved surface is generated by the motion of some plane curve (generatrix) along another arbitrary curve (directrix) so that the generatrix curve lies in the normal plane of the directrix line and is rigidly connected with it. However, it should be noted that the latter condition is necessary, but not sufficient for the formation of a carved surface. To achieve this goal, methods of differential geometry are used. In this article, the equation of the generatrix curve in the polar coordinate system is used and the parametric equation of carved Monge surfaces is specified that allows to more study their inner and outer geometries. New equations are obtained for more kinds of carved Monge surfaces that are classified and plotted by mean of the software Mathcad. These new forms of carved Monge surfaces can be used as middle surfaces for thin elastic shells that will be expressive and cover large spans. © Published under licence by IOP Publishing Ltd.