Last update: Sept. 11, 2003

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References for Computer Aided Geometric Design by H. Pottmann et al.:

• Geometry of the squared distance function to curves and surfaces, 2003.
• Computational Line Geometry, 2001.
• Error propagation in geometric constructions, 2000
• A Laguerre geometric approach to rational offsets, 1998.
• Applications of Laguerre geometry in CAGD, 1998.

BibTeX references

This book for the first time studies line geometry from the viewpoint of scientific computation and shows the interplay between theory and numerous applications. On the one hand, the reader will find a modern presentation of `classical' material. On the other hand we show how the methods of line geometry enable an elegant approach to many problems whose connection to line geometry is not obvious at first sight.

The geometry of lines occurs naturally in such different areas as sculptured surface machining, computation of offsets and medial axes, surface reconstruction for reverse engineering, geometrical optics, kinematics and motion design, and modeling of developable surfaces. This book covers line geometry from various viewpoints and aims towards computation and visualization. Besides applications, it contains a tutorial on projective geometry and an introduction into the theory of smooth and algebraic manifolds of lines. It will be useful to researchers, graduate students, and anyone interested either in the theory or in computational aspects in general, or in applications in particular.

1. Fundamentals
   1. Real Projective Geometry
   2. Basic Projective Differential Geometry
   3. Elementary Concepts of Algebraic Geometry
   4. Rational Curves and Surfaces in Geometric Design
2. Models of Line Space
   1. The Klein Model
   2. The Grassmann Algebra
   3. The Study Sphere
3. Linear Complexes
   1. The Structure of a Linear Complex
   2. Linear Manifolds of Complexes
   3. Reguli and Bundles of Linear Complexes
   4. Applications
4. Approximation in Line Space
   1. Fitting Linear Complexes
   2. Kinematic Surfaces
   3. Approximation via Local Mappings into Euclidean 4-Space
   4. Approximation in the Set of Line Segments
5. Ruled Surfaces
   1. Projective Differential Geometry of Ruled Surfaces
   2. Algebraic Ruled Surfaces
   3. Euclidean Geometry of Ruled Surfaces
   4. Numerical Geometry of Ruled Surfaces
6. Developable Surfaces
   1. Differential Geometry of Developable Surfaces
   2. Dual Representation
   3. Developable Surfaces of Constant Slope and Applications
   4. Connecting Developables and Applications
   5. Developable Surfaces with Creases
7. Line Congruences and Line Complexes
   1. Line Congruences
   2. Line Complexes
8. Linear Line Mappings --- Computational Kinematics
   1. Linear Line Mappings and Visualization of the Klein Model
   2. Kinematic Mappings
   3. Motion Design

The book is a milestone of geometry in a modern style, realising an excellent synthesis of theory and practice for CAD and computational purposes. `Line geometry' therein is considered as a method to deal with geometric problems rather than as a certain area of it. So the book covers a large spectrum of modern computational geometry and CAGD methods, with a thorough theoretical background and an immense number of special problems and examples, mostly illustrated by well-designed figures.

As line geometry stems from projective geometry, one finds a concise, nevertheless precise, introduction to projective geometry (being often neglected in academic courses), to the basic concepts of projective differential geometry and to those of algebraic geometry (as far as needed in the sequel). Though some deeper results of the latter are quoted without proofs, the reader finds all necessary information to understand algebraic varieties, Gröbner bases and their usefulness in elimination of variables or in implicitation of parametrized varieties. All these topics lead to a deep understanding of rational curves and surfaces and their usual CAD representations (Bézier, B-spline, NURBS, etc.). Whenever seeming useful, their duals are also taken into consideration.

In the next chapters the kernel information on line geometry (Klein quadric in projective 5-space, Klein mapping and Plücker coordinates) and the specialisation to the Euclidean case are given. The authors emphasize the central role of linear complexes and the derived notions of linear congruences and reguli.

The remainder of the book (about two third) is devoted to ruled surfaces (skew and developable), to line congruences and to linear line mappings and their various applications. A few keywords must be sufficient to give an impression of this stuff:

• Spatial kinematics and robotics; the linear complex of path normals of the momentaneous helical motion; application to screw theory and six-leg manipulators (platforms).
• Approximation of lines using the normalized polar form of the Klein quadric; finding the best fitting linear complex for a set (of more than five) lines; application to reverse engineering (e.g. reconstruction of a surface of revolution from scattered data).
• Ruled surfaces in projective and Euclidean space, first and second order invariants, striction line, Sannia frame; algebraic and rational ruled surfaces, skew cubic surfaces and their linear congruence; Bézier representation in the model space and geometric continuity; discrete approximation and variational design of ruled surfaces.
• Developable surfaces and their development; the osculating cone; singular generators; rational and algebraic developable surfaces; dual control structures; interpolation and approximation with developable surfaces; developable surfaces with constant slope; the cyclographic mapping and Laguerre geometry with application to offset curves and medial axis transform; developable surfaces with creases.
• Line congruences, focal surfaces, Euclidean invariants, central surface; normal congruences with application to geometrical optices anticaustics of a mirror surface (mirror design); milling problems (singular positions) via the normal congruence; sculptured surface machining.
• Linear line mappings (computational kinematics); the mappings of Hopf and those of Blaschke Grünwald, Clifford parallelity, theorem of Ivory, trilateration problem, spherical motions, motion design.

The book can be recommended with emphasis to any scientist (academic as well as industrial) working in a CAD environment. It does not require many prerequisites beyond the usual techniques in this field, but some maturity in geometric reasoning and the ability of seeing what the analytic representations of geometric objects really mean is absolutely necessary because of the condensed style.

There are almost no items of criticism to be mentioned. Only at some places references to original papers or to classical books are missing. Sometimes the relation of an application to the theoretical passage seems rather loose. Summarizing, `Computational Line Geometry' is an excellent book, also with respect to its production quality, and anyone who will read it will certainly draw much gain (and pleasure on geometry) from it.

In this paper we consider error propagation in geometric constructions from a geometric viewpoint. First we study affine combinations of convex bodies: this has many applications concerning spline curves and surfaces defined by control points. Second, we study in detail the circumcircle of three points in the Euclidean plane. It turns out that the right geometric setting for this problem is Laguerre geometry and the cyclographic mapping, which provides a point model for sets of circles or spheres.

Keywords: Error propagation; Convex combination; Spline curve; Apollonius problem.

Laguerre geometry provides a simple approach to the design of rational curves and surfaces with rational offsets. These so-called PH curves and PN surfaces can be constructed from arbitrary rational curves or surfaces with help of a geometric transformation which describes a change between two models of Laguerre geometry. Closely related to that is their optical interpretation as anticaustics of arbitrary rational curves/surfaces for parallel illumination. A theorem on rational parametrizations for envelopes of natural quadrics leads to algorithms for the computation of rational parametrizations of surfaces; those include canal surfaces with rational spine curve and rational radius function, offsets of rational ruled surfaces or quadrics, and surfaces generated by peripheral milling with a cylindrical or conical cutter. Laguerre geometry is also useful for the construction of PN surfaces with rational principal curvature lines. New families of such principal PN surfaces are determined.

Keywords: Laguerre geometry; NC milling; geometrical optics; rational curve; rational surface; offset; rational offset; Pythagorean-hodograph curve; principal patch; principal curvature line.

We briefly introduce to the basics of Laguerre geometry and then show that this classical sphere geometry can be applied to solve various problems in geometric design. In the present part, we focus on applications of the cyclographic model of Laguerre geometry and the cyclographic map. It relates the medial axis and Voronoi curves/surfaces to special surface/surface intersections and the corresponding trimming procedures to hidden line removal. Rational canal surfaces are treated as cyclographic images of rational curves in R4. This leads to a simple control structure for rational canal surfaces. Its practical use is demonstrated at hand of modeling techniques with Dupin cyclides.

Keywords: Laguerre geometry; NC milling; Blending; Canal surface; Dupin cyclide; Geometrical optics; Medial axis; Rational curve; Rational surface; Voronoi curve; Voronoi surface

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