Przemysław Kiciak

Okładka2 Book (in Polish) Konstrukcje powierzchni gładko wypełniających wielokątne otwory
(Constructions of surfaces filling smoothly polygonal holes),
Prace Naukowe, Elektronika, z. 159, Oficyna Wydawnicza Politechniki Warszawskiej, Warszawa 2007.

The book is accompanied with a CD with a package of several hundred procedures: graphical, numerical and geometric.

Errata (a .PDF file) to download from this page. The readers, who find errors not listed there, are kindly requested to report them to me, via the e-mail.

Summary

The possibility of designing smooth surfaces with a complicated topology is essential for the usefulness of CAD systems for practical purposes. The simple methods of joining B-spline patches, ensuring the continuity of derivatives of their constant parameter curves at the junction points, make it possible to obtain smooth surfaces with polygonal holes. This study is devoted to constructions of surfaces filling these holes. Such constructions have to ensure that the filling surfaces have the geometric continuity of the appropriate order and satisfy some aesthetic criteria; as the diameters of the holes are often small, it is usually desirable to fill the holes so as to make them invisible.

This study contains theoretical foundations for constructions of smooth surfaces consisting of parametric patches, a survey of constructions known from existing publications, and new constructions, developed by the author. These constructions make it possible to fill holes in surfaces made of bicubic B-spline patches (with continuous curvature), with preserving the tangent plane continuity and curvature continuity. A filling surface being result of the construction consists of tensor product Bézier patches and it is obtained by optimisation with respect to some aesthetic criterion, which in many cases results in a satisfactory final effect. In addition there is a possibility of imposing constraints, being interpolation conditions, which allow the user of a CAD system to make corrections specific for the project. The implementations of those constructions (C language procedures) are available on the enclosed disc.

The theory described in this study, whose central notion is the space of class G^n, covers the background for constructing such spaces (geometric continuity equations, compatibility conditions etc.). The theory includes also the geometric interpretation of the optimisation criteria used in the construction and the analysis of the existence and uniqueness of solutions of the optimisation problems. There is also an analysis of independence of the interpolation conditions (constraints), which may be imposed at a common corner of patches. This theory may find its applications also in other constructions of smooth surfaces.
Updated: