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2000
AI
This book introduces geometric concepts and tools essential for addressing problems in computer graphics, geometric modeling, computer vision, and motion planning. It aims to bridge the gap between theoretical and application-oriented resources by providing a coherent view on geometric methods and algorithms related to polynomial curves and surfaces, catering to a broad audience from students to professionals in computer science, mathematics, and engineering.
Computer-Aided Design, 1989
It is a classical principle in mathematics that polynomials in a single variable of degree n are essentially equivalent to symmetric polynomials in n variables that are linear in each variable separately. We shall apply this principle to the Bezier and B-spline curves and surfaces that are used in computer aided geometric design. The main result is a method of labeling the Bezier points that control a curve segment or surface patch or the de Boor points that control a B-spline curve with symmetric, multivariate labels. The properties of these labels make it simple to understand or to reconstruct the basic algorithms in this area, such as the de Casteljau Algorithm and the de Boor Algorithm.
ACM Transactions on Graphics, 1984
2006
In this technical report we present a rational reconstruction of the area method (developed by Chou, Gao and Zhang) for automated theorem proving for Euclidean geometry. Our rational reconstruction covers all relevant lemmas proved in full details and also full details of required algebraic reasoning (missing from the papers introducint the area method). We also present our implementation of this algorithm, made within the program GCLC. The area method main idea is to express the hypothesis of a theorem using a set of constructive statements each of then introducing a new point, and to express the conclusion by a polynomial in a some geometry quantities, without any relation to a given system of coordinates. The proof is then developed by eliminating, in reverse order, the point introduced before, using for that purpose a set of lemmas. After eliminating all the introduced points the polynomial is just an equality between two rational expression in independent variables. Hence if they are equal the statement is true, otherwise it is false. The proofs generated by the prover (developed as a part of GCLC) are generally short and readable. The program can prove many non-trivial theorems in a very efficient way.
Sailing Routes in the World of Computation, 2018
It is well known that several classical geometry problems (e.g., angle trisection) are unsolvable by compass and straightedge constructions. But what kind of object is proven to be non-existing by usual arguments? These arguments refer to an intuitive idea of a geometric construction as a special kind of an "algorithm" using restricted means (straightedge and/or compass). However, the formalization is not obvious, and different descriptions existing in the literature are far from being complete and clear. We discuss the history of this notion and a possible definition in terms of a simple game.
IEEE Computer Graphics and Applications, 2002
T he traditional method of defining geometrical objects in fields like computer graphics, robotics, and computer vision routinely uses vectors to characterize constructions. Doing this effectively means extending the basic concept of a vector as an element of a linear space by an inner product and a cross product and by some additional constructions such as homogeneous coordinates. This compactly encodes the intersection of, for instance, offset planes in space. Many of these techniques work well in 3D space, but some problems exist, such as the difference between vectors and points 1 and characterizing planes by normal vectors (which may require extra computation after linear transformations because a transformed plane's normal vector is not the normal vector's transform). Application programmers typically fix these problems by introducing data structures and combination rules, possibly using objectoriented programming to implement this patch. 2 Yet, deeper issues in programming geometry exist that many practitioners still accept. For instance, when intersecting linear subspaces, it seems unavoidable that we need to split our intersection algorithms to treat the intersection of lines and planes, planes and planes, lines and lines, and so on in separate pieces of code. After all, the outcomes themselves can be points, lines, or planes, which are essentially different in their further processing.
Several fields of mathematics are relevant to computer aided design and other software systems involving solid object geometry, topology, dif-ferential and algebraic geometry being particularly important. This paper discusses some of this mathematics in order to provide a theoretical foun-dation for geometric modelling kernels that support non-manifold objects with an internal cellular structure and subsets of different dimensions. The paper shows relationships between relevant concepts from topol-ogy, differential geometry and computer aided geometric design that are not widely known in the CAD community. It also discusses semialgebraic, semianalytic and subanalytic sets as candidates for object representation. Stratifications of such sets are proposed for an object's cellular structure and new stratification concepts are introduced to support candidate ap-plications. Acknowledgements This paper was motivated by the Djinn research project funded by the Engineering and Physical ...
The Visual Computer, 1987
Until relatively recently, researchers in computer graphics paid scant attention to the numerics of their computations. Computation was used as a simple tool to evaluate algorithms or transform data into some appropriate pictoral representation. Thus standard computer graphics texts have little to say about numerical methods, just as earlier numerical analysis textbooks had little to say about computer graphics. This is now changing, for the important reasons outlined in this paper.
Applications of Geometric Algebra in Computer Science and Engineering, 2002
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riccardo.migliari.it
Descriptive geometry has a historically consolidated relationship with the art and the construction in general, and it could therefore not fail to be affected by the technological evolution we mentioned earlier. The classical corpus of texts on the discipline, based first of all on the representation methods, understood as the theories of the construction of the encoded image, appeared to be completely inadequate compared with the contemporary project procedure and, what is worse, it seemed unrelated to the new representation techniques of the space, while these last, at the same time, did not seem to have a basis theory of general character, but only the algorithms that permit to solve this or that particular problem. In the academic circles, the architects who, due to changing historical events, today are the repository of the discipline of descriptive geometry, have finally seized the wish for renewal that came above all from the youth, faced, on the one hand, with a theory that does not seem to have any more applications and, on the other, with a technique that is incomprehensible, precisely because it is lacking the general concepts of a theory; now, finally, is ongoing a process of revision and renewal of the classic descriptive geometry, which is based on new definitions of the fundamentals and fulfilled through integrations and transformations of the corpus of texts on the discipline. As we will see in a while, the integrations concern, essentially, the representation methods, while the transformations involve above all the construction procedures of the geometric shapes. The representation methods, in general, are distinguishable for two essential reasons: the first, and the most important, is that each of the methods is able to record the characteristics of the shape and the dimension of an object in the space and, at the same time, it is able to transfer the object back into the space once it has been represented. A method, to be considered as such, must be able to perform this path, in both directions, autonomously, that is, without turning to other methods. The second reason that permits to distinguish the methods, the one from the other, concerns the use of each of them within the planning activity: the metric control, as in the case of the representation in plan and elevation, or the formal and perceptual control, as in the case of the perspective. The information systems make use, basically, of two digital representation methods that have been called: mathematical representation and numerical or polygonal representation. The digital representation methods have some other extremely innovative characteristics, which have remarkable impacts on the planning as well as on the theoretical evolution of descriptive geometry and these are: the accuracy and the spatiality. This modality of the geometric expression, which uses the analogy to describe the result of a digital process, may appear to be hybrid, too, and not purely geometric. But, as a matter of fact, we are simply dealing with the fulfilment of one of Monge’s wishes, the synergies between analysis and synthesis, between symbolic languages and graphic languages, which the French mathematician already clearly expressed in the first edition of his most famous work (1798): the analytic geometry and descriptive geometry should be ‘cultivated’ together, since in this way the descriptive geometry would bring its own evidence into the most complex analytical processes and, in turn, the analysis would bring the generality and accuracy of the results to the descriptive geometry. After all, Monge’s idea has not been abandoned: this way of making research, open-minded, deliberately disrespectful of the fences that surround the disciplinary fields of mathematics, has illustrious supporters: from Guido Castelnuovo (1903) to Harold Scott Mc Donald Coxeter (1961). In former times, when electronic computers were still unknown or had far from reached their present-day world-wide distribution, the schools of mathematics have created a rich selection of three-dimensional models, maquettes in plaster, wood and other materials very similar to those which decorate the studio of an architect or a designer. Recently these collections have been subject to studies and additions carried out, precisely, with virtual models. René Thom (1977), while on the one hand he considers the descriptive geometry as an obsolete science, and affirms that he would like to free himself from the manipulation of the Euclidean three-dimensional bodies of the space, on the other hand he recognizes to the qualitative sciences the capability to construct models that are the only ones able to explain our surrounding world, even if they cannot give quantitative results. So, it really looks like as if the synergy hoped for by Monge, and today fully expressed by the union between the concise reasoning of the geometry and the electronic calculation, is able to create a synthesis between quantity and quality, producing results that, while they describe with audacious analogies and with great immediacy the reality of a phenomenon, they also allow a quantitative analysis with controlled accuracy.
In this technical report we present a rational reconstruction of the area method (developed by Chou, Gao and Zhang) for automated theorem proving for Euclidean geometry. Our rational reconstruction covers all relevant lemmas proved in full details and also full details of required algebraic reasoning (missing from the papers introducint the area method). We also present our implementation of this algorithm, made within the program GCLC.
Handbook of Computer Aided Geometric Design, 2002
The concepts and methods of algebra and algebraic geometry have found significant applications in many disciplines. This chapter presents a collection of gleanings from algebra or algebraic geometry that hold practical value for the field of computer aided geometric design. We focus on the insights, algorithm enhancements and practical capabilities that algebraic methods have contributed to CAGD. Specifically, we examine resultants and Gröbner basis, and discuss their applications in implicitization, inversion, parametrization and intersection algorithms. Other topics of CAGD research work using algebraic methods are also outlined.
Cahiers de Topologie et Géométrie Différentielle Catégoriques, 1986
© Andrée C. Ehresmann et les auteurs, 1986, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
ACM Transactions on Graphics, 1995
This article presents a functional programming approach to geometric design with embedded polyhedral complexes. Its main goals are to show the expressive power of the language as well as its usefulness for geometric design. The language, named PIASM (the ProgrammingIAnguage for Solid Modeling), introduces a very high level approach tn "constructive" or "generative" modeling. Geometrical objects are generated by evaluating some suitable language expressions. Because generating expressions can be easily combined, the language also extends the standard variational geometry approach by supporting classes of geometric objects with varying topology and shape. The design language PLASM can be roughly considered as a geometry-oriented extension of a subset of the functional language FL. The language takes a dimension-independent approach to geometry representation and algorithms. In particular it implements an algebraic calculus over embedded polyhedra of any dimension. The generated objects are always geometrically consistent because the validity of geometry is guaranteed at a syntactical level. Such an approach allows one to use a representation scheme which is weaker than those usually adopted in solid modelers, thus encompassing a broader geometric domain, which contains solids, surfaces, and wire-frames, as well as higher-dimensional objects. Geometric Programming . 267 Son of man,. . . . show them the design and plan of the Temple, its exits and entrances, its shape, how all of it is arranged, the entire design and all its principles, Give them all this in writing so that they can see and take note of its design and the wa.v it is all arranged and carry it out.
arXiv (Cornell University), 2017
Designing software systems for Geometric Computing applications can be a challenging task. Software engineers typically use software abstractions to hide and manage the high complexity of such systems. Without the presence of a unifying algebraic system to describe geometric models, the use of software abstractions alone can result in many design and maintenance problems. Geometric Algebra (GA) can be a universal abstract algebraic language for software engineering geometric computing applications. Few sources, however, provide enough information about GA-based software implementations targeting the software engineering community. In particular, successfully introducing GA to software engineers requires quite different approaches from introducing GA to mathematicians or physicists. This article provides a high-level introduction to the abstract concepts and algebraic representations behind the elegant GA mathematical structure. The article focuses on the conceptual and representational abstraction levels behind GA mathematics with sufficient references for more details. In addition, the article strongly recommends applying the methods of Computational Thinking in both introducing GA to software engineers, and in using GA as a mathematical language for developing Geometric Computing software systems.
2019
© Andrée C. Ehresmann et les auteurs, 1988, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
Natural Computing, 2010
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