Hartmut Prautzsch

Geometric concepts for geometric design

This book is a comprehensive tool both for self-study and for use as a text in classical geometry. It explains the concepts that form the basis for computer-aided geometric design.

Degree elevation of B-spline curves

An algorithm is presented which expresses any given linear combination of B-splines of order k as a linear combination of B-splines of order k + 1. This algorithm consists of two parts: (1) A simple representation of the curve by a sum of B-splines of order k + 1, and (2) a representation of these splines on a common knot vector by inserting new knots.

The insertion algorithm

Abstract A complete formulation is given of a multiple knot insertion algorithm. The presented algorithm compares favourably with other existing multiple knot insertion algorithms.

A short proof of the Oslo algorithm

Abstract A short proof of the Oslo algorithm is presented that uses a simple comparison of coefficients

Chapter 10 – Box Splines

This chapter introduces Box and half-box Splines with particular focus on triangular splines and surface design. A particular example of box splines is B-splines with equidistant knots. Box splines consist of regularly arranged polynomial pieces and have a useful geometric interpretation. They can be viewed as density functions of the shadows of higher dimensional boxes and half-boxes. Geometric Design uses box spline surfaces that consist of triangular polynomial pieces. These box spline surfaces have planar domains, but it is quite simple to construct arbitrary two-dimensional surfaces, that is, manifolds, with these box splines. The chapter describes how to compute the Bezier points of a box spline surface over a regular triangular grid. It discusses the properties of half-box splines.

Métodos de Bézier y B-splines

Este libro provee una base solida para la teoria de curvas de Bezier y B-spline, revelando su elegante estructura matematica. En el texto se hace enfasis en las nociones centrales del Diseno Geometrico Asistido por Computadora con la intencion de dar un tratamiento analiticamente claro y geometricamente intuitivo de los principios basicos del area. Tambien contiene material avanzado incluyendo splines multivariados, tecnicas de subdivision y la construccion a mano alzada de superficies con cualquier grado de suavidad. El libro esta excelentemente bien ilustrado con diagramas y figuras que aluden directamente al material que se desarrolla en el texto y complementan su caracter constructivo. This book provides a solid and uniform derivation of the various properties Bezier and B-spline representations have, and shows the beauty of this underlying rich mathematical structure. The book focuses on the core concepts of Computer Aided Geometric design with the intention to give a clear and illustrative presentation of the basic principles, as well as a treatment of advanced material including multivariate splines, some subdivision techniques and constructions of free form surfaces with arbitrary smoothness. The text is beautifully illustrated with many excellent fiugres to emphasize the geometric constructive approach of this book. In diesem Buch werden die grundlegenden Konzepte des Geometrischen Designs (CAGD) dargestellt. Die Eigenschaften von Bezier- und B-Spline Darstellungen werden mit Hilfe von Polarformen einheitlich und stringent hergeleitet. Daruber hinaus werden Konstruktionen von Freiformflachen beliebiger Glattheitsordnung, Unterteilungsalgorithmen, Boxsplines, Simplexsplines und multivariate Splines behandelt. Der Text ist mit vielen hervorragenden Abbildungen illustriert, die den geometrisch konstruktiven Zugang des Buches deutlich hervorheben.

Stationary subdivision for regular nets

Under a stationary subdivision scheme, a regular control net is transformed into a regular control net whose vertices are affine combinations of the initial control points. The weights of these affine combinations can be given by masks or algebraically by a characteristic polynomial, as in the case of curve algorithms.

G k -constructions

Two patches join smoothly if they can be (re-)parametrized so that their derivatives up to some order are identical along a common boundary curve. For any fixed reparametrization, this smoothness condition means that the derivatives of both patches at any common point are related by a linear transformation. This is analogous to the curve case.

Constructing smooth surfaces

The simple C 1 joint discussed in 11.7 is too restrictive for modelling smooth regular surfaces of arbitrary shape. Here, we present general C 1-conditions and a interpolation scheme that allows to design regular surfaces of arbitrary topology with triangular patches.

Tensor product surfaces

The easiest way to build a surface is to sweep a curve through space such that its control points move along some curves. The control points of these control curves control the surface. The surface representation by these control points has properties analogous to those of a univariate curve (e.g., Bezier or B-spline) representation. This is due to the fact that one can deal with these surfaces by applying just curve algorithms. Similarly, one can build multidimensional volumes by sweeping a surface or volume through space such that its control points move along curves. Again, one obtains control nets having properties analogous to those of the underlying curve representations.