Theories and Methods for Development of Developable Ruled Surfaces and Approximate Flattening of Non-developable Surfaces

Geometric genesis of surfaces and knowledge of their properties are basis for solving many problems, both constructive and mea-surement. A developable surface can be manufactured starting from a flat “strip”, using a flexible and non-deformable material. This is a very important feature of the surface. Geometry studies the properties that don’t change and, therefore, the shape of the “strip” to obtain a certain configuration, after a series of rigid movements. The paper addresses different methods to define the development of developable surfaces and non-developable surface “flattening”, or approximate development. The aim is to study the relationships between methods, illustrated in some treatises, and the applications that can derive from the use of parametric tools. We are going to create an overview of different approaches, that have defined the bases of differential calculus for the study of ruled surfaces properties, and of different methodologies that allow to determine their development. Starting from the first definition of surface by Aristotle in De Anima (384-322 BC) and the ambiguous definitions by Amédée François Frézier (1682-1773), we analyzed the studies of Leonhard Euler (1707-1783) and Monge’s main work on developable surfaces (1795)