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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]

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]

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]

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]

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.

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]

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.)

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.

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.

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.

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]

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]

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]

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]

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]

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.

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]

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]

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]

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]

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]

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]

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]