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**Computational line geometry.**
*(English)*
Zbl 1006.51015

Mathematics and Visualization. Berlin: Springer. ix, 563 p. (2001).

Mainly line geometry deals with subsets of the line set of the (real or complex extended) three-dimensional space from a differential geometric point of view. Line geometry is an active branch of mathematical research for its own as well as it provides advanced tools for many applications (e.g. spatial kinematics and robotics, scientific computation and visualisation as well as special problems of reverse engineering and geometric modelling).

The didactically well-considered and marvellous illustrated book is self-contained at least for readers acquainted with basic concepts of (classical) differential geometry and CAGD, and, of course, with (advanced) linear algebra and some calculus. The reader of this demanding textbook is richly rewarded by applicable knowledge, and he surely will enjoy the elegance of geometric reasoning and the effectiveness of line-geometric calculations.

The first chapter, “Fundamentals”, introduces to real projective geometry, (projective) differential geometry, elementary algebraic geometry and to computer aided curve and surface design. Here the authors win maximum effect by minimal efforts: They introduce to geometry in general in a very broad sense, and one surely can use the book as introductory reference for each of the mentioned topics independent of their line-geometric context.

The second chapter, “Models of Line Space”, deals with Plücher coordinates of a line and their interpretation as coordinates of a point on the Klein hyperquadric or the Study sphere, while the third, “Linear Complexes”, treats linear subspaces of lines from the projective geometric point of view as well es from the euclidean one. Here motions and sets of motions forge first links between pure theory and applications.

In chapter 4, “Approximation in Line Space”, the authors come along with some of their own research material. This material, published for the first time in a textbook of line geometry, is fundamental for many modern applications of line geometry, as is illutrated impressively in the following chapters.

In chapters 5 and 6, “Ruled Surfaces” and “Developable Surfaces”, (continuous) 1-parameter sets of lines are discussed. Besides the classical differential geometric treatment leading to the well-known Fundamental Theorem of Kruppa-Sannia and the inner-geometric concepts of ruled surfaces as 2-manifolds, the reader will meet concepts of CAGD modified and applied to ruled surfaces and, of course, many practical problems presented as elaborated demonstrations. (In chapter 6 one even will learn about the so-called Cyclographic Mapping, another half-forgotten classical geometric method, which turns out to be a mighty and broadly applicable tool.)

Chapter 7, “Line Congruences and Line Complexes”, shows 2- and 3-parameter sets of lines. Also here the authors leave traditional paths aiming at applications in sculptured surface machining, again topics of their own research.

The last chapter 8 deals with “general” linear mappings and their representations in different models. Generalisations of the classical Kinematic Mapping of Blaschke-Grünwald are used in Motion Design and lead the authors to coin the concept “Computational Kinematics”.

The thoroughly elaborated book ends with a comprehensive list of references, a “List of Symbols” and a detailed “Index”.

The didactically well-considered and marvellous illustrated book is self-contained at least for readers acquainted with basic concepts of (classical) differential geometry and CAGD, and, of course, with (advanced) linear algebra and some calculus. The reader of this demanding textbook is richly rewarded by applicable knowledge, and he surely will enjoy the elegance of geometric reasoning and the effectiveness of line-geometric calculations.

The first chapter, “Fundamentals”, introduces to real projective geometry, (projective) differential geometry, elementary algebraic geometry and to computer aided curve and surface design. Here the authors win maximum effect by minimal efforts: They introduce to geometry in general in a very broad sense, and one surely can use the book as introductory reference for each of the mentioned topics independent of their line-geometric context.

The second chapter, “Models of Line Space”, deals with Plücher coordinates of a line and their interpretation as coordinates of a point on the Klein hyperquadric or the Study sphere, while the third, “Linear Complexes”, treats linear subspaces of lines from the projective geometric point of view as well es from the euclidean one. Here motions and sets of motions forge first links between pure theory and applications.

In chapter 4, “Approximation in Line Space”, the authors come along with some of their own research material. This material, published for the first time in a textbook of line geometry, is fundamental for many modern applications of line geometry, as is illutrated impressively in the following chapters.

In chapters 5 and 6, “Ruled Surfaces” and “Developable Surfaces”, (continuous) 1-parameter sets of lines are discussed. Besides the classical differential geometric treatment leading to the well-known Fundamental Theorem of Kruppa-Sannia and the inner-geometric concepts of ruled surfaces as 2-manifolds, the reader will meet concepts of CAGD modified and applied to ruled surfaces and, of course, many practical problems presented as elaborated demonstrations. (In chapter 6 one even will learn about the so-called Cyclographic Mapping, another half-forgotten classical geometric method, which turns out to be a mighty and broadly applicable tool.)

Chapter 7, “Line Congruences and Line Complexes”, shows 2- and 3-parameter sets of lines. Also here the authors leave traditional paths aiming at applications in sculptured surface machining, again topics of their own research.

The last chapter 8 deals with “general” linear mappings and their representations in different models. Generalisations of the classical Kinematic Mapping of Blaschke-Grünwald are used in Motion Design and lead the authors to coin the concept “Computational Kinematics”.

The thoroughly elaborated book ends with a comprehensive list of references, a “List of Symbols” and a detailed “Index”.

Reviewer: Gunter Weiß (Dresden)

### MSC:

51M30 | Line geometries and their generalizations |

53A25 | Differential line geometry |

51J15 | Kinematic spaces |

70B10 | Kinematics of a rigid body |

65D18 | Numerical aspects of computer graphics, image analysis, and computational geometry |

68U05 | Computer graphics; computational geometry (digital and algorithmic aspects) |

51-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to geometry |

53-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry |

65-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis |