Mathematical Aspects of the Geometric Structures used for the Automatic Generation of Hull Surface

Abstract: Traditionally, determination of the hull shapes and their description have always been a very important aspect of ship design and construction. The availability and the growing power of computers and appropriate software have made it possible to tackle the complex problem of fitting a surface to a hull the geometrical characteristics of which have previously been defined by means of traditional drawings or by offset points. The mathematical tools used by computer have evolved in time from the simulation of the behaviour of battens and manual drawing techniques to the adoption of more complex analytical structures but more easy to use. The mathematical structures used by almost all the most recent software are Bezier, B-splines and NURBS (non-uniform rational B-splines). There's not lack of literature about Bezier,B-splines and NURBS curves, but mathematical aspects of the theory are still neglected. Books and articles on these topics are mainly devoted to practical development of these fundamental tools, but it is often worthwhile to investigate theoretical aspects which can simplify algorithms and description of the tools. We have tried a critical review of some of the theories involved and especially a definition of the underlying concepts of NURBS curves , not easy to find, in the pertinent literature, expressed in a clear and accurate form.