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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.

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.

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.

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.

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.

Machined surface topography prediction is an important and useful tool for optimizing cutting parameters. However, accurate prediction of machined surface topography in ball-end milling operation has been extremely challenging, due to the complexity in tool-workpiece interaction induced by the trochoidal motion of cutting edge and computing burden. In this present research, a modified iterative approach was proposed to solve the intersections between the cutting-edge sweeping surface and the discrete Z-vector model of workpiece, which were used to predict the machined surface topography in ball-end milling operation. Firstly, the accurate model of cutting-edge sweeping surface was established utilizing homogeneous coordinate transformation, in which the tool runout was considered. Secondly, the cutting-edge sweeping surface was dispersed into a series of patches in accordance with equal parameter interval, and the in-cut patch was extracted by using the minimum and maximum axial immersion angle of the cutting edge. Thirdly, the intersection between each in-cut patch and discrete Z-vector was solved using the Newton’s method, which was used to update the endpoint of the corresponding discrete Z-vector. Finally, ball-end milling experiments of AISI P20 steel were carried out to validate the proposed approach as well as investigate the effect of cutting parameters on the machined surface topography and roughness. The predicted machined surface topography and roughness were in good agreement with the measured results. Moreover, the proposed approach needs less computing time than the traditional iterative approaches at the same predicting accuracy. This research also provides guidance for optimizing cutting parameters to control surface quality in ball-end milling operation.

We present an algorithm in homogeneous coordinates for factoring a homography $$\mathbf{h}$$ on $$\mathbb {R}P^2$$ as $$\mathbf{h}=\mathbf{p}\circ \mathbf{s}$$ , where $$\mathbf{p}$$ is a perspective collineation and $$\mathbf{s}$$ is a similarity. We use the factorization to derive a function that measures the local projective distortion of a non-affine homography $$\mathbf{h}$$ at ordinary points of $$\mathbb {R}P^2$$ , and discover interesting geometric structures associated with $$\mathbf{h}$$ , somewhat like the centers and axes of perspective collineations. These results reveal infinite families of circle pairs, and neighborhoods of certain points associated with $$\mathbf{h}$$ , that give the appearance of suffering no projective distortion, even though $$\mathbf{h}$$ is not a similarity. In fact, when the factor $$\mathbf{p}$$ is a perspective collineation, every non-empty subset $$\mathcal{S}$$ of $$\mathbb {R}P^2$$ has an “ $$\mathbf{h}$$ -conjugate” set $$\mathcal{S}^*$$ such that $$\mathbf{h}(\mathcal{S}\cup \mathcal{S}^*)=\mathbf{s}(\mathcal{S}\cup \mathcal{S}^*)$$ , even though $$\mathbf{h}$$ and $$\mathbf{s}$$ do not agree on $$\mathcal{S}\cup \mathcal{S}^*$$ . We include examples from photography as well as connections to Apollonian circles, Mobius transformations and stereographic projections.

Strict requirements for concentricity of the multistage high pressure rotor of an aero-engine are employed to guarantee performances such as vibration. Tedious and time-wasting trial assembly by adjusting the installation angles of stages is needed to meet the requirements due to the lack of effective analysis methods. Furthermore, there is no quick way to find out where the problem is and how to repair the parts when the installation-angle-adjusting method fails. This article focuses on a solution to optimize the installation angle of each stage and to make repair decisions in the assembly process. The run-out data are processed by least square method to get the spatial positions and attitudes of flanges and a deviation propagation analysis model is built by virtue of homogeneous coordinate transformation theory to predict the accumulative errors of each stage. The eccentricities of stages are evaluated with reference to the common axis and the installation angles of stages are optimized by minimizing the sum of eccentricities. Sensitivities of eccentricity, eccentric angle and parallelism of each stage are analyzed and repair decisions for parts are made to meet more strict requirements. An example of a three-stage subassembly is presented to demonstrate the solution.

By mapping the homogeneous coordinates of two points in space to the Plucker coordinates of theline they determine, any transformation of type $SL(4)$ upon points in space is mapped to a transformationof type $SO_0(3,3)$, the latter being the connected componentcontaining the identity of the special orthogonal transformation group of the linear spacespanned by Plucker coordinates. This is the classical Plucker correspondence, two-to-one and onto. It has importantapplications in line geometry and projective transformations. While the explicit form of Plucker correspondence is trivialto present, its inverse in explicit form, which is also important in application, is not found in the literature.In this paper, we present a simple and unified formula for the inverse of the Plucker correspondence.

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. 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 techniques are applicable to many types of matrix eigenvalue problems; standard deflation is possible only for linear one-parameter problems. The methods are easy to understand and implement. Although divided differences are 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., Comment: 17 pages; revised version

This work presents a new solution for coordinated search with a team of heterogeneous robots executing a time-critical mission. It is challenging to specify and represent search locations (targets) in known but dynamic environments as well as to find robotic paths to visit the locations. We propose a technique to construct an information map that includes locations of uncertain targets, and generate optimal paths. We especially focus on combining a satellite map that has global coordinates with local images gathered from an aerial robot. Specific targets are represented on a homogeneous coordinate system, so that different types of robots, capable to gather necessary information, may cooperatively conduct a mission. Once a homogeneous map is constructed, a centralized pathfinding algorithm can be applied. Our path-finding algorithm is to choose a set of paths, suggesting a proper number of robots along with their initial locations. In our work, robots can independently travel search locations, which may have dynamics or changes, but collaboratively cover all target locations. Through the experiments with real robotic platforms, we validate the generation of a map including targets and a choice of paths, and compare with existing algorithms.

The automation of tolerance analysis is the ambitious goal of the researcher. In order to achieve this aim, the homogeneous coordinate transformation is a usual method for the position calculation of geometric features and parts in the assembly tolerance analysis method. The hierarchy definition and the automatic creation of the coordinate system are the key work for the automation of tolerance analysis. In this paper, the position calculation equations of the geometric feature in assembly model are described firstly, then the error propagation relation graph for geometric features in the part model is analyzed based on the datum–target mechanism of geometric tolerance. The assembly errors propagation graph of the part in the machine model is described based on the assembly relation and assembly precedence. The controlling points variation model (CPVM) of a geometric feature is used to generate and represent the position of the substitute geometry of the geometric feature, and the position parameters of substitute geometry are generated using Monte Carlo simulations. Based on the CPVM model and datum–target mechanism, the hierarchy of the coordinate system for a geometric feature is defined on the part level. Based on the assembly relations and the real part model, the assembly coordinate system between the locating part and the assembling part is established. According to these two errors propagation graphs, the homogeneous coordinate transformation matrix between the related coordinate systems is generated automatically. Finally, a case study is provided to illustrate the proposed method.

We present in this paper a framework which leverages the underlying topology of a data set, in order to produce appropriate coordinate representations. In particular, we show how to construct maps to real and complex projective spaces, given appropriate persistent cohomology classes. An initial map is obtained in two steps: First, the persistent cohomology of a sparse filtration is used to compute systems of transition functions for (real and complex) line bundles over neighborhoods of the data. Next, the transition functions are used to produce explicit classifying maps for the induced bundles. A framework for dimensionality reduction in projective space (Principal Projective Components) is also developed, aimed at decreasing the target dimension of the original map. Several examples are provided as well as theorems addressing choices in the construction., Final version to appear in Discrete & Computational Geometry

Ship geometry reconstruction technology is a key technique and research focus in the ship design. And the optimization algorithm used in the ship design is always a hot research area. To establish the control points’ distribution model database of characteristic curves, based on the NURBS control mesh of the hull surface, an approach to approximate the initial stem and stern contours of hull form modeling using the concept of parametric generation is proposed. The optimization model is constructed with the homogeneous coordinate component of the fitted contour’s control points as the design variables. The objective function is set to minimize the maximum relative difference among the longitudinal coordinates of the approximated contour and the original ones corresponding to the same drafts. The appropriate constraints are set according to the characteristics of the contours and the geometrical characteristic of NURBS. Considering a number of empirical knowledge existing in the process of ship design, and the conventional quantum-behaved particle swarm optimization (QPSO) algorithm depending upon good initial population, the immune quantum-behaved particle swarm optimization (IQPSO) is specifically designed and used to solve this complex nonlinear constrained optimization problem through the immune operator of the immune genetic algorithm combining with the QPSO algorithm. Meanwhile, the self-adaptive constraint evolution strategy is proposed to improve the IQPSO algorithm. The simulation results of the full-scale ship’s contours demonstrate the feasibility and effectiveness of the proposed algorithm and method.

This paper presents an imaginary planar rack cutter having two parameters of motion to create a spherical gear pair with two degrees of freedom. The working region of the imaginary planar rack cutter is that of symmetrical straight lines. The relationship between the coordinate system of the imaginary planar rack cutter and that of the spherical gear pair having two parameters of motion is illustrated. The two-parameter family of imaginary planar rack cutter surfaces is obtained by using the homogeneous coordinate transformation matrix to transfer the coordinate system of the planar rack cutter to that of the spherical pinion or the spherical gear. Based on the two-parameter enveloping theory, the mathematical models of the spherical gear pair with two degrees of freedom were proposed. The developed mathematical model of the gear pair with two degrees of freedom can be used to analyze undercutting analysis of the imaginary planar rack cutter. Based on the undercutting equation, a minimum number of teeth of the spherical pinion are determined. A line of singular points on the spherical pinion’s teeth are displayed. Results show that a minimum number of teeth of the spherical pinion can be determined in order to avoid undercutting.

We study the existence of invariants for the family of systems in an open domain \(\mathcal {D}\) of \(\mathbb {R}^n\) or \(\mathbb {C}^n\) whose components are linear fractionals sharing denominator. Such systems can be written with the aid of homogeneous coordinates as the composition of a linear map in \(\mathbb {K}^{n+1}\) with a certain projection and their behaviour is strongly determined by the spectral properties of the corresponding linear map.The paper is committed to prove that if \(n\ge 2\) then every system of this kind admits an invariant, both in the real and in the complex case. In fact, for a sufficiently large n several functionally independent invariants can be obtained and, in many cases, the invariant can be chosen as the quotient of two quadratic polynomials.

We explore a specific discretization of isothermic nets in the plane which can also be interpreted as a discrete holomorphic map. The discrete orthogonality of the quadrilateral net is achieved by the so called conical condition imposed on vertex stars. That is, the sums of opposite angles between edges around all vertices are equal. This conical condition makes it possible to define a family of underlying circle patterns for which we can show invariance under Mobius transformations. Furthermore, we use the underlying circle pattern to characterize discrete isothermic nets in the projective model of Mobius geometry, and as Moutard nets in homogeneous coordinates in relation to the light cone in this model. We further investigate some examples of discrete isothermic nets and apply the Christoffel dual construction to obtain discrete minimal surfaces of conical type.

When the numerical Newton-Raphson method is applied to find the intersections of two algebraic curves (that is, the roots of a pair of bivariate polynomials), some difficulties appear when the value of a denominator of the corresponding bivariate rational functions is zero. In this paper we give a solution to these problems by using adequate homogeneous coordinates and extending the domain of the iteration function. The iteration of a map given by a pair of bivariate rational maps is analyzed by taking a canonical extension, which is defined on the product of two copies of an (real or complex) augmented projective line. This method gives a global description of the basins of attraction of fixed points of an iteration. In particular, the attraction basins of fixed points associated to the intersection points of the two algebraic curves is obtained. As a consequence of our techniques, we are able to plot the attraction basins of a real root either in a torus (considered as a compactification of the real plane) or in an open square, which is homeomorphic to the global real plane containing the two algebraic curves.

A method is proposed for the construction of conservative flow difference schemes that compute compressible viscous gas flows in curvilinear coordinates based on an arbitrary Cartesian Godunov-type scheme. The method is designed for easy use in the sense that the implementation of the scheme in curvilinear coordinates requires only minimal additions to the program code of the basic Cartesian scheme. The scheme construction procedure is illustrated for the case of cylindrical coordinates. A cylindrical scheme is constructed to second order approximation in space. Results of test calculations are reported.

We show how to construct semi-invariants and integrals of the full symmetric \({\mathfrak{sl}_n}\) Toda lattice for all n. Using the Toda equations for the Lax eigenvector matrix we prove the existence of semi-invariants which are homogeneous coordinates in the corresponding projective spaces. Then we use these semi-invariants to construct the integrals. The existence of additional integrals which constitute a full set of independent non-involutive integrals was known but the chopping and Kostant procedures have crucial computational complexities already for low-rank Lax matrices and are practically not applicable for higher ranks. Our new approach solves this problem and results in simple explicit formulae for the full set of independent semi-invariants and integrals expressed in terms of the Lax matrix and its eigenvectors, and of eigenvalue matrices for the full symmetric \({\mathfrak{sl}_n}\) Toda lattice.

The paper is devoted to the solution of straight and inverse geometrical tasks of five link mechanism with two degrees of freedom. The solution of the mentioned problem is very important in order to determine kinematic parameters of actuators. The problem can be divided into two parts. The first part is considered when we are given the coordinates of the output link of the mechanism and the necessity arises of determining the angles of rotation of actuators. On the other hand, it is very important to determine the position of the output link when the angles of rotation of the actuators are known. Here we consider that the mechanism is composed only of five classes of rotating kinematic pairs and the actuators are situated at the junctions of frames and links of the examined mechanism. The solution of the said problem is based on utilization of homogenous coordinates. On the basis of the obtained equations of motion, one can calculate the trajectories of motion of the output link as well angles of rotation of the actuators by taking into consideration preliminary given kinematic parameters of the mechanism. Here we also obtain equations for calculating values of speed and acceleration of the links of the mechanism. The calculations differ from known methods in simplicity and high performance, which would be useful for programming actuators mounted in the joints of the linkage.

Recently E. Feigin introduced the $\mathbb G_a^N$-degenerations of semisimple algebraic groups and their associated degenerate flag varieties. It has been shown by Feigin, Finkelberg, and Littelmann that the degenerate flag varieties in types $A_n$ and $C_n$ are Frobenius split. In this paper we construct an associated degeneration of homogeneous coordinate rings of classical flag varieties in all types and show that these rings are Frobenius split in most types. It follows that the degenerate flag varieties of types $A_n$, $C_n$ and $G_2$ are Frobenius split. In particular we obtain an alternate proof of splitting in types $A_n$ and $C_n$; the case $G_2$ was not previously known. We also give a representation-theoretic condition on PBW-graded versions of Weyl modules which is equivalent to the existence of a Frobenius splitting of the classical flag variety that maximally compatibly splits the identity., Comment: Bug fixes; final version. To appear in Selecta Mathematica

Conventionally, image object recognition and pose estimation are two independent components in machine vision. This paper presented a simple but effective method KNN-SNG, which tightly couples these two components within a single algorithm framework. The basic idea of this method came from the bionic pattern recognition and the manifold ways of perception. Firstly, the shortest neighborhood graphs (SNG) are established for each registered object. SNG can be regarded as a covering and triangulation for a hypersurface on which the training data are distributed. Then for recognition task, the determined test image lies on which SNG by employing the parameter “k”, which could be calculated adaptively. Finally, the local linear approximation method was adopted to build a local map between high-dimensional image space and low-dimensional manifold for pose estimation. The projective coordinates on manifold can depict the pose of object. Experiment results manifested the effectiveness of the method.

NURBS were introduced into CAD/CAM systems predominantly for the representation of conventional objects, such as conics and quadrics. Among these, the circle played a critical role in representing a myriad of parts used in the every-day practice. Being the most universally used object, the circle has enjoyed the most popularity in science and engineering. It is an essential entity in both design as well as manufacturing and hence, its representation within CAD/CAM systems requires careful attention. Although the circle enjoys both smoothness as well as a uniform parametrization, its de facto mathematical form, the NURBS form, does not provide either sufficient smoothness or uniform parametrization. On top of all this, NURBS are rational forms requiring homogeneous coordinates in the four-dimensional space, whereas the engineering entity is only Euclidean in 3D. This multiple representational glitch, 3D presence and 4D storage, has given rise to enough headache to warrant reconsideration of the validity of the rational form in engineering design. This paper argues that it is time to drop the "R" from NURBS and to switch to integral splines with approximations where necessary. We also argue that it is time to bury dumb algorithms and consider human-based computing, i.e. our algorithms should be biologically inspired and not based on pure number crunching.

In this paper, we present the generalized Huff curves that contain Huff’s model as a special case. First, it is proved that every elliptic curve with three points of order 2 is isomorphic to a generalized Huff curve. Then, the fast and explicit formulae are derived for generalized Huff curves in projective coordinates. This paper also enumerates the number of isomorphism classes of generalized Huff curves over finite fields. Finally, the explicit formulae are presented for the doubling step and addition step in Miller’s algorithm to compute the Tate pairing on generalized Huff elliptic curves.

A bifunctional construction depending on a specified density–potential pair and an approximate guiding electron density functional is presented. The proposed bifunctional construction properly transforms under homogeneous coordinate scaling and yields the specified functional derivative, which determines the electron density. Whereas the method is general and applicable to all functional types, it will prove especially helpful in the context of orbital-free density functional theory, where most existing approximate density functionals predict inaccurate potentials.

The paper presents a methodology for modeling profile of two circular-arc cams, which is based on the application of homogeneous coordinate transformation method. Analytical expressions that allow complete determining of all design parameters and coordinates of all the points of two circular-arc cams profile are defined. Also, analytical expressions for determining the kinematic performance such as displacement, velocity, acceleration and jerk are derived. The defined methodology is applied on concrete examples for which the analysis of the kinematic performance is also performed. Additionally, it is shown that a proper choice of the design parameters can significantly affect the improvement of the kinematic performance, especially the reduction of jerk as one of the most unfavorable occurrences in these mechanisms. The given methodology allows very efficient determination of all input data for parametric modeling and production of two circular-arc cams on numerical machines. This approach allows the fulfillment of one of the most important requirements of modern production–reduction of time necessary for design and manufacturing which causes reduction of costs.

In this paper, we study the non-degenerate holomorphic S 2 in the complex Grassmann manifold G(k, n), 2k ≤ n, by the method of moving frame. For a non-degenerate holomorphic one, there exists globally defined positive functions λ 1, ..., λ k on S 2. We first show that the holomorphic S 2 in G(k, 2k) is totally geodesic if these λ i are all equal. Conversely, for any totally geodesic immersion f from S 2 into G(k, n), we prove that f(S 2) ⊂ G(k, 2k) up to U(n)-transformation, $\lambda_i=\frac{1}{\sqrt{k}}$ , the Gaussian curvature $K=\frac{4}{k}$ and f is given by (z 0, z 1)↦(z 0 I k , z 1 I k , 0), in terms of homogeneous coordinate.

Let R be a chain ring with four elements. In this paper, we present two new constructions of R-linear codes that contain a subcode associated with a simplex code over the ring R. The simplex codes are defined as the codes generated by a matrix having as columns the homogeneous coordinates of all points in some projective Hjelmslev geometry PHG(R k ). The first construction generalizes a recent result by Kiermaier and Zwanzger to codes of arbitrary dimension. We provide a geometric interpretation of their construction which is then extended to projective Hjelmslev spaces of arbitrary dimension. The second construction exploits the possibility of adding two non-free rows to the generator matrix of a linear code over R associated with a given point set. Though the construction works over both chain rings with four elements, the better codes are obtained for $${R=\mathbb{Z}_4}$$ .

$\SOL$ geometry is one of the eight homogeneous Thurston 3-geomet-ri-es $$\EUC, \SPH, \HYP, \SXR, \HXR, \SLR, \NIL, \SOL.$$ In \cite{Sz10} the {\it densest lattice-like translation ball packings} to a type (type {\bf I/1} in this paper) of $\SOL$ lattices has been determined. Some basic concept of $\SOL$ were defined by {\sc{P. Scott}} in \cite{S}, in general. In our present work we shall classify $\SOL$ lattices in an algorithmic way into 17 (seventeen) types, in analogy of the 14 Bravais types of the Euclidean 3-lattices, but infinitely many $\SOL$ affine equivalence classes, in each type. Then the discrete isometry groups of compact fundamental domain (crystallographic groups) can also be classified into infinitely many classes but finitely many types, left to other publication. To this we shall study relations between $\SOL$ lattices and lattices of the pseudoeuclidean (or here rather called Minkowskian) plane \cite{AQ}. Moreover, we introduce the notion of $\SOL$ parallelepiped to every lattice type. From our new results we emphasize Theorems 3-4-5-6. In this paper we shall use the affine model of $\SOL$ space through affine-projective homogeneous coordinates \cite{M97} which gives a unified way of investigating and visualizing homogeneous spaces, in general., Comment: 34 pages, 9 figures

The representation method of heterogeneous material information is one of the key technologies of heterogeneous object modeling, but almost all the existing methods cannot represent non-uniform rational B-spline (NURBS) entity. According to the characteristics of NURBS, a novel data structure, named NURBS material data structure, is proposed, in which the geometrical coordinates, weights and material coordinates of NURBS heterogeneous objects can be represented simultaneously. Based on this data structure, both direct representation method and inverse construction method of heterogeneous NURBS objects are introduced. In the direct representation method, three forms of NURBS heterogeneous objects are introduced by giving the geometry and material information of control points, among which the homogeneous coordinates form is employed for its brevity and easy programming. In the inverse construction method, continuous heterogeneous curves and surfaces can be obtained by interpolating discrete points and curves with specified material information. Some examples are given to show the effectiveness of the proposed methods.

Applying the distance function between two B-spline curves with respect to the L2 norm as the approximate error, we investigate the problem of approximate merging of two adjacent B-spline curves into one B-spline curve. Then this method can be easily extended to the approximate merging problem of multiple B-spline curves and of two adjacent surfaces. After minimizing the approximate error between curves or surfaces, the approximate merging problem can be transformed into equations solving. We express both the new control points and the precise error of approximation explicitly in matrix form. Based on homogeneous coordinates and quadratic programming, we also introduce a new framework for approximate merging of two adjacent NURBS curves. Finally, several numerical examples demonstrate the effectiveness and validity of the algorithm.

Scalar multiplication [n]P is the kernel and the most time-consuming operation in elliptic curve cryptosystems. In order to improve scalar multiplication, in this paper, we propose a tripling algorithm using Lopez and Dahab projective coordinates, in which there are 3 field multiplications and 3 field squarings less than that in the Jacobian projective tripling algorithm. Furthermore, we map P to ϕe−1(P), and compute [n]ϕe−1 (P) on elliptic curve Ee, which is faster than computing [n]P on E, where ϕe is an isomorphism. Finally we calculate ϕe([n]ϕe−1(P)) = [n]P. Combined with our efficient point tripling formula, this method leads scalar multiplication using double bases to achieve about 23% improvement, compared with Jacobian projective coordinates.

Computer graphics is ostensibly based on projective geometry. The graphics pipeline—the sequence of functions applied to 3D geometric primitives to determine a 2D image—is described in the graphics literature as taking the primitives from Euclidean to projective space, and then back to Euclidean space. This is a weak foundation for computer graphics. An instructor is at a loss: one day entering the classroom and invoking the established and venerable theory of projective geometry while asserting that projective spaces are not separable, and then entering the classroom the following week to tell the students that the standard graphics pipeline performs clipping not in Euclidean, but in projective space—precisely the operation (deciding sidedness, which depends on separability) that was deemed nonsensical. But there is no need to present Blinn and Newell’s algorithm (Comput. Graph. 12, 245–251, 1978; Commun. ACM 17, 32–42, 1974)—the crucial clipping step in the graphics pipeline and, perhaps, the most original knowledge a student learns in a fourth-year computer graphics class—as a clever trick that just works. Jorge Stolfi described in 1991 oriented projective geometry. By declaring the two vectors $(x,y,z,w)^{{\ensuremath {\mathsf {T}}}}$and $(-x,-y,-z,-w)^{{\ensuremath {\mathsf {T}}}}$distinct, Blinn and Newell were already unknowingly working in oriented projective space. This paper presents the graphics pipeline on this stronger foundation.

In the early eighties, the author and co-workers created and further developed the natural coordinates to describe the motion of 2-D and 3-D multibody systems. Natural coordinates do not need angles or angular parameters to define orientation, leading to constant inertia matrices and to the simplest form of the constraint equations. Natural coordinates are composed by the Cartesian coordinates of some points and the Cartesian components of some unit vectors distributed on the different bodies of the system. The points and vectors can be located in the joints, being shared by contiguous bodies, decreasing or even eliminating the need to set joint constraints and reducing the total number of variables. However, other authors prefer not to share variables in order to get even simpler equations and to keep a bigger decoupling of equations, which is preferable in some cases. In this paper the history of natural coordinates is reviewed, as well as the main contributions coming from other research groups. In the second part of the paper some application areas in which natural coordinates can be particularly advantageous are examined.

An algorithmic approach to degree reduction of rational Bezier curves is presented. The algorithms are based on the degree reduction of polynomial Bezier curves. The method is introduced with the following steps: (a) convert the rational Bezier curve to polynomial Bezier curve by using homogenous coordinates, (b) reduce the degree of polynomial Bezier curve, (c) determine weights of degree reduced curve, (d) convert the Bezier curve obtained through step (b) to rational Bezier curve with weights in step (c).

The ideal class group of hyperelliptic curves can be used in cryptosystems based on the discrete logarithm problem. In this article we present explicit formulae to perform the group operations for genus 2 curves. The formulae are completely general but to achieve the lowest number of operations we treat odd and even characteristic separately. We present 3 different coordinate systems which are suitable for different environments, e. g. on a smart card we should avoid inversions while in software a limited number is acceptable. The presented formulae render genus two hyperelliptic curves very useful in practice. The first system are affine coordinates where each group operation needs one inversion. Then we consider projective coordinates avoiding inversions on the cost of more multiplications and a further coordinate. Finally, we introduce a new system of coordinates and state algorithms showing that doublings are comparably cheap and no inversions are needed. A comparison between the systems concludes the paper.

In this note we are interested in the graded modulesMk=I(k)/Ik and\(N_k : = \overline {I^k } /I^k \), whereI is a saturated ideal in the homogeneous coordinate ringS=K[x0,…,xn] of ℙn,I(k) is the symbolic power and\(\overline {I^k } \) is the saturation of the ordinary power. Very little is known about these modules, and we provide a bound on their diameters, we compute the Hilbert functions and we study some characteristic submodules in the special case ofn+1 general points in ℙn.

In the literature of triangle geometry, there are several kinds of conjugacies. The purpose of this paper is to recognize and generalize the functional forms for these conjugacies as represented in homogeneous coordinates. We introduce the U-Hirst inverse of X, which is a conjugacy given for points U = u : v : w and X = x : y : z by \( U \diamond X = vwx^2 - yzu^2:wuy^2 - zxv^2 : uvz^2 - xyw^2 \). Since \( U \diamond X = X \diamond U \), the operation \( \diamond \) is commutative. Another such case is isoconjugacy, given by \( U\,cX = vwyz\,:\,wuzx\,:\,uvxy \) and exemplified by the well-known isogonal and isotomic conjugacies. Others, such as line conjugacies and Ceva conjugacies, are represented by noncommutative operations. Diamond and line conjugacies are members of a two-parameter family of online conjugacies, whereas isoconjugacy and Ceva conjugacy are members of a two-parameter family of offline conjugacies. To the extent that these conjugacies are projective involutions, the underlying projective theory is well known.

We consider the blowing up of ℙk/n−1 along a closed subscheme defined by a homogeneous idealI ∪A=k[X1, …,Xn] generated by forms of degree ≤d, and its projective embeddings by the linear systems corresponding to (Ie)c, forc≥de+1. The homogeneous coordinate rings of these embeddings arek[(Ie)c]. One wants to study the Cohen-Macaulay property of these rings. We will prove that if the Rees algebraRA(I) is Cohen-Macaulay, thenk[(Ie)c] are Cohen-Macaulay forc>>e>0, thus proving a conjecture stated by A. Conca, J. Herzog, N.V. Trung and G. Valla.

Given multiple image data from a set of points in 3D, there are two fundamental questions that can be addressed: What is the structure of the set of points in 3D? What are the positions of the cameras relative to the points? In this paper we show that, for projective views and with structure and position defined projectively, these problems are dual because they can be solved using constraint equations where space points and camera positions occur in a reciprocal way. More specifically, by using canonical projective reference frames for all points in space and images, the imaging of point sets in space by multiple cameras can be captured by constraint relations involving three different kinds of parameters only, coordinates of: (1) space points, (2) camera positions (3) image points. The duality implies that the problem of computing camera positions fromp points in q views can be solved with the same algorithm as the problem of directly reconstructing q+4 points in p-4 views. This unifies different approaches to projective reconstruction: methods based on external calibration and direct methods exploiting constraints that exist between shape and image invariants.

The perspective 4 point (P4P) problem - also called the three-dimensional resection problem - is solved by means of a new algorithm: At first the unknown Cartesian coordinates of the perspective center are computed by means of Mobius barycentric coordinates. Secondly these coordinates are represented in terms of observables, namely space angles in the five-dimensional simplex generated by the unknown point and the four known points. Substitution of Mobius barycentric coordinates leads to the unknown Cartesian coordinates (2.8)–(2.10) of Box 2.2. The unknown distances within the five-dimensional simplex are determined by solving the Grunert equations, namely by forward reduction to one algebraic equation (3.8) of order four and backward linear substitution. Tables 1.–4. contain a numerical example. Finally we give a reference to the solution of the 3 point (P3P) problem, the two-dimensional resection problem, namely to the Ansermet barycentric coordinates initiated by C.F. Gaus (1842), A. Schreiber (1908) and A.␣Ansermet (1910).

Analysis and computations are presented for singularities in the solution of the steady Boussinesq equations for two-dimensional, stratified flow. The results show that for codimension 1 singularities, there are two generic singularity types for general solutions, and only one generic singularity type if there is a certain symmetry present. The analysis depends on a special choice of coordinates, which greatly simplifies the equations, showing that the type is exactly that of one dimensional Legendrian singularities, generalized so that the velocity can be infinite at the singularity. The solution is viewed as a surface in an appropriate compactified jet space. Smoothness of the solution surface is proved using the Cauchy-Kowalewski Theorem, which also shows that these singularity types are realizable. Numerical results from a special, highly accurate numerical method demonstrate the validity of this geometric analysis. A new analysis of general Legendrian singularities with blowup, i.e., at which the derivative may be infinite, is also presented, using projective coordinates.

We deal with the homographic and affine change of parameter of rational curves represented as BR-curves. Forcomputational purposes in CAGD and CAM we determine the new massic polygon through four different methods and we study the particular case of Bezier curves among the applications.

In this paper, we present an up-to-date survey of various numerically efficient methods for solving the problem of computing manipulator inverse dynamics. The literature on this subject is extensive. However, in this paper, we review only those algorithms which have been derived based on the Euler—Lagrange, Newton—Euler and Kane's formulations of the dynamic equations of motion and are applicable to rigid-link open-chain robot manipulators. In particular, for each of these formulations we present a chronological account of the development of the most important algorithms which compute manipulator inverse dynamics. In this process some ‘classical’ algorithms are given and a number of issues which make it possible to reduce their computational complexity are emphasized. Also, the most efficient algorithms currently available are compared in terms of their computational complexity.

The application of the mortar method using Lagrange multiplier to the waveguides is the basic objective of the present paper. The application of the analytical integration technique is made with integrals written in homogeneous coordinates of the finite element. In such a way, the matrices are calculated once, being independent of the element dimensions and dependent only on the type and the order of the used approximation. To confirm, the cutoff wavelengths of the dominant and first high-order mode of the horseshoe-shaped waveguide are calculated.

A new axiomatization involving incidence and remoteness of planes with nondivision coordinate rings is introduced and a coordinatization theorem is obtained. A geometric process of splitting points and lines to obtain another plane with the same coordinates is described. It is also shown that a group of Steinberg type is parametrized by a nonassociative ring. The notion of elementary basis sets for an associative ring is introduced and constructions of projective and affine planes are given. A plane with reflections determining a system of rotations is shown to have commutative, associative coordinates.