W O L F G A N G   B O E H M

My Professional  Goals     In the early fifties of the last century I studied mathematics with special emphasis on geometry and numerical analysis. As a consequence my later research interests can be grouped into the following fields: classical geometry, numerical mathematics and computer aided geometric design.     In the early period of my research work I was interested in unsolved problems of classical synthetic geometry. Among the papers I have written in this time two papers are very characteristic of this period: my doctoral thesis of 1953 on "String constructions of quadratic surfaces" and my habilitation thesis of 1964 on "Eight-flat structures and the theorem of Ivory (1809)", both concerned with new purely geometric proofs of the involved theorems. My research carried out in this period and my experiences in numerical methods proved to be useful in my further research on computer aided geometry.     In 1965 I accepted an appointment to the Technical University of Braunschweig to teach on Mathematics with the special emphasis of Constructive Geometry and Practical Mathematics.                                                    -------------------     The past three decades of computer science have been marked by the rapid profileration of CAD techniques in industry. This has resulted in a strong requirement being able to describe curved surfaces in a way natural to the computer and the man as well. A new branch of mathematics, the computational geometry, was growing up which I joined in 1971. The Bernstein-Bézier techniques,  B-spline curves and surfaces,  and the relation between them, e.g. how to express a B-spline curve or surface in terms of Bézier control points, were in the focus of my work until 1980.     In 1980 I discovered the so-called "Insertion Algorithm" which was concerned with the problem of finding the new control points of B-spline curves if additional knots were inserted. In contrast to the so-called "Oslo-Algorithm", which was discovered nearly the same time by E. Cohen et al. and which can bee seen as a generalisation of the well-known de Boor's algorithm, the insertion algorithm turned out to be another interpretation of de Boor's algorithm only. This further interpretation of this well-known algorithm proved to be a successfull tool in geometric modeling, in particular, subdividing techniques, interactive styling systems, intersections and boolean combinations, as well as the use of the Bernstein-Bézier technique.     Since 1980 substantial effort has also been made in multivariate spline research. Multivariate splines carry promise of contributing to a further improvement of flexibility and smoothness of sculptured surfaces. In 1983 at the R.P.I. I gave a first talk on new subdivision algorithms of this kind of splines and the famous ideas of H. Prautzsch. These algorithms and further developments made an important part of the multivariate spline package developed at the University of Braunschweig in 1984-1986.     Since the mideighties the interest in geometric continuity was growing up. In 1985 I generalized the so-called Beta-Splines to the more flexible Gamma-splines and torsion continuous splines. I also discussed higher geometric and visual continuity and their applications. Further interest was focussed also to the use of algebraic surfaces and higher primitives in Solid Modeling.     Later I devoted to classic and non-classic principles underlying the methods and algorithms of Geometric Design. Topics were the representation and the use of cyclides, the representation of quadrics, and the use of osculants, the last giving the geometric foundation of the new so-called principle of blossoming.                                                    -------------------     My group's scientific and public activities at the Technical University of Braunschweig have aquired appreciation among computer sientists and mathematicians in Germany and other countries as well. Even in 1972 I was invited to Moskow to give a lecture on methods in CAGD for members of the Academy of Science. From the early eighties I was invited to give presentations of our work and tutorials on CAGD in the United States and Europe. A close cooperation was developed with the CAGD groups of the Arizona State University, the Rensselaer Polytechnic Institute, the CAGD group of the Eurographics Association, and later to the Universidad Central de Venezuela.     In the academic year 1986/87 I got an appointment as a visiting and research professor at the Rensselaer Polytechnic Institute in Troy, N.Y. and in fall 1987 an invitation as a visiting professor at the CAD/CAM Center of the Northwestern Polytechnical University in Xian, P.R. China. In fall 1990 I accepted an invitation to the R.P.I. again.     In 1989 the special group for "Applied Geometry and Computer Graphics" at our Technical University of Braunschweig was official founded. The task of this group was the research, development and teaching of Geometrical and Mathematical Methods in CAGD, Geometric Modeling and Computergraphics.     In spring 1992 we expanded the field of activities of our group to the Universidad Central de Venezuela in Caracas. Since this time there is a close cooperation between the "Centro de Computación Gráphicas y Geometría" there and my group in Braunschweig in the field of the applications of Algebraic Geometry and CAGD.                                                    -------------------     In 1984 together with R.E. Barnhill I founded the international journal "Computer Aided Geometric Design"  published by North-Holland (Elsevier), Amsterdam, and  now edited by G. Farin and H.Prautzsch.     Since 1984, commonly with J.Hoschek et al.,  I organized nine international conferences on CAGD, e.g. at the Mathematics Research Institute Oberwolfach, at the famous Herzog August Bibliothek in Wolfenbüttel, at the Hebrew University in Jerusalem, and at Monte Erice on Sicily.     I retired in fall 1993, but was immediately asked to continue my work at the University of Braunschweig leading the special group for Applied Geometry and Computer Graphics till the end of the last century.                                                    -------------------     At the end I want to thank all my students,  partners and friends who have accompanied my scientific and teaching life and who have helped to consolidate the success and reputation of our group.  I am feeling unable to list them all, but I am very thankful to all of them. CAGD-Kelch, nach einer  Zeichnung von Paolo Uccello (1397-1475)