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Book review

Author(s) Abramowski, S.
Müller, H.
Title Geometrisches Modellieren
Publisher BI-Wiss.-Verlag
Year of publication 1991
Reviewed by Bazil Pârv

This book, the 75th issue of the series "Computer Science" has for purpose the fundamentals of geometric modelling. Geometric modelling offers a mathematical tool for graphic processing, with applications in computer aided design/manufacturing/architecture, geosciences, biology, computer animation, medicine. Geometric models of real objects are used in numerical simulation techniques needed for solving real world problems.

The book is structured in 11 chapters, followed by a very consistent list of references (17 pages of text) and a subject index. Each chapter follows the pattern: definitions (of the mathematical objects involved), properties (theorems and proofs related to defined objects), algorithms (given in Pascal-like pseudocode). At the end of every chapter, a short history and a bibliographical discussion are given.

Taking into account the mathematical tools involved, geometric modelling can be split into several domains: differential-geometric modelling, trigonometric modelling, combinatoric (combinatorial) modelling, raster-geometric modelling, and fractal modelling. First chapter of the book, an introductory one, states the scope of geometric modelling and presents the fundamental concepts and algorithms of graphic processing.

Differential geometry has for purpose the study of geometric forms from the differential calculus point of view. The next 5 chapters of the book cover this approach, including quality criteria (chapter 3 refers to graphical objects representation, differential geometry methods for curve and surface study); techniques based on such models like Bezier (chapter 2), B-spline (chapter 4), interpolation (chapter 5), and based on implicit representations ones (chapter 6). Applications of such models are found in computer-aided design and manufacturing (automobile, airplane, boat).

Trigonometric modelling is adequate in modelling water or earth surfaces. Chapter 7 of the book is dedicated to this kind of models, which use trigonometric functions (sine and cosine ones).

Combinatorial models represent surfaces or bodies like unions of disjunct cells, used especially in numerical simulation techniques with finite or boundary elements. Computational geometry concepts and techniques (like convex hull, Voronoi diagram, Delaunay triangularisation and so on) falls into this. There exist different combinatorial models: based on set theory, Minkowski, iterative division (chapter 8) and syntactic ones (chapter 9). Combinatorial models are used for body modelling, while syntactic modelling is adequate to represent plants and trees.

As a particular case of combinatorial geometry, raster geometry (chapter 10) considers the geometric form as set of raster elements from a raster-world, with applications in computer tomography. Finally, fractal modelling (chapter 11) offers computer-generated pictures of exquisite beauty, resembling real-world landscapes, trees and flowers.

The book is well produced, like other mathematical textbooks edited by B.I. Wissenschaftsverlag Mannheim. The good understanding of the presented mathematical models needs some mathematical background; so, I consider this book very valuable for graduate students and researchers in the field of computer graphics. On the other hand, the clarity of the text, the figures presented and the unambiguous specification and explanation of algorithms makes it attractive, despite the high-level mathematics involved.