Since their first appearance in 1974, subdivision algorithms for generating surfaces of arbitrary topology have gained widespread popularity in computer graphics and are being evaluated in engineering applications. This development was complemented by ongoing efforts to develop appropriate mathematical tools for a thorough analysis, and today, many of the fascinating properties of subdivision are well understood.
This book summarizes the current knowledge on the subject. It contains both meanwhile classical results as well as brand-new, unpublished material, such as a new framework...
Since their first appearance in 1974, subdivision algorithms for generating surfaces of arbitrary topology have gained widespread popularity in com...
The central object of this book is the measure of geometric quantities describing N a subset of the Euclidean space (E, ), endowed with its standard scalar product. Let us state precisely what we mean by a geometric quantity. Consider a subset N S of points of the N-dimensional Euclidean space E, endowed with its standard N scalar product. LetG be the group of rigid motions of E . We say that a 0 quantity Q(S) associated toS is geometric with respect toG if the corresponding 0 quantity Q g(S)] associated to g(S) equals Q(S), for all g?G . For instance, the 0 diameter ofS and the area of the...
The central object of this book is the measure of geometric quantities describing N a subset of the Euclidean space (E, ), endowed with its standard s...
The application of geometric algebra to the engineering sciences is a young, active subject of research. The promise of this field is that the mathematical structure of geometric algebra together with its descriptive power will result in intuitive and more robust algorithms.
This book examines all aspects essential for a successful application of geometric algebra: the theoretical foundations, the representation of geometric constraints, and the numerical estimation from uncertain data. Formally, the book consists of two parts: theoretical foundations and applications. The first...
The application of geometric algebra to the engineering sciences is a young, active subject of research. The promise of this field is that the math...
papers, illustrated with examples. They include wavelet bases, implicit functions de ned on a space grid, etc. It appears that a common pattern is the recovery of a controllable model of the scene, such that the resulting images can be edited (interaction). Changing the viewpoint is only one (important) aspect, but changing the lighting and action is equally important 2]. Recording and representing three-dimensional scenes is an emerging technology made possible by the convergence of optics, geometry and computer science, with many applications in the movie industry, and more generally in...
papers, illustrated with examples. They include wavelet bases, implicit functions de ned on a space grid, etc. It appears that a common pattern is the...
Since their first appearance in 1974, subdivision algorithms for generating surfaces of arbitrary topology have gained widespread popularity in computer graphics and are being evaluated in engineering applications. This development was complemented by ongoing efforts to develop appropriate mathematical tools for a thorough analysis, and today, many of the fascinating properties of subdivision are well understood.
This book summarizes the current knowledge on the subject. It contains both meanwhile classical results as well as brand-new, unpublished material, such as a new framework...
Since their first appearance in 1974, subdivision algorithms for generating surfaces of arbitrary topology have gained widespread popularity in com...
The application of geometric algebra to the engineering sciences is a young, active subject of research. The promise of this field is that the mathematical structure of geometric algebra together with its descriptive power will result in intuitive and more robust algorithms.
This book examines all aspects essential for a successful application of geometric algebra: the theoretical foundations, the representation of geometric constraints, and the numerical estimation from uncertain data. Formally, the book consists of two parts: theoretical foundations and applications. The first...
The application of geometric algebra to the engineering sciences is a young, active subject of research. The promise of this field is that the math...
By virtue of their special algebraic structures, Pythagorean-hodograph (PH) curves offer unique advantages for computer-aided design and manufacturing, robotics, motion control, path planning, computer graphics, animation, and related fields. This book offers a comprehensive and self-contained treatment of the mathematical theory of PH curves, including algorithms for their construction and examples of their practical applications. Special features include an emphasis on the interplay of ideas from algebra and geometry and their historical origins, detailed algorithm descriptions, and many...
By virtue of their special algebraic structures, Pythagorean-hodograph (PH) curves offer unique advantages for computer-aided design and manufactur...
This welcome boon for students of algebraic topology cuts a much-needed central path between other texts whose treatment of the classification theorem for compact surfaces is either too formalized and complex for those without detailed background knowledge, or too informal to afford students a comprehensive insight into the subject. Its dedicated, student-centred approach details a near-complete proof of this theorem, widely admired for its efficacy and formal beauty. The authors present the technical tools needed to deploy the method effectively as well as demonstrating their use in a...
This welcome boon for students of algebraic topology cuts a much-needed central path between other texts whose treatment of the classification theorem...
Subdivision is a way of representing smooth shapes in a computer. A curve or surface (both of which contain an in?nite number of points) is described in terms of two objects. One object is a sequence of vertices, which we visualise as a polygon, for curves, or a network of vertices, which we visualise by drawing the edges or faces of the network, for surfaces. The other object is a set of rules for making denser sequences or networks. When applied repeatedly, the denser and denser sequences are claimed to converge to a limit, which is the curve or surface that we want to represent. This book...
Subdivision is a way of representing smooth shapes in a computer. A curve or surface (both of which contain an in?nite number of points) is described ...
This book summarizes research carried out in workshops of the SAGA project, an Initial Training Network exploring the interplay of Shapes, Algebra, Geometry and Algorithms.
Written by a combination of young and experienced researchers, the book introduces new ideas in an established context. Among the central topics are approximate and sparse implicitization and surface parametrization; algebraic tools for geometric computing; algebraic geometry for computer aided design applications and problems with industrial applications.
Readers will encounter new methods for the...
This book summarizes research carried out in workshops of the SAGA project, an Initial Training Network exploring the interplay of Shapes, Algebra,...
