In view of the meaning of ruled surfaces in aesthetics, statics, scale and manufacturing technologies, we point out the possibility of a mathematical analysis in the case of infinitesimal deformations by considering the rigidity of surfaces. In case of bendable surfaces it is useful to discuss the variation of magnitudes such as the shape operator. The shape operator is a good way to measure how a regular surface S bends in R3 by valuation how the surface normal ? changes from point to point. In this work we consider variations of the shape operator, the normal curvature and the principal curvatures of helicoid under infinitesimal bending of the surface.

In this paper we consider general conoid surfaces. We consider mathematical and constructional aspects of these surfaces. In the constructional meaning conoids are thin shells which are used for spatial roof structures either in complete form or in parts. In mathematical sense it is a geometrical surface on which it is possible to determine rotational and translational fields and check the rigidity