Simple Geometric Algorithm Research Articles

Providing a simple and efficient algorithm for passive sound localization

The sound source localization (SSL) system is one of the main approaches for detecting targets. This kind of localization method is carried out in a passive way, without the emission of any signals, and cannot be detected easily. This paper investigates the two-dimensional SSL of stationary targets using only three receivers. Given that, flow-induced noise of the bluff-bodies in viscous fluids is one of the most important noise generation sources, firstly, this noise is simulated numerically. Due to the distance between the receivers in an array, the sound reaches the receiver with time delay. High accuracy SSL needs highly accurate time delays measurement. In the following, time delays are obtained by calculating the cross-correlation in a time domain of sound signals arriving at the receiver array, and next, the bearing angle of the sound source is estimated. Later, using a new, simple (easy installation), fast (real-time), and accurate pure geometric algorithm, the source candidate locations are obtained. Then, using an elimination algorithm, the ambiguity of the noise source location is removed, and source imaginary locations are removed. Finally, this process is validated by a physical algorithm. Numerical results verify the effectiveness of this geometric algorithm.

Improvements in micro-CT method for characterizing X-ray monocapillary optics

Accurate characterization of the inner surface of X-ray monocapillary optics (XMCO) is of great significance in X-ray optics research. Compared with other characterization methods, the micro computed tomography (micro-CT) method has its unique advantages but also has some disadvantages, such as a long scanning time, long image reconstruction time, and inconvenient scanning process. In this paper, sparse sampling was proposed to shorten the scanning time, GPU acceleration technology was used to improve the speed of image reconstruction, and a simple geometric calibration algorithm was proposed to avoid the calibration phantom and simplify the scanning process. These methodologies will popularize the use of the micro-CT method in XMCO characterization.

A digital twin study for immediate design / redesign of impellers and blades: Part 1: CAD modelling and tool path simulation

This paper presents a digital twining study conducted for an immediate design / redesign and manufacturing of on impellers & blades. It is by accomplished by developing (i) CAD automation methods, based on the standard modelling procedures and (ii) Manufacturing automation based on the 3/3 + 1/3 + 2/5 axis milling process. Initially, the CAD model of impeller / blade is created by utilizing the dimensional parameters obtained through standard design calculations / data. It is then parametrized and converted to an automated model through simple dimensional rules and geometric algorithms developed for the purpose. After this stage, the CAD model is analyzed for manufacturing automation where the process planning data comprising cutting tools, process parameters and setups are selected. Here, the tool paths are generated for 3/3 + 1/3 + 2/5 axis milling considering a CNC Vertical Machining Center (VMC) to digitally twin milling process. Both the CAD modelling and manufacturing process plans including tool path generation are captured through journaling and customized / improved using the Application Programmable Interface’s (API’s) to suit the present scope. In this paper, the first part on CAD modelling and manufacturing simulation methodologies are discussed through validating the digital twining concept in a virtual environment. The work is developed with the focus to help industries moving towards Industry 4.0 and requiring a constant design improvement in their products. It is by emphasizing the importance of digital twinning concept where a concurrent verification of design and manufacturing process can be achieved.

Moving Mesh Non-standard Finite Difference Method for Non-linear Heat Transfer in a Thin Finite Rod

In this paper, a moving mesh technique and a non-standard finite difference method are combined, and a moving mesh non-standard finite difference (MMNSFD) method is developed to solve an initial boundary value problem involving a quartic nonlinearity that arises in heat transfer with thermal radiation. In this method, the moving spatial grid is obtained by a simple geometric adaptive algorithm to preserve stability. Moreover, it uses variable time steps to protect the positivity condition of the solution. The results of this computational technique are compared with the corresponding uniform mesh non-standard finite difference scheme. The simulations show that the presented method is efficient and applicable, and approximates the solutions well, while because of producing unreal solution, the corresponding uniform mesh non-standard finite difference fails.

Landau singularities from the amplituhedron

We propose a simple geometric algorithm for determining the complete set of branch points of amplitudes in planar mathcal{N} = 4 super-Yang-Mills theory directly from the amplituhedron, without resorting to any particular representation in terms of local Feynman integrals. This represents a step towards translating integrands directly into integrals. In particular, the algorithm provides information about the symbol alphabets of general amplitudes. We illustrate the algorithm applied to the one- and two-loop MHV amplitudes.

Reconstruction of Low Degree B-spline Surfaces with Arbitrary Topology Using Inverse Subdivision Scheme

Multivariate B-spline surfaces over triangular parametric domain have many interesting properties in the construction of smooth free-form surfaces. This paper introduces a novel approach to reconstruct triangular B-splines from a set of data points using inverse subdivision scheme. Our proposed method consists of two major steps. First, a control polyhedron of the triangular B-spline surface is created by applying the inverse subdivision scheme on an initial triangular mesh. Second, all control points of this B-spline surface, as well as knotclouds of its parametric domain are iteratively adjusted locally by a simple geometric fitting algorithm to increase the accuracy of the obtained B-spline. The reconstructed B-spline having the low degree along with arbitrary topology is interpolative to most of the given data points after some fitting steps without solving any linear system. Some concrete experimental examples are also provided to demonstrate the effectiveness of the proposed method. Results show that this approach is simple, fast, flexible and can be successfully applied to a variety of surface shapes.

Implementation of surface tension force in fluid flow during reactive rotational molding

During Reactive Rotational Molding (RRM), it is very important to predict the fluid flow in order to obtain the piece with homogeneous shape and high quality. This prediction may be possible by simulation the fluid flow during rotational molding. In this study we have used a mixture of isocyanate and polyol as reactive system. The kinetic rheological behaviors of thermoset polyurethane are investigated in anisothermal conditions. Thanks to these, rheokinetik model of polyurethane was identified. Then, to simulate the RRM, we have applied Smoothed Particles Hydrodynamics (SPH) method which is suited method to simulate the fluid flow with free surface such as occurs at RRM. Modelling and simulating reactive system flow depend on different parameters; one of them is the surface tension of reactive fluid. To implement force tension surface, the interface between polymer and air is dynamically tracked by finding the particles on this border. First, the boundary particles are detected by free-surface detection algorithm developed by Barecasco, Terissa and NAA [1, 2] in two and three dimension. Then, analytical and geometrical algorithms have been used for interface reconstructions. The aim of this work is the implementation of surface tension force in the SPH solver applied to RRM. To illustrate that, we used novel and simple geometric algorithm fitting circle and fitting sphere, in two and three dimensional configurations, respectively. The model has been validated using a well-known dam break test case which covered the experimental data.

Topological relations between separating circles

The family of separating circles of two finite sets in the plane consists of all the circles that enclose the first set but exclude the second set. We prove some theoretical results on distances between families of circles, and properties about enclosure and intersection. Most of these results state that a property that involves one or more infinite families of circles can be verified by examining a finite subcollection of circles. As a result enclosure and intersection can be decided, and distances can be computed with simple geometric algorithms. Furthermore, the circles of the finite subcollections correspond to the vertices of a polytope in the parameter space of separating circles. A polytope of separating circle parameters is well-known computational geometry, but we prove some new properties and we introduce the concept of an elementary circular separation as a concise way to define such a polytope.

A generally suited method to get initial guess solutions for one-step and multi-step simulations of tube and profile forming based on parameterization

In one-step and multi-step simulations, how to get initial guess solutions effectively is a key point, which has gained great attention during the past decades. However, most of the achievements have been targeted at sheet metal forming, rarely at tube or profile forming, which hinders the applications of one- and multi-step approach to tube and profile forming. A new method, based on the parameterization in computer graphics, is presented in this paper, in which a sort of parameterized computation defined in the cylindrical coordinate system is utilized to produce initial guess solution for initial configuration, and a simple but effective geometric algorithm on the basis of the parameterization is proposed for intermediate configuration. Different from the original plane parameterization, the parameterized computation in this method is to map mesh onto initial straight tube or profile surface instead of plane. The method is highly efficient since only one set of large sparse linear equations needs to be solved in the parameterized computation and the geometric algorithm for intermediate configuration is also fast in mesh-mapping. In order to verify the validity and efficiency of the presented method, several numerical examples are provided and the obtained initial guess meshes are satisfactory. Copyright © 2010 John Wiley & Sons, Ltd.

Sequential simplex optimization of recombinant biotinylated survivin production by Escherichia coli using mineral supplementation

The sequential simplex method is a useful tool for rapidly optimizing a process by moving through factor space via a relatively simple geometric algorithm. This method was employed to optimize growth and survivin-BCCP production in Escherichia coli Origami B using zinc sulphate supplemented culture medium. Five experimental parameters were investigated: concentration of zinc sulphate and IPTG, pH, temperature, and agitation rate. Optimized conditions were as follows: 190 μM zinc sulphate with 246 μM IPTG, pH 7, at a temperature of 23.5 °C and agitation rate of 345 rpm. The results for cell density and survivin-BCCP production were, 17% and 140% higher than in non-zinc supplemented medium and 12.6% and 25.5% higher than could be obtained by varying one-factor-at-a-time. The present work demonstrates the effectiveness of zinc sulphate supplementation and the efficiency of the sequential simplex optimization method in obtaining high yields.

A simple geometric algorithm to predict optimal starting gantry angles using equiangular‐spaced beams for intensity modulated radiation therapy of prostate cancer

A fast, geometric beam angle optimization (BAO) algorithm for clinical intensity-modulated radiation therapy (IMRT) was implemented on ten localized prostate cancer patients on the Radiation Therapy Oncology Group (RTOG) 0126 protocol. The BAO algorithm computed the beam intersection volume (BIV) within the rectum and bladder using five and seven equiangular-spaced beams as a function of starting gantry angle for comparison to the V 75 Gy and V 70 Gy. A mathematical theory was presented to explain the correlation of BIV with dose and dose-volume metrics. The class solution 'W' pattern in the rectal V 75 Gy and V 70 Gy as a function of starting gantry angle using five equiangular-spaced beams (with two separate minima centered near 20 degrees and 50 degrees) was reproduced by the 5 BIV within the rectum. A strong correlation was found between the rectal 5 BIV and the rectal V 75 Gy and V 70 Gy as a function of starting gantry angle. The BAO algorithm predicted the location of the two dosimetric minima in rectal V 75 Gy and V 70 Gy (optimal starting gantry angles) to within 5 degrees. It was demonstrated that the BIV geometric variations for seven equiangular-spaced beams were too small to translate into a strong dosimetric effect in the rectal V 75 Gy and V 70 Gy. The relatively flat distribution with starting gantry angle of the bladder V 75 Gy and V 70 Gy was reproduced by the bladder five and seven BIV for each patient. A geometric BAO method based on BIV has the advantage over dosimetric BAO methods of simplicity and rapid computation time. This algorithm can be used as a standalone optimization method or act as a rapid calculation filter to reduce the search space for a dosimetric BAO method. Given the clinically infeasible computation times of many dosimetric beam orientation optimization algorithms, this robust geometric BIV algorithm has the potential to facilitate beam angle selection for prostate IMRT in clinical practice.

SU-FF-T-42: A Simple Geometric Algorithm to Predict Optimal Starting Gantry Angles Using Equiangular-Spaced Beams for IMRT of Prostate Cancer

Purpose: A fast, geometric beam angle optimization (BAO) algorithm for intensity-modulated radiation therapy (IMRT) was implemented on ten localized prostate cancer patients on the Radiation Therapy Oncology Group 0126 protocol. Method and Materials: Fifteen segmental IMRT plans per patient were generated in Pinnacle3 using 5 equiangular-spaced beams with 5° increments of the starting gantry angles. Constant target coverage was ensured for all plans in order to isolate the variation in the rectal dose metrics as a function of starting gantry angle. A geometric BAO algorithm computed the beam intersection volume (BIV) within the rectum using 5 equiangular-spaced beams as a function of starting gantry angle for comparison to the rectal V 75 Gy and V 70 Gy. Results: The variations in rectal V 75 Gy and V 70 Gy as a function of starting gantry angle using 5 equiangular-spaced beams were statistically significant (p < 0.001) with dosimetric importance. The class solution ‘W’ pattern in the rectal V 75 Gy and V 70 Gy (with two separate minima centered near 20° and 50°) was reproduced by the 5 BIV within the rectum. A strong correlation (Pearson's correlation coefficient > 0.70) was found between the rectal 5 BIV and the rectal V 75 Gy and V 70 Gy. The geometric BAO algorithm predicted the location of the two dosimetric minima in rectal V 75 Gy and V 70 Gy (optimal starting gantry angles) to within 5°. Conclusions: This BAO method is the first geometric algorithm capable of predicting an optimal IMRT dose distribution with typical computation times of only a few minutes. Given the clinically infeasible computation times of many dosimetric BAO algorithms, this robust geometric BIV algorithm has the potential to facilitate beam angle selection for prostate IMRT in clinical practice.

A two-step algorithm of smooth spline generation on Riemannian manifolds

This paper presents a simple geometric algorithm to generate splines of arbitrary degree of smoothness in Euclidean spaces. Unlike other existing methods, this simple geometric algorithm does not require a recursive procedure and, consequently, introduces a significant reduction in calculation time. The algorithm is then extended to other complete Riemannian manifolds, namely to matrix Lie groups and spheres.

LeptonicCPviolation in a two parameter model

We further study the ``complementary'' ansatz, $\mathrm{Tr}({M}_{\ensuremath{\nu}})=0$, for a prediagonal light Majorana type neutrino mass matrix. Previously, this was studied for the $CP$ conserving case and the case where the two Majorana type $CP$ violating phases were present but the Dirac type $CP$ violating phase was neglected. Here we employ a simple geometric algorithm which enables us to ``solve'' the ansatz including all three $CP$ violating phases. Specifically, given the known neutrino oscillation data and an assumed two parameter (the third neutrino mass ${m}_{3}$ and the Dirac $CP$ phase $\ensuremath{\delta}$) family of inputs we predict the neutrino masses and Majorana $CP$ phases. Despite the two parameter ambiguity, interesting statements emerge. There is a characteristic pattern of interconnected masses and $CP$ phases. For large ${m}_{3}$ the three neutrinos are approximately degenerate. The only possibility for a mass hierarchy is to have ${m}_{3}$ smaller than the other two. A hierarchy with ${m}_{3}$ largest is not allowed. Small $CP$ violation is possible only near two special values of ${m}_{3}$. Also, the neutrinoless double beta decay parameter is approximately bounded as $0.020\text{ }\text{ }\mathrm{eV}&lt;|{m}_{ee}|&lt;0.185\text{ }\text{ }\mathrm{eV}$. As a by-product of looking at physical amplitudes we discuss an alternative parametrization of the lepton mixing matrix which results in simpler formulas. The physical meaning of this parametrization is explained.

Constructing global bifurcation diagrams by the parametric representation method

The parameter dependence of the solution x of equation f 0( x)+ u 1 f 1( x)+ u 2 f 2( x)=0 is considered. Our aim is to divide the parameter plane ( u 1, u 2) according to the number of the solutions, that is to construct a bifurcation curve. This curve is given by the singularity set, but in practice it is difficult to depict it, because it is often derived in implicit form. Here we apply the parametric representation method which has the following advantages: (1) the singularity set can be easily constructed as a curve parametrized by x, called D-curve; (2) the solutions belonging to a given parameter pair can be determined by a simple geometric algorithm based on the tangential property; (3) the global bifurcation diagram, that divides the parameter plane according to the number of solutions can be geometrically constructed with the aid of the D-curve.

A geometric theory for formulation and evaluation of form and profile tolerances

This paper develops a geometric theory which unities the formulation and computation of form and profile tolerances stipulated in ANSI Y14.5M standard. The theory rests on an important observation that the configuration space of a toleranced feature can be identified with the homogeneous space SE (3)/ G 0 of the Euclidean group, where G 0 is the symmetry group of the underlying feature. We show that all cases of form and profile tolerances can be formulated as a minimization or constrained minimization problem in the Space SE (3)/ G 0 . Using properties of SE (3)/ G 0 we develop a simple geometric algorithm, called the Symmetric Minimum Zone (SMZ) algorithm to unify the computation of form and profile tolerances.

A geometric theory of form, profile, and orientation tolerances

This article develops a geometric theory which unifies the formulation and computation of form (straightness, flatness, cylindricity, and circularity), profile and orientation tolerances stipulated in ANSI Y14.5M standard. The theory rests on an important observation that a toleranced feature exhibits a symmetry subgroup G 0 under the action of the Euclidean group, SE(3). Thus, the configuration space of a toleranced (or a symmetric) feature can be identified with the homogeneous space SE(3)/ G 0 of the Euclidean group. Geometric properties of SE(3)/ G 0, especially its exponential coordinates carried over from that of SE(3), are analyzed. We show that all cases of form, profile and orientation tolerances can be formulated as a minimization or constrained minimization problem on the space SE(3)/ G 0, with G 0 being the symmetry subgroup of the underlying feature. We develop a simple geometric algorithm, called the Symmetric Minimum Zone (SMZ) algorithm, to unify the computation of form, profile, and orientation tolerances. Finally, we use numerical simulation results comparing the performances of the SMZ algorithm against the best known algorithms in the literature.

Forward kinematics for Hunt–Primrose parallel manipulators

This paper gives closed-form solutions (requiring linear and quadratic equations only) for the forward displacement and velocity kinematics of two parallel kinematic chain designs presented by Hunt and Primrose in 1993, [4]. The solutions rely on a simple geometric algorithm that finds the top of a tetrahedron if the base triangle and the lengths of all of its sides are known. The forward kinematics has eight configurations.

Simple and practical geometric algorithms

Simple and practical geometric algorithms

A shortest path algorithm for a nonrotating object among obstacles of arbitrary shapes

A shortest path algorithm for motion planning problem via mathematical morphology is proposed. The moving object and obstacles can be of arbitrary shapes of finite size. To speed up the computation of the free positions of a moving object, the idea of shape decomposition is developed. It is proven that the decomposition always reduces the number of iterations. In addition, a simple geometric algorithm is proposed to check the existence of safe paths between the given initial and final positions. Once the existence condition is verified, a modified grass fire algorithm via the grayscale morphology is then used to compute the shortest path. Finally, examples of various descriptions of moving objects among obstacles are demonstrated. >