768 results on '"Homogeneous coordinates"'
Search Results
2. Paraxial Geometric Optics in 3D Through Point-Based Geometric Algebra
- Author
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Dorst, Leo, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Sheng, Bin, editor, Bi, Lei, editor, Kim, Jinman, editor, Magnenat-Thalmann, Nadia, editor, and Thalmann, Daniel, editor
- Published
- 2024
- Full Text
- View/download PDF
3. An interconnected system of energy.
- Author
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Wadi, Husam
- Subjects
- *
CLASSICAL mechanics , *BOLTZMANN machine , *NON-Euclidean geometry , *SPECIAL relativity (Physics) - Abstract
The purpose of this paper is to describe a system of interconnected energy models. These models produce inferences that may enhance our understanding of classical and quantum mechanics. From the formulated models, an empirical test is derived to validate the proposed light projection hypothesis. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
4. ROBUST LINE-CONVEX POLYGON INTERSECTION COMPUTATION IN E2 USING PROJECTIVE SPACE REPRESENTATION.
- Author
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Skala, Vaclav
- Subjects
COMPUTER graphics ,GRAPHICS processing units ,COMPUTER algorithms ,PROJECTIVE spaces ,ROBUST control - Abstract
This paper describes modified robust algorithms for a line clipping by a convex polygon in E2 and a convex polyhedron in E3. The proposed algorithm is based on the Cyrus-Beck algorithm and uses homogeneous coordinates to increase the robustness of computation. The algorithm enables computation fully in the projective space using the homogeneous coordinates and the line can be given in the projective space, in general. If the result can remain in projective space, no division operation is needed. It supports the use of vector-vector operations, SSE/AVX instructions, and GPU. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
5. Numerical Solution of the Inverse Kinematics Problem on the Example of a 6-DOF Robot
- Author
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Karabanov, Georgy, Selyukov, Alexander, Krakhmalev, Oleg, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Ronzhin, Andrey, editor, Meshcheryakov, Roman, editor, and Xiantong, Zhen, editor
- Published
- 2022
- Full Text
- View/download PDF
6. A Brief Survey of Clipping and Intersection Algorithms with a List of References (including Triangle-Triangle Intersections)✩.
- Author
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Skala, Vaclav
- Subjects
- *
ALGORITHMS , *TRIANGLES , *PROJECTIVE spaces - Abstract
This contribution presents a brief survey of clipping and intersection algorithms in E 2 and E 3 with a nearly complete list of relevant references. Some algorithms use the projective extension of the Euclidean space and vector-vector operations, which support GPU and SSE use. This survey is intended to help researchers, students, and practitioners dealing with intersection and clipping algorithms. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
7. A Brief Survey of Clipping and Intersection Algorithms with a List of References (including Triangle-Triangle Intersections)✩.
- Author
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Skala, Vaclav
- Subjects
ALGORITHMS ,TRIANGLES ,PROJECTIVE spaces - Abstract
This contribution presents a brief survey of clipping and intersection algorithms in E 2 and E 3 with a nearly complete list of relevant references. Some algorithms use the projective extension of the Euclidean space and vector-vector operations, which support GPU and SSE use. This survey is intended to help researchers, students, and practitioners dealing with intersection and clipping algorithms. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
8. A Novel Line Convex Polygon Clipping Algorithm in E2 with Parallel Processing Modification
- Author
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Skala, Vaclav, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Woeginger, Gerhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Gervasi, Osvaldo, editor, Murgante, Beniamino, editor, Misra, Sanjay, editor, Garau, Chiara, editor, Blečić, Ivan, editor, Taniar, David, editor, Apduhan, Bernady O., editor, Rocha, Ana Maria A. C., editor, Tarantino, Eufemia, editor, and Torre, Carmelo Maria, editor
- Published
- 2021
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- View/download PDF
9. Methods for Determining the Size of three-Dimensional Objects in Images
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L. V. Serebryanaya and V. N. Breshko
- Subjects
image ,three-dimensional model ,computer vision ,singular points ,homogeneous coordinates ,descriptor ,Information technology ,T58.5-58.64 - Abstract
Computer vision technology is used to obtain the necessary information from images. The two-dimensional image was transformed into its corresponding three-dimensional structure based on the reconstruction process and singular points. A review of software tools designed for scanning objects and measuring their dimensions is performed. A mathematical model and algorithm for constructing three-dimensional structures of objects in images are presented. Based on the design pattern, the architecture of the software application is developed. It consists of an image processing module, a module for calculating the size of objects, and a module for storing the received data. The fastest algorithm for finding singular points and their descriptors is used. In this paper, both homogeneous and Euclidean coordinates of characteristic points are calculated. The method of using the mobile application in automatic and manual modes is described. The results of scanning objects with the camera of a mobile device and determining their size in real time are presented.
- Published
- 2021
10. 3D Transformation Matrix Synthesis in Column Major and Row Major Forms: Applicative Perspective for 3D Object Representation.
- Author
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ShwetaChaku, Sainger, Monika, Bhatnagar, Amrita, RaginiKarwayun, and SoumiGhosh
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IMAGE processing ,COORDINATES ,THREE-dimensional imaging ,ROTATIONAL motion ,IMAGING systems - Abstract
The Transformation of a 2D or 3D objects are the effective means of shifting or changing the dimensions and orientations of images in the most effective way. If we fail to transform the object in terms of displacement,enlargement, orientation, we may land up in creating something that is distorted and processessing a distorted object is not acceptable in the various critical fields like Medical and Forensics. The usual practice of defining transformations is straight forward. The transformed object can be obtained by coupling original object with the transformation vectors. The main challenge is how to evaluate it. The usual practice is standard Column Vector form. The alternative Row Vector Form is also known approach but what matters is the sequence of operations that make these both approaches worth mentioning.While doing so our analysis on 3D content keeps our knowledge flawless and takes it a step further as far as Image Processing is concerned. Such analytical study is very vital since most of the content created, acquired, reproduced, and visualized in 3D needs to be mapped on to 3D. This paper describes the transformations(Translation, Scaling and Rotation) in the both Column and Row Vectar Approach. This paper aims in providing a clear sequence of calculations which differ in both approaches. [ABSTRACT FROM AUTHOR]
- Published
- 2022
11. 三次加权 Lupaş q-Bézier 曲线表示的圆锥曲线.
- Author
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申学超 and 韩力文
- Abstract
Copyright of Journal of Computer-Aided Design & Computer Graphics / Jisuanji Fuzhu Sheji Yu Tuxingxue Xuebao is the property of Gai Kan Bian Wei Hui and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
- Published
- 2022
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12. Rational swept surface constructions based on differential and integral sweep curve properties
- Author
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Farouki, Rida T and Nittler, Kevin M
- Subjects
Swept surface ,Profile curve ,Sweep curve ,Homogeneous coordinates ,Rational surface ,Pythagorean-hodograph curve ,Mathematical Sciences ,Information and Computing Sciences ,Engineering ,Software Engineering - Abstract
A swept surface is generated from a profile curve and a sweep curve by employing the latter to define a continuous family of transformations of the former. By using polynomial or rational curves, and specifying the homogeneous coordinates of the swept surface as bilinear forms in the profile and sweep curve homogeneous coordinates, the outcome is guaranteed to be a rational surface compatible with the prevailing data types of CAD systems. However, this approach does not accommodate many geometrically intuitive sweep operations based on differential or integral properties of the sweep curve - such as the parametric speed, tangent, normal, curvature, arc length, and offset curves - since they do not ordinarily have a rational dependence on the curve parameter. The use of Pythagorean-hodograph (PH) sweep curves surmounts this limitation, and thus makes possible a much richer spectrum of rational swept surface types. A number of representative examples are used to illustrate the diversity of these novel swept surface forms - including the oriented-translation sweep, offset-translation sweep, generalized conical sweep, and oriented-involute sweep. In many cases of practical interest, these forms also have rational offset surfaces. Considerations related to the automated CNC machining of these surfaces, using only their high-level procedural definitions, are also briefly discussed.
- Published
- 2015
13. Rational swept surface constructions based on differential and integral sweep curve properties
- Author
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Farouki, RT and Nittler, KM
- Subjects
Swept surface ,Profile curve ,Sweep curve ,Homogeneous coordinates ,Rational surface ,Pythagorean-hodograph curve ,Software Engineering ,Mathematical Sciences ,Information and Computing Sciences ,Engineering - Abstract
A swept surface is generated from a profile curve and a sweep curve by employing the latter to define a continuous family of transformations of the former. By using polynomial or rational curves, and specifying the homogeneous coordinates of the swept surface as bilinear forms in the profile and sweep curve homogeneous coordinates, the outcome is guaranteed to be a rational surface compatible with the prevailing data types of CAD systems. However, this approach does not accommodate many geometrically intuitive sweep operations based on differential or integral properties of the sweep curve - such as the parametric speed, tangent, normal, curvature, arc length, and offset curves - since they do not ordinarily have a rational dependence on the curve parameter. The use of Pythagorean-hodograph (PH) sweep curves surmounts this limitation, and thus makes possible a much richer spectrum of rational swept surface types. A number of representative examples are used to illustrate the diversity of these novel swept surface forms - including the oriented-translation sweep, offset-translation sweep, generalized conical sweep, and oriented-involute sweep. In many cases of practical interest, these forms also have rational offset surfaces. Considerations related to the automated CNC machining of these surfaces, using only their high-level procedural definitions, are also briefly discussed.
- Published
- 2015
14. Geometric Algebra, Extended Cross-Product and Laplace Transform for Multidimensional Dynamical Systems
- Author
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Skala, Vaclav, Kacprzyk, Janusz, Series editor, Pal, Nikhil R., Advisory editor, Bello Perez, Rafael, Advisory editor, Corchado, Emilio S., Advisory editor, Hagras, Hani, Advisory editor, Kóczy, László T., Advisory editor, Kreinovich, Vladik, Advisory editor, Lin, Chin-Teng, Advisory editor, Lu, Jie, Advisory editor, Melin, Patricia, Advisory editor, Nedjah, Nadia, Advisory editor, Nguyen, Ngoc Thanh, Advisory editor, Wang, Jun, Advisory editor, Silhavy, Radek, editor, Silhavy, Petr, editor, and Prokopova, Zdenka, editor
- Published
- 2018
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15. Geometrical Transformations
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Ammeraal, Leen, Zhang, Kang, Ammeraal, Leen, and Zhang, Kang
- Published
- 2017
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16. THE INFLUENCE OF THE HORIZONTAL LOAD VIBRATIONS OF THE CRANE-PIPE LAYING MACHINE GIBBET ON THE LOAD MOMENT STABILITY
- Author
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R. Yu. Sukharev and V. V. Tanskiy
- Subjects
crane-pipe laying machine ,vibrations ,cargo ,load moment ,stability ,design scheme ,homogeneous coordinates ,microrelief ,load deflection ,Transportation engineering ,TA1001-1280 - Abstract
Introduction: the main feature of operation of cranes-pipe layers is work in difficult ground conditions, which significantly affects the operating mode of the machine. This factor is one of the main reason leading to the rocking of the load on the crane-pipe laying machine gibbet and as a consequence it leads to emergency and contingency situations. In the research the forced vibration of the load on the crane-pipe layer gibbet and the causes of their occurrence and the problems to which they lead are observed. Engineering solutions of predecessors are considered. A new approach to the solution of these problems is proved.Materials and methods: the design scheme of the crane-pipe laying machine is justified, assumptions are accepted, coordinate systems are introduced, and a mathematical model of the crane-pipe-laying machine is compiled.Results: the following time dependences are constructed: the deflection of the cargo in the transverse plane of the crane-pipe layer (horizontalvibrations), the deflection of the cargo in the longitudinal plane of the pipe laying crane, the roll of the base machine, the change in the load moment are also constructed. The influence of the horizontal vibrations of the load on the change in the load moment and the occurrence of vertical vibrations is determined.Discussion and conclusion: the effect of horizontal vibrations on the change of the load moment is evaluated and as a consequence of the negative effect on the stability of the crane-pipe laying machine is presented.
- Published
- 2018
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17. Optimized line and line segment clipping in E2 and Geometric Algebra.
- Author
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Skala, Vaclav
- Subjects
- *
ALGEBRA , *HOMOGENEOUS spaces , *COMPUTER graphics , *PROJECTIVE spaces , *ALGORITHMS - Abstract
Algorithms for line and line segment clipping are well known algorithms especially in the field of computer graphics. They are formulated for the Euclidean space representation. However, computer graphics uses the projective extension of the Euclidean space and homogeneous coordinates for representation geometric transformations with points in the E² or E³ space. The projection operation from the E³ to the E² space leads to the necessity to convert coordinates to the Euclidean space if the clipping operation is to be used. In this contribution, an optimized simple algorithm for line and line segment clipping in the E² space, which works directly with homogeneous representation and not requiring the conversion to the Euclidean space, is described. It is based on Geometric Algebra (GA) formulation for projective representation. The proposed algorithm is simple, efficient and easy to implement. The algorithm can be efficiently modified for the SSE4 instruction use or the GPU application, too. [ABSTRACT FROM AUTHOR]
- Published
- 2020
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- View/download PDF
18. Computing several eigenvalues of nonlinear eigenvalue problems by selection.
- Author
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Hochstenbach, Michiel E. and Plestenjak, Bor
- Subjects
- *
NONLINEAR equations , *EIGENVALUES , *POLYNOMIALS , *EIGENVECTORS - Abstract
Computing more than one eigenvalue for (large sparse) one-parameter polynomial and general nonlinear eigenproblems, as well as for multiparameter linear and nonlinear eigenproblems, is a much harder task than for standard eigenvalue problems. We present simple but efficient selection methods based on divided differences to do this. Selection means that the approximate eigenpair is picked from candidate pairs that satisfy a certain suitable criterion. The goal of this procedure is to steer the process away from already detected pairs. In contrast to locking techniques, it is not necessary to keep converged eigenvectors in the search space, so that the entire search space may be devoted to new information. The selection techniques are applicable to many types of matrix eigenvalue problems; standard deflation is feasible only for linear one-parameter problems. The methods are easy to understand and implement. Although the use of divided differences is well known in the context of nonlinear eigenproblems, the proposed selection techniques are new for one-parameter problems. For multiparameter problems, we improve on and generalize our previous work. We also show how to use divided differences in the framework of homogeneous coordinates, which may be appropriate for generalized eigenvalue problems with infinite eigenvalues. While the approaches are valuable alternatives for one-parameter nonlinear eigenproblems, they seem the only option for multiparameter problems. [ABSTRACT FROM AUTHOR]
- Published
- 2020
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19. Scientific Computing and Computer Graphics with GPU: Application of Projective Geometry and Principle of Duality.
- Author
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Skala, V., Karim, S. A. A., and Kadir, E. A.
- Subjects
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PROJECTIVE geometry , *SCIENTIFIC computing , *VECTOR algebra , *ALGEBRA , *LINEAR equations , *COMPUTER graphics - Abstract
Geometric problems are usually solved in the Euclidean space by using the standard vector algebra techniques. In this study, principles of the projective geometry and geometric algebra will be introduced via a novel method that significantly simplifies the solution of geometrical problems. Also, it supports the GPU parallel computation application. Besides that, an application of the principle of duality leads to a simple solution of the dual problems. We show that, the equivalence of the extended cross-product (outer product) and the solution of the system of linear equations. This gives a direct impact to scientific computation, solution of geometrical problems, robotics, computer graphics algorithms and virtual reality via fast computation through GPU parallel systems. Some numerical and graphical results are presented. [ABSTRACT FROM AUTHOR]
- Published
- 2020
20. Associate Submersions and Qualitative Properties of Nonlinear Circuits with Implicit Characteristics.
- Author
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Riaza, Ricardo
- Subjects
- *
PROJECTIVE spaces , *NONLINEAR equations , *NONLINEAR theories , *MATHEMATICAL equivalence , *ERROR rates - Abstract
We introduce in this paper an equivalence notion for submersions U → ℝ , U open in ℝ 2 , which makes it possible to identify a smooth planar curve with a unique class of submersions. This idea, which extends to the nonlinear setting the construction of a dual projective space, provides a systematic way to handle global implicit descriptions of smooth planar curves. We then apply this framework to model nonlinear electrical devices as classes of equivalent functions. In this setting, linearization naturally accommodates incremental resistances (and other analogous notions) in homogeneous terms. This approach, combined with a projectively-weighted version of the matrix-tree theorem, makes it possible to formulate and address in great generality several problems in nonlinear circuit theory. In particular, we tackle unique solvability problems in resistive circuits, and discuss a general expression for the characteristic polynomial of dynamic circuits at equilibria. Previously known results, which were derived in the literature under unnecessarily restrictive working assumptions, are simply obtained here by using dehomogenization. Our results are shown to apply also to circuits with memristors. We finally present a detailed, graph-theoretic study of certain stationary bifurcations in nonlinear circuits using the formalism here introduced. [ABSTRACT FROM AUTHOR]
- Published
- 2020
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21. Exercises on Projective Spaces
- Author
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Fortuna, Elisabetta, Frigerio, Roberto, Pardini, Rita, Fortuna, Elisabetta, Frigerio, Roberto, and Pardini, Rita
- Published
- 2016
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22. Algorithms for computing basins of attraction associated with a rational self-map of the Hopf fibration based on Lyapunov exponents.
- Author
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Álvarez-Aparicio, V., García-Calcines, J.M., Hernández-Paricio, L.J., and Rivas-Rodríguez, M.T.
- Subjects
- *
ENDOMORPHISMS , *NEWTON-Raphson method , *LYAPUNOV exponents , *LYAPUNOV functions , *ALGORITHMS - Abstract
The objective of this work is twofold. One, we develop a theory of iteration for complex rational functions so that the problems of overflow and indeterminacy caused by null, or almost null, denominators can be avoided when developing implementations. Second, we present easily implementable methods that allow the calculation of attracting cycles as well as the graphical representation of their basins of attraction. In order to deal with our first goal we work with homogeneous complex coordinates and we take the complex projective line as a model, which is expressed as a quotient of the 3-sphere through the Hopf fibration. An irreducible representation of a rational function can now be presented as a Hopf fibration endomorphism. As well as our second goal is concerned, we use Lyapunov functions and exponents to calculate the cycles associated with endomorphisms and to graphically represent the corresponding basins of attraction. We point out that our algorithms are based on the calculation of a finite set of non-negative real constants and their calculation does not depend on the previous determination of the fixed points or attracting cycles. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
23. A Graphic Method for Detecting Multiple Roots Based on Self-Maps of the Hopf Fibration and Nullity Tolerances
- Author
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José Ignacio Extreminana-Aldana, José Manuel Gutiérrez-Jiménez, Luis Javier Hernández-Paricio, and María Teresa Rivas-Rodríguéz
- Subjects
Newton’s method ,Hopf fibration ,multiple roots ,rational functions ,homogeneous coordinates ,Riemann sphere ,Mathematics ,QA1-939 - Abstract
The aim of this paper is to study, from a topological and geometrical point of view, the iteration map obtained by the application of iterative methods (Newton or relaxed Newton’s method) to a polynomial equation. In fact, we present a collection of algorithms that avoid the problem of overflows caused by denominators close to zero and the problem of indetermination which appears when simultaneously the numerator and denominator are equal to zero. This is solved by working with homogeneous coordinates and the iteration of self-maps of the Hopf fibration. As an application, our algorithms can be used to check the existence of multiple roots for polynomial equations as well as to give a graphical representation of the union of the basins of attraction of simple roots and the union of the basins of multiple roots. Finally, we would like to highlight that all the algorithms developed in this work have been implemented in Julia, a programming language with increasing use in the mathematical community.
- Published
- 2021
- Full Text
- View/download PDF
24. Geometric Transformations and Duality for Virtual Reality and Haptic Systems
- Author
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Skala, Vaclav, Junqueira Barbosa, Simone Diniz, editor, Chen, Phoebe, editor, Cuzzocrea, Alfredo, editor, Du, Xiaoyong, editor, Filipe, Joaquim, editor, Kara, Orhun, editor, Kotenko, Igor, editor, Sivalingam, Krishna M., editor, Ślęzak, Dominik, editor, Washio, Takashi, editor, Yang, Xiaokang, editor, and Stephanidis, Constantine, editor
- Published
- 2014
- Full Text
- View/download PDF
25. Error-free measurement method for 6-DOF systems via matrix decomposition approach
- Author
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Chung-Yu Tsai
- Subjects
Nonlinear system ,Matrix (mathematics) ,Transformation matrix ,Homogeneous coordinates ,Linearization ,Computer science ,Applied Mathematics ,Modeling and Simulation ,Applied mathematics ,Rotation matrix ,Rotation (mathematics) ,Matrix decomposition - Abstract
A six-degree-of-freedom measurement system possessing a pure algebraic and error-free calculation algorithm is proposed based on the rotation/translation matrix decomposition approach. Generally, such complicated mathematical models are simplified through linearization. However, this inevitably leads to calculation errors. Accordingly, in the method proposed in this study, the object motion behavior in three-dimensional space is described instead using a pose-change homogeneous coordinate transformation matrix consisting of two separate matrices, namely a rotation matrix which describes the nonlinear component of the object motion and a translation matrix which describes the linear component. In the solution process, the image-orientation-change method is first used to determine the rotation matrix. The translation matrix is then obtained using the singular-value-decomposition least-squares method. Finally, the two matrices are multiplied together to obtain the desired pose-change matrix. The validity of the proposed approach is demonstrated by means of an illustrative numerical example and a simple experimental trial. It is shown that, since the calculation algorithm is a purely analytical and algebraic method and involves no approximations or omissions (i.e., the relevant equations are not simplified through linearization and thus retain their original nonlinear integrity), the derived solutions are error-free if the errors introduced by the measurement technology are ignored.
- Published
- 2022
26. The Analytic Approach
- Author
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Lord, Eric and Lord, Eric
- Published
- 2013
- Full Text
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27. 2D Matrix Transforms
- Author
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Vince, John and Vince, John
- Published
- 2012
- Full Text
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28. The Poincaré problem for foliations on compact toric orbifolds
- Author
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Miguel Rodríguez Peña
- Subjects
Pure mathematics ,Homogeneous coordinates ,Mathematics::Commutative Algebra ,Mathematics::Complex Variables ,Holomorphic function ,Toric variety ,Algebraic geometry ,Mathematics::Algebraic Geometry ,Differential geometry ,Foliation (geology) ,Mathematics::Differential Geometry ,Geometry and Topology ,Invariant (mathematics) ,Mathematics::Symplectic Geometry ,Orbifold ,Mathematics - Abstract
We give an optimal upper bound of the degree of quasi-smooth hypersurfaces which are invariant by a one-dimensional holomorphic foliation on a compact toric orbifold, i.e. on a complete simplicial toric variety. This bound depends only on the degree of the foliation and of the degrees of the toric homogeneous coordinates.
- Published
- 2021
29. The Fučík spectrum of the discrete Dirichlet operator.
- Author
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Looseová, Iveta and Nečesal, Petr
- Subjects
- *
SPECTRAL theory , *DISCRETE systems , *DIRICHLET problem , *OPERATOR theory , *MODULES (Algebra) , *ALGEBRAIC curves - Abstract
In this paper, we deal with the discrete Dirichlet operator of the second order and we investigate its Fučík spectrum, which consists of a finite number of algebraic curves. For each non-trivial Fučík curve, we are able to detect a finite number of its points, which are given explicitely. We provide the exact implicit description of all non-trivial Fučík curves in terms of Chebyshev polynomials of the second kind. Moreover, for each non-trivial Fučík curve, we give several different implicit descriptions, which differ in the level of depth of used nested functions. Our approach is based on the Möbius transformation and on the appropriate continuous extension of solutions of the discrete problem. Let us note that all presented descriptions of Fučík curves have the form of necessary and sufficient conditions. Finally, our approach can be also directly used in the case of difference operators of the second order with other local boundary conditions. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
30. Multi‐directional colour edge detector using LQS convolution.
- Author
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Yasmin, Shagufta and Sangwine, Stephen J.
- Abstract
A new linear colour image filter based on linear quaternion systems (LQSs) is introduced. It detects horizontal, vertical, left‐ and right‐diagonal edges with a single LQS convolution mask. The proposed filter is a canonic minimal filter of four LQS filters, each with different angles of rotation combined parallel wise. Different angles of rotation are a key features of the new filter such that horizontal, vertical, left, and right‐diagonal LQS filter masks rotate pixels through angles π/2, 5π/2, 3π/2, and 7π/2, respectively. Although, the four LQS masks are combined parallel to make a single LQS mask but derived using four quaternion convolutions, one for each direction of edges, the LQS filter produces a result without the combination of results from four separate edge detectors. This methodology could be generalised to design more elaborate LQS filters to perform other geometric operations on colour image pixels. The proposed filter translates smoothly changing colours to different shades of grey and produces coloured edges in multiple directions, where there is a sudden change of colour in the original image. Another key idea of the proposed filter is that it is linear because it operates in homogeneous coordinates. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
31. Algebraic Clustering of Affine Subspaces.
- Author
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Tsakiris, Manolis C. and Vidal, Rene
- Subjects
- *
MACHINE learning , *SUBSPACES (Mathematics) , *IMAGE analysis , *AFFINE geometry , *POLYNOMIALS - Abstract
Subspace clustering is an important problem in machine learning with many applications in computer vision and pattern recognition. Prior work has studied this problem using algebraic, iterative, statistical, low-rank and sparse representation techniques. While these methods have been applied to both linear and affine subspaces, theoretical results have only been established in the case of linear subspaces. For example, algebraic subspace clustering (ASC) is guaranteed to provide the correct clustering when the data points are in general position and the union of subspaces is transversal. In this paper we study in a rigorous fashion the properties of ASC in the case of affine subspaces. Using notions from algebraic geometry, we prove that the homogenization trick , which embeds points in a union of affine subspaces into points in a union of linear subspaces, preserves the general position of the points and the transversality of the union of subspaces in the embedded space, thus establishing the correctness of ASC for affine subspaces. [ABSTRACT FROM PUBLISHER]
- Published
- 2018
- Full Text
- View/download PDF
32. Image Pose Estimation Based on Procrustes Theory.
- Author
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Xu, Zhenliang, Sun, Yi, Ma, Zhenling, and Li, Yanhuan
- Abstract
This paper has established a high-precision hierarchical estimated pose parameters of image. Firstly, we select corresponding three image points of 3D points which constitute the largest area in image as a base, in order to estimate the depth and translate information; then based on the above method, we obtain the scale parameter of camera exterior information. And finally, the topic is transformed to a problem of estimating rotation relationship by vector, using Procrustes theory to obtain the best estimate of the angle elements of exterior parameters. The method can effectively solve problems which depth and coupling pose parameters cannot deal with. Experimental results show that this method of determining position and orientation parameter estimation model is of briefness, easy convergence and it can also achieve higher parameter estimation accuracy than the direct projection matrix factorization. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
33. Online Template Attacks: Revisited
- Author
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Billy Bob Brumley, Alejandro Cabrera Aldaya, Tampere University, and Computing Sciences
- Subjects
FOS: Computer and information sciences ,Computer engineering. Computer hardware ,Computer Science - Cryptography and Security ,Computer science ,side-channel analysis ,Information technology ,Scalar multiplication ,TK7885-7895 ,Public-key cryptography ,microarchitecture attacks ,applied cryptography ,Elliptic curve cryptography ,TRACE (psycholinguistics) ,Homogeneous coordinates ,business.industry ,213 Electronic, automation and communications engineering, electronics ,T58.5-58.64 ,public key cryptography ,Microarchitecture ,EdDSA ,elliptic curve cryptography ,Curve25519 ,online template attacks ,business ,Cryptography and Security (cs.CR) ,Algorithm - Abstract
An online template attack (OTA) is a powerful technique previously used to attack elliptic curve scalar multiplication algorithms. This attack has only been analyzed in the realm of power consumption and EM side channels, where the signals leak related to the value being processed. However, microarchitecture signals have no such feature, invalidating some assumptions from previous OTA works. In this paper, we revisit previous OTA descriptions, proposing a generic framework and evaluation metrics for any side-channel signal. Our analysis reveals OTA features not previously considered, increasing its application scenarios and requiring a fresh countermeasure analysis to prevent it. In this regard, we demonstrate that OTAs can work in the backward direction, allowing to mount an augmented projective coordinates attack with respect to the proposal by Naccache, Smart and Stern (Eurocrypt 2004). This demonstrates that randomizing the initial targeted algorithm state does not prevent the attack as believed in previous works. We analyze three libraries libgcrypt, mbedTLS, and wolfSSL using two microarchitecture side channels. For the libgcrypt case, we target its EdDSA implementation using Curve25519 twist curve. We obtain similar results for mbedTLS and wolfSSL with curve secp256r1. For each library, we execute extensive attack instances that are able to recover the complete scalar in all cases using a single trace. This work demonstrates that microarchitecture online template attacks are also very powerful in this scenario, recovering secret information without knowing a leakage model. This highlights the importance of developing secure-by-default implementations, instead of fix-on-demand ones. publishedVersion
- Published
- 2021
34. Mathematical model of S-shaped gear surface
- Author
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Po-Yi Tsai and Hsueh-Cheng Yang
- Subjects
Surface (mathematics) ,Homogeneous coordinates ,business.product_category ,Mechanical Engineering ,Mathematical analysis ,Contact analysis ,Kinematics ,Rack ,Transformation matrix ,Differential geometry ,Mechanics of Materials ,business ,Pinion ,Mathematics - Abstract
In this study, an imaginary S-shaped surface rack cutter was used to create a gear pair with S-shaped surface. First, a mathematical model of the imaginary S-shaped surface rack cutter was constructed by using the geometry. Then, a family of the imaginary rack cutter surfaces was obtained through the homogeneous coordinate transformation matrix. The equation of meshing was calculated by using differential geometry. A mathematical model for the S-shaped surface gear pair was determined by substituting the equation of meshing into the family of imaginary rack-cutter surfaces. The kinematic errors of the gear pair were calculated by setting the assembly error and using tooth contact analysis. Contact between the gear and the pinion was simulated by using the interference function of a computer-aided design software package. To investigate the real contact between the pinion and the gear, a rapid prototype machine was used to manufacture a real gear pair. Three of the teeth on the real gear were coated with dyes. After running the gear for long time, the dyes on the teeth were scraped and rubbed, and they were evenly rolled over the teeth.
- Published
- 2021
35. Resistance of the Montgomery Ladder Against Simple SCA: Theory and Practice
- Author
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Ievgen Kabin, Peter Langendoerfer, Zoya Dyka, Dan Klann, and Marcin Aftowicz
- Subjects
Homogeneous coordinates ,Computer science ,020208 electrical & electronic engineering ,Scalar (physics) ,02 engineering and technology ,020202 computer hardware & architecture ,Power (physics) ,Elliptic curve ,Application-specific integrated circuit ,VHDL ,0202 electrical engineering, electronic engineering, information engineering ,NIST ,Electrical and Electronic Engineering ,Field-programmable gate array ,computer ,Algorithm ,computer.programming_language - Abstract
The Montgomery kP algorithm i.e. the Montgomery ladder is reported in literature as resistant against simple SCA due to the fact that the processing of each key bit value of the scalar k is done using the same sequence of operations. We implemented the Montgomery kP algorithm using Lopez-Dahab projective coordinates for the NIST elliptic curve B-233. We instantiated the same VHDL code for a wide range of clock frequencies for the same target FPGA and using the same compiler options. We measured electromagnetic traces of the kP executions using the same input data, i.e. scalar k and elliptic curve point P, and measurement setup. Additionally, we synthesized the same VHDL code for two IHP CMOS technologies, for a broad spectrum of frequencies. We simulated the power consumption of each synthesized design during an execution of the kP operation, always using the same scalar k and elliptic curve point P as inputs. Our experiments clearly show that the success of simple electromagnetic analysis attacks against FPGA implementations as well as the one of simple power analysis attacks against synthesized ASIC designs depends on the target frequency for which the design was implemented and at which it is executed significantly. In our experiments the scalar k was successfully revealed via simple visual inspection of the electromagnetic traces of the FPGA for frequencies from 40 to 100 MHz when standard compile options were used as well as from 50 MHz up to 240 MHz when performance optimizing compile options were used. We obtained similar results attacking the power traces simulated for the ASIC. Despite the significant differences of the here investigated technologies the designs’ resistance against the attacks performed is similar: only a few points in the traces represent strong leakage sources allowing to reveal the key at very low and very high frequencies. For the “middle” frequencies the number of points which allow to successfully reveal the key increases when increasing the frequency.
- Published
- 2021
36. Problem of Differentiating Homogeneous Coordinate Adverbial Modifiers (by the Material of the French and Russian Languages)
- Author
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Magomed Gazilovich Gazilov
- Subjects
Homogeneous coordinates ,Computer science ,Linguistics ,Adverbial - Published
- 2021
37. Use of Block Toeplitz Matrix in the Study of Möbius Pairs of Simplexes in Higher-Dimensional Projective Space
- Author
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Panda, Golak Bihari and Misra, Saroj Kanta
- Published
- 2020
- Full Text
- View/download PDF
38. Conformal Geometry, Euclidean Space and Geometric Algebra
- Author
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Doran, Chris, Lasenby, Anthony, Lasenby, Joan, Winkler, Joab, editor, and Niranjan, Mahesan, editor
- Published
- 2002
- Full Text
- View/download PDF
39. Evaluation of the efficiency of differential addition of points of curves in the generalized Edwards form
- Author
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N.V. Kuchynska, L.V. Kovalchuk, and A.V. Bessalov
- Subjects
Homogeneous coordinates ,Finite field ,Twisted Edwards curve ,Edwards curve ,Scalar (mathematics) ,Montgomery curve ,Applied mathematics ,Field (mathematics) ,General Medicine ,Square (algebra) ,Mathematics - Abstract
A survey of the main properties of three classes of curves in the generalized Edwards form is given: complete, quadratic and twisted Edwards curves. The analysis of the Montgomery algorithm for differential addition of points for the Montgomery curve is carried out. An estimation of the record low cost of computing the scalar product kP of a point P is given, which is equal to 5M+4S+1U on one step of the iterative cycle (M is the cost of finite field multiplication, S is the cost of squaring, U is the cost of field multiplication by a known constant). A detailed derivation of the formulas for addition-subtraction and doubling points for the curve in the generalized Edwards form in projective coordinates of Farashahi-Hosseini is carried out. Moving from three-dimensional projective coordinates (X: Y: Z) to two-dimensional coordinates (W: Z) allows achieving the same minimum computational cost for the Edwards curves as for the Montgomery curve. Aspects of the choice of an Edwards-form curve acceptable for cryptography and its parameters optimization in the problem of differential addition of points are discussed. Twisted Edwards curves with the order of NE=4n (n is prime) at p≡5mod8 are recommended, minimizing the parameters a and d allows achieving the minimum cost estimation 5M+4S for one step of computing the point product. It is shown that the transition from the Weierstrass curves (the form used in modern cryptographic standards) to the Edwards curves makes it possible to obtain a potential gain in the speed of computing the scalar product of the point by a factor of 3.09.
- Published
- 2020
40. Operator-based homogeneous coordinates: application in camera document scanning.
- Author
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Juarez-Salazar, Rigoberto and Diaz-Ramirez, Victor H.
- Subjects
- *
PROJECTIVE geometry , *HOMOGRAPHY (Computer vision) , *CAMERA calibration - Abstract
An operator-based approach for the study of homogeneous coordinates and projective geometry is proposed. First, some basic geometrical concepts and properties of the operators are investigated in the one-and two-dimensional cases. Then, the pinhole camera model is derived, and a simple method for homography estimation and camera calibration is explained. The usefulness of the analyzed theoretical framework is exemplified by addressing the perspective correction problem for a camera document scanning application. Several experimental results are provided for illustrative purposes. The proposed approach is expected to provide practical insights for inexperienced students on camera calibration, computer vision, and optical metrology among others. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
41. 3D Oriented Projective Geometry Through Versors of $${\mathbb{R}^{3,3}}$$.
- Author
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Dorst, Leo
- Abstract
It is possible to set up a correspondence between 3D space and $${\mathbb{R}^{3,3}}$$ , interpretable as the space of oriented lines (and screws), such that special projective collineations of the 3D space become represented as rotors in the geometric algebra of $${\mathbb{R}^{3,3}}$$ . We show explicitly how various primitive projective transformations (translations, rotations, scalings, perspectivities, Lorentz transformations) are represented, in geometrically meaningful parameterizations of the rotors by their bivectors. Odd versors of this representation represent projective correlations, so (oriented) reflections can only be represented in a non-versor manner. Specifically, we show how a new and useful 'oriented reflection' can be defined directly on lines. We compare the resulting framework to the unoriented $${\mathbb{R}^{3,3}}$$ approach of Klawitter (Adv Appl Clifford Algebra, 24:713-736, 2014), and the $${\mathbb{R}^{4,4}}$$ rotor-based approach by Goldman et al. (Adv Appl Clifford Algebra, 25(1):113-149, 2015) in terms of expressiveness and efficiency. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
42. Use of Block Toeplitz Matrix in the Study of Möbius Pairs of Simplexes in Higher-Dimensional Projective Space
- Author
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Saroj Kanta Misra and Golak Bihari Panda
- Subjects
Combinatorics ,Matrix (mathematics) ,Simplex ,Homogeneous coordinates ,Chain (algebraic topology) ,General Mathematics ,Block (permutation group theory) ,Mathematics::Metric Geometry ,Projective space ,Toeplitz matrix ,Vertex (geometry) ,Mathematics - Abstract
A simplex in a projective space of dimension n is expressed by a matrix of order n + 1, where each row represents the homogeneous coordinates of a vertex of the simplex with respect to a reference frame. In the present study, a block Toeplitz matrix is used to express a simplex which forms a Mobius pair along with the reference simplex. A pair of mutually inscribed, circumscribed tetrahedrons is called a Mobius pair. The existence of such pairs of simplexes in higher-dimensional (odd) projective spaces is already established. In the present study an existence of an infinite chain of simplexes in a five-dimensional projective space is established where any two simplexes from the chain form a Mobius pair in some order of their vertices. This is studied with the help of powers of a block Toeplitz matrix. Also, attempt has been made to generalize this result to 2n + 1-dimensional projective space.
- Published
- 2020
43. A SINGLE HIGH-PRECISION HOMOGENEOUS COORDINATE SPACE OF TERRITORIES AND LOCAL COORDINATE SYSTEMS: WAYS OF HARMONIZATION
- Author
-
Vladimir I. Obidenko
- Subjects
Homogeneous coordinates ,Computer science ,Coordinate system ,Harmonization ,Topology ,Space (mathematics) - Published
- 2020
44. Projective Geometry, Duality and Plucker Coordinates for Geometric Computations with Determinants on GPUs
- Author
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Vaclav Skala
- Subjects
FOS: Computer and information sciences ,Homogeneous coordinates ,Floating point ,Plücker coordinates ,Topology ,Graphics (cs.GR) ,Significand ,Computer Science - Graphics ,Grassmannian ,Duality (projective geometry) ,Algorithm ,Mathematics ,Projective representation ,Projective geometry ,68xx, 68U05 - Abstract
Many algorithms used are based on geometrical computation. There are several criteria in selecting appropriate algorithm from already known. Recently, the fastest algorithms have been preferred. Nowadays, algorithms with a high stability are preferred. Also technology and computer architecture, like GPU etc., plays a significant role for large data processing. However, some algorithms are ill-conditioned due to numerical representation used; result of the floating point representation. In this paper, relations between projective representation, duality and Plucker coordinates will be explored with demonstration on simple geometric examples. The presented approach is convenient especially for application on GPUs or vector-vector computational architectures, Comment: arXiv admin note: substantial text overlap with arXiv:1708.06684
- Published
- 2022
- Full Text
- View/download PDF
45. Liouville geometry of classical thermodynamics
- Author
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Arjan van der Schaft and Systems, Control and Applied Analysis
- Subjects
Mathematics - Differential Geometry ,General Physics and Astronomy ,Thermodynamics ,FOS: Physical sciences ,Geometry ,Homogeneous Hamiltonian vector field ,Symplectic and contact geometry ,Homogeneity (physics) ,FOS: Mathematics ,Trigonometric functions ,Geometric framework ,Representation (mathematics) ,Mathematics - Optimization and Control ,Mathematics::Symplectic Geometry ,Mathematical Physics ,Mathematics ,Homogeneous coordinates ,Mathematical Physics (math-ph) ,Manifold ,Gibbs-Duhem relation ,Differential Geometry (math.DG) ,Homogeneous ,Optimization and Control (math.OC) ,Cotangent bundle ,Geometric thermodynamics ,Geometry and Topology - Abstract
In the contact-geometric formulation of classical thermodynamics distinction is made between the energy and entropy representation, which can be resolved by taking homogeneous coordinates for the intensive variables. This results in a geometric formulation on the cotangent bundle of the manifold of extensive variables, where all geometric objects are homogeneous in the cotangent variables. The resulting geometry is studied in-depth. Additional homogeneity with respect to the extensive variables, corresponding to the classical Gibbs-Duhem relation, is treated within the same geometric framework., Comment: 27 pages
- Published
- 2021
46. A New Approach to Line - Sphere and Line - Quadrics Intersection Detection and Computation.
- Author
-
Skala, Vaclav
- Subjects
- *
QUADRICS , *COMPUTATIONAL mathematics , *POLYHEDRA , *ALGORITHMS , *GEOMETRIC analysis - Abstract
Line intersection with convex and un-convex polygons or polyhedron algorithms are well known as line clipping algorithms and very often used in computer graphics. Rendering of geometrical problems often leads to ray tracing techniques, when an intersection of many lines with spheres or quadrics is a critical issue due to ray-tracing algorithm complexity. A new formulation of detection and computation of the intersection of line (ray) with a quadric surface is presented, which separates geometric properties of the line and quadrics that enables pre-computation. The presented approach is especially convenient for implementation with SSE instructions or on GPU. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
47. Machine Learning Attacks and Countermeasures on Hardware Binary Edwards Curve Scalar Multipliers
- Author
-
Odysseas Koufopavlou, Charis Dimopoulos, and Apostolos P. Fournaris
- Subjects
Hardware security module ,Technology ,Control and Optimization ,Computer Networks and Communications ,Computer science ,Edwards curve ,02 engineering and technology ,Design strategy ,Scalar multiplication ,Machine learning ,computer.software_genre ,side channel attacks ,0202 electrical engineering, electronic engineering, information engineering ,Side channel attack ,Elliptic curve cryptography ,Instrumentation ,Homogeneous coordinates ,Elliptic Curve Cryptography ,business.industry ,Deep learning ,embedded devices ,020202 computer hardware & architecture ,machine learning ,hardware security ,020201 artificial intelligence & image processing ,Artificial intelligence ,business ,computer - Abstract
Machine Learning techniques have proven effective in Side Channel Analysis (SCA), enabling multiple improvements over the already-established profiling process of Template Attacks. Focusing on the need to mitigate their impact on embedded devices, a design model and strategy is proposed that can effectively be used as a backbone for introducing SCA countermeasures on Elliptic Curve Cryptography (ECC) scalar multipliers. The proposed design strategy is based on the decomposition of the round calculations of the Montgomery Power Ladder (MPL) algorithm and the Scalar Multiplication (SM) algorithm into the underlined finite field operations, and their restructuring into parallel-processed operation sets. Having as a basis the proposed design strategy, we showcase how advanced SCA countermeasures can be easily introduced, focusing on randomizing the projective coordinates of the MPL round’s ECC point results. To evaluate the design approach and its SCA countermeasures, several simple ML-based SCAs are performed, and an attack roadmap is provided. The proposed roadmap assumes attackers that do not have access to a huge number of leakage traces, and that have limited resources with which to mount Deep Learning attacks. The trained models’ performance reveals a high level of resistance against ML-based SCAs when including SCA countermeasures in the proposed design strategy.
- Published
- 2021
- Full Text
- View/download PDF
48. A Graphic Method for Detecting Multiple Roots Based on Self-Maps of the Hopf Fibration and Nullity Tolerances
- Author
-
María Teresa Rivas-Rodríguéz, José Ignacio Extreminana-Aldana, Luis Javier Hernández-Paricio, and José Manuel Gutiérrez-Jiménez
- Subjects
Polynomial ,Homogeneous coordinates ,Iterative method ,General Mathematics ,multiple roots ,Newton’s method ,Zero (complex analysis) ,rational functions ,Hopf fibration ,Rational function ,Algebra ,symbols.namesake ,Simple (abstract algebra) ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,QA1-939 ,Computer Science (miscellaneous) ,symbols ,Riemann sphere ,homogeneous coordinates ,Engineering (miscellaneous) ,Newton's method ,Mathematics - Abstract
The aim of this paper is to study, from a topological and geometrical point of view, the iteration map obtained by the application of iterative methods (Newton or relaxed Newton’s method) to a polynomial equation. In fact, we present a collection of algorithms that avoid the problem of overflows caused by denominators close to zero and the problem of indetermination which appears when simultaneously the numerator and denominator are equal to zero. This is solved by working with homogeneous coordinates and the iteration of self-maps of the Hopf fibration. As an application, our algorithms can be used to check the existence of multiple roots for polynomial equations as well as to give a graphical representation of the union of the basins of attraction of simple roots and the union of the basins of multiple roots. Finally, we would like to highlight that all the algorithms developed in this work have been implemented in Julia, a programming language with increasing use in the mathematical community.
- Published
- 2021
- Full Text
- View/download PDF
49. Batch advection for the piecewise linear vector field on simplicial grids.
- Author
-
Wang, Wentao, Wang, Wenke, and Li, Sikun
- Subjects
- *
VECTOR fields , *DATA visualization , *NUMERICAL grid generation (Numerical analysis) , *ALGORITHMS , *RUNGE-Kutta formulas , *TIME-varying systems - Abstract
In practice, a sampled vector field is viewed as a piecewise linear field, which simplifies the interpolation of curve integration, and this assumption is widely adopted in most visualization algorithms. However, existing works omit the local coherent nature of this simplification, which could be leveraged to improve the efficiency of computation. In this paper, we reformulate the scheme of linear interpolation and integration for the simplicial mesh grid with homogeneous coordinates, and find that we can benefit significantly by simultaneously integrating multiple curves passing through a single cell. Based on this observation, we revise the 4th-order Runge–Kutta method and propose an efficient cell-wise and step-wise batch advection scheme for both 2D and 3D datasets, which is useful in the multiple particle tracing for the integral curves, e.g. streamline, pathline, timeline and streakline. We apply our method to some applications for a series of datasets, and the results show that our batch advection method can significantly speed up the curve integration while comparing with the traditional 4th-order Runge–Kutta. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
50. Synchronous motion error identification method of dual-five-axis CNC machine tool based on R-test
- Author
-
Gang Liu, Dawen Ruan, LiMa, and Jian Mao
- Subjects
Homogeneous coordinates ,business.product_category ,Series (mathematics) ,Computer science ,Mechanical Engineering ,Kinematics ,Industrial and Manufacturing Engineering ,Computer Science Applications ,Machine tool ,Transformation (function) ,Control and Systems Engineering ,Numerical control ,Trajectory ,Measuring instrument ,business ,Algorithm ,Software ,ComputingMethodologies_COMPUTERGRAPHICS - Abstract
In order to quickly and conveniently identify the synchronous motion errors of dual-five-axis CNC machine tool, this paper proposes an effective error measurement and identification method, which consists of an error identification model and a new error measurement trajectory for the dual-five-axis CNC machine tool based on R-test measuring instrument. According to the theory of multi-body system and the method of homogeneous coordinate transformation, this paper developed the kinematic model of machine tool to identify the synchronous motion error of machine tool and a new error measuring track of spherical “8” shape. Thus, three kinds of synchronous motion errors of A1 / A2 axis, A1 / B2 axis, and A2 / B2 / C1 axis can be identified respectively. In addition, a series of milling experiment have been performed on the dual-five-axis CNC machine tool to verify the feasibility and effectiveness of the propsoed model. The experimental results show that the proposed model and method are more efficient.
- Published
- 2021
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