Geometry Construction Research Articles

Applying Dynamic Geometry Environment Software as a Visualization Tool for Teaching Planimetry Construction Tasks

The present study addresses a notable gap in the current pedagogical research by rigorously examining the impact of Augmented Reality (AR) on mathematics education, specifically focusing on two-dimensional (2D) plane geometry construction tasks. Our primary aim is to discern and articulate the unique contributions of AR as a pedagogical tool in this domain. Employing a quasi-experimental design, we engaged with 141 high school students, dividing them into two distinct groups: one receiving instruction through AR tools and the other via traditional teaching methods. Performance metrics were derived from pre- and post-construction tests as well as an attitudinal survey gauging students’ perceptions of AR. Our findings reveal that the AR-facilitated group showcased a marked improvement in post-test scores and exhibited a more positive attitude towards AR technology compared to the traditionally taught cohort. This point towards AR’s potential to significantly elevate comprehension and engagement levels in 2D geometry construction. The study further highlights the transformative potential of AR in bolstering students’ enthusiasm and curiosity in mathematical endeavors. The results presented herein emphasize the pressing need for extended research to gauge AR’s broader and sustained effects in mathematics education and its applicability across various mathematical disciplines.

An automatic parallel scheme to design an augmented hydrokinetic river turbine using a simulation-based optimization approach

The effect of diffuser-enhanced flow on the energy extraction performance of hydrokinetic turbines is widely reported in the literature. In this context, this work combines automatic scripting for geometry construction and meshing, together with a simulation-based approach using particle-swarm optimization with Computational Fluid Dynamics (CFD) to design an augmented hydrokinetic river turbine (HKRT). The goal is to achieve an increase in mass flow in the throat of the diffuser and then to design an ad-hoc set of rotor blades. The optimization is performed in two separate stages: Continuous optimization for the diffuser and discrete optimization for the rotor blades. The complex shape of the rotor blades in terms of the chord, twist, and thickness distribution was parametrized with second-order Bezier curves, which facilitates the correlation between the blade’s shape and its performance, proving to be more intuitive to the experienced designer. Furthermore, a novel search and discrete arrangement of feasible design variables are proposed, which allows for reducing the computational costs of this large optimization problem. Results show that the optimized diffuser achieved a flow rate increase of 279%, and the ad-hoc 1.5 [m] in diameter rotor delivers a power of 11.98 [kW] with an efficiency of 0.3385%.

Characterizing Learners’ Construction of Geometry Diagrams in Dynamic Geometry Environments

As a distinct mode of mathematical representation and communication, geometry diagrams are both signs that represent abstract geometric concepts and visual images that offer graphical-spatial properties. Dynamic geometry environment offers a wide range of tools for its users to create and interact with geometry diagrams. In this paper, I propose a framework to characterize geometry diagrams that learners create in DGEs. The framework considers a learner’s approach to creating a geometry diagram (i.e., perceptualbased, measurement-based, construction-based, and transformation-based), the driving force that guides the learner’s specific actions in a DGE (i.e., tool-driven, and property-driven), and the constraints in the resulting diagram (i.e., drawing, underconstrained, overconstrained, and appropriately constrained). Examples of student work on two geometry construction problems are used to illustrate the use of the framework. These examples show that the proposed framework provides a useful tool to characterize geometry diagrams in DGEs.

Low Impact Velocity Modeling of 3D Printed Spatially Graded Elastomeric Lattices.

Additive manufacturing technologies have facilitated the construction of intricate geometries, which otherwise would be an extenuating task to accomplish by using traditional processes. Particularly, this work addresses the manufacturing, testing, and modeling of thermoplastic polyurethane (TPU) lattices. Here, a discussion of different unit cells found in the literature is presented, along with the based materials used by other authors and the tests performed in diverse studies, from which a necessity to improve the dynamic modeling of polymeric lattices was identified. This research focused on the experimental and numerical analysis of elastomeric lattices under quasi-static and dynamic compressive loads, using a Kelvin unit cell to design and build non-graded and spatially side-graded lattices. The base material behavior was fitted to an Ogden 3rd-order hyperelastic material model and used as input for the numerical work through finite element analysis (FEA). The quasi-static and impact loading FEA results from the lattices showed a good agreement with the experimental data, and by using the validated simulation methodology, additional special cases were simulated and compared. Finally, the information extracted from FEA allowed for a comparison of the performance of the lattice configurations considered herein.

High resolution DLP stereolithography to fabricate biocompatible hydroxyapatite structures that support osteogenesis.

Lithography based additive manufacturing techniques, specifically digital light processing (DLP), are considered innovative manufacturing techniques for orthopaedic implants because of their potential for construction of complex geometries using polymers, metals, and ceramics. Hydroxyapatite (HA) coupons, printed using DLP, were evaluated for biological performance in supporting viability, proliferation, and osteogenic differentiation of the human cell line U2OS and human mesenchymal stem cells (MSCs) up to 35 days in culture to determine feasibility for future use in development of complex scaffold geometries. Contact angle, profilometry, and scanning electron microscopy (SEM) measurements showed the HA coupons to be hydrophilic, porous, and having micro size surface roughness, all within favourable cell culture ranges. The study found no impact of leachable and extractables form the DLP printing process. Cells seeded on coupons exhibited morphologies comparable to conventional tissue culture polystyrene plates. Cell proliferation rates, as determined by direct cell count and the RealTime-GloTM MT Cell Viability Assay, were similar on HA coupons and standard tissue culture polystyrene plates). Osteogenic differentiation of human MSCs on HA coupons was confirmed using alkaline phosphatase, Alizarin Red S and von Kossa staining. The morphology of MSCs cultured in osteogenic medium for 14 to 35 days was similar on HA coupons and tissue culture polystyrene plates, with osteogenic (geometric, cuboidal morphology with dark nodules) and adipogenic (lipid vesicles and deposits) features. We conclude that the DLP process and LithaBone HA400 slurry are biocompatible and are suitable for osteogenic applications. Coupons served as an effective evaluation design in the characterization and visualization of cell responses on DLP printed HA material. Results support the feasibility of future technical development for 3D printing of sophisticated scaffold designs, which can be constructed to meet the mechanical, chemical, and porosity requirements of an artificial bone scaffold.

Fast identification, and construction of adsorbate-adsorbent geometries for high throughput computational applications: The Automatic Surface Adsorbate Structure Provider (ASAP) algorithm

High through-put computation (HTC) of material properties requires databases or code to generate candidate structures. The complexity of interfaces has stymied the creation of code to generally build candidate sorption geometries, which is required for studying surface, and porous material chemistry. The Automatic Surface Adsorbate Structure Provider (ASAP) constructs candidate atomic structures for any ad/absorbent-ad/absorbate pair. ASAP identifies all unique sites in/on the sorbent, places the binding atom of the sorbate at open sites in unique atom’s coordination shell, and rotates the sorbate to minimize steric overlap. It only requires the geometric structure of the solid and sorbate, the bonding atom, a bonding distance, and a minimum inter-atomic distance. Thus, ASAP is a new tool that removes the need for researchers to rigorously identify and create the unique absorbate sites by hand and accelerates HTC investigations of surface, intra-pore, and interstitial chemistry and physics.

Free and Non-Free Vortex Design for Axial Fans with Circumferential Sweep through CFD Techniques

The sound generated by axial flow fans has become increasingly important in several industrial areas. These devices represent considerable noise sources that must be considered from the design stage. The scope of this research work is to present an axial fan design methodology based on the theory of the wing lift and to compare the aerodynamic and aeroacoustics behavior with the free vortex and the non-free vortex conditions. The effect of forward circumferential sweep on the rotor blades is analyzed in both conditions, which consists of a displacement in the tangential direction in the direction of rotor rotation. A cubic polynomial function is proposed for the construction of the blade geometry with circumferential sweep. It defines the center line of the blade and it is taken at 50% of the leading edge of each blade profile. The methodology for analyzing aeroacoustics behavior in fans is based on the integration of Computational Aeroacoustics and Computational Fluid Dynamics in order to analyze aerodynamic behavior, sound power level and sound pressure level at the preliminary design stage. It was verified that although both rotors present a similar aerodynamic behavior (the NFV rotor has a slightly lower performance), according to the analysis of local sound sources, the NFV rotor presents a slight decrease in the sound sources. It was also verified that the NFV rotor presents 2.2 dB less in sound pressure level than the FV rotor.

GeoGebra in Improving Students’ Learning Ability of Geometry Construction

GeoGebra in Improving Students’ Learning Ability of Geometry Construction

RETRACTED: Geometric Construction of Video Stereo Grid Space

The construction of digital twin cities is a current research hotspot. Video data are one of the important aspects of digital twin cities, and their digital modeling is one of the important foundations of its construction. For this reason, the construction and digital analysis of video data space has become an urgent problem to be solved. After in-depth research, this study found that the existing video space construction methods have three shortcomings: first, the problem of high requirements for objective conditions or low accuracy; second, the lack of easy and efficient mapping algorithms from 2D video pixel coordinates to 3D; and third, the lack of efficient correlation mechanisms between video space and external geographic information, making it difficult to integrate video space with external information, and thus prevent a more effective analysis. In view of the above problems, this paper proposes a video stereo grid geometric space construction method based on GeoSOT-3D stereo grid coding and a camera imaging model to form a video stereo grid space model. Finally, targeted experiments of video stereo grid space geometry construction were conducted to analyze the experimental results before and after optimization and compare the variance size to verify the feasibility and effectiveness of the model.

The proof of hypothesis regarding distortion of time and space using the nuclear fusion model

The study is dedicated to modern topic: the analysis of conditions that lead to distortion of the time and space coordinates which results from the general theory of relativity. The main goal of this research is to prove the hypothesis regarding distortion of time and space using nuclear fusion model. For this purpose the simulation instrument is used to imitate a moving proton that hits an electron of a hydrogen atom. The methodology of simulation is based upon calculation of the probabilities of elastic scattering and charge exchange of a proton with a target electron. The distortion is modeled by the functions that relate time and space logarithmically for distorted time and exponentially for distorted space. Such geometry construction is described by the Schrödinger equation using the electron wave function. Then the probability of charge exchange is calculated as the squared coefficient of this wave function in the negative side of the geometry that is divided by the sum of the squared coefficients of all the terms of the equation. Thus, the calculation result shows that the calculated probability of the charge exchange is high when the time and space are not distorted. However, when time and space are distorted it decreases, and the probability of elastic scattering is growing. The achieved result also indicates that the discrete energy levels of electrons in hydrogen atoms shift when the distortion of time and space occurs in the nuclear fusion.

Содержание курса «Геометрическое моделирование» для направления подготовки «Математика и компьютерные науки»

In this paper has been considered the main content and distinctive features of the “Geometric Modeling” training course for the “Mathematics and Computer Science” training program 02.03.01 (“Mathematical and Computer Modeling” specialization).
 The goal of the “Geometric Modeling” course study is the assimilation of mathematical methods for construction of geometric objects with complex curved shapes, and techniques for their computer visualization by using polygons of curves and surfaces. Methods for construction of structures’ curved shapes using spline representations, as well as techniques for construction of surfaces and volumetric geometries using motion operations and basic logical operations on geometric objects are considered. The spline representations include linear and bilinear splines, Hermite cubic splines and Hermite surfaces, natural cubic and bicubic interpolation splines, Bezier curves and surfaces, rational Bezier splines, B-splines and B-spline surfaces, NURBS-curves and NURBS-surfaces, transfinite interpolation methods, and splines of surfaces with triangular form. Logical operations for intersection of two spline curves, and intersection of two parametric surfaces are considered. The principles of scientific visualization and computer animation are considered in this course as well.
 Some examples for visualization of initial data and results of curves and surfaces construction in two- and three-dimensional spaces through the software shell developed by authors and used by students while doing tests have been demonstrated. The software shell has a web interface with the WebGL library graphic support. Tasks for four practical studies in a computer classroom, as well as several variations of homework are represented.
 The problems occurring in preparation materials for some course sections are discussed, as well as the practical importance of acquired knowledge for the further progress of students.
 The paper may be interesting for teachers of “Geometric Modeling” and “Computer Graphics” courses aimed to students with a specialization in mathematics and information, as well as to those who independently develop software interfaces for algorithms of geometric modeling.

Geometry and Actual Construction in Brick Vaults by Slices: The Case of Carranque in Spain

This paper deals with the study of a rectangular plan sail vault built by brick slices in the Roman villa of Carranque in Spain in the fifth century, in the context of a research project on the constructive configuration of Mediterranean vaults of this kind. The project aims to identify technical links to trace their expansion and examine the possibilities of using this technique in present-day building practice. The case at Carranque confirms the arrival of this vaulting technique to the Iberian Peninsula prior to a possible diffusion through the Arab world. The analysis of a 3d photogrammetric model of the remains allows posing that the solution used to solve the slices meeting at the diagonal is different from Byzantine ones; the large perimetral arches were lowered, almost matching the height of the small ones, and a vertical area was placed, being arranged as if it were part of the vault itself. With this design, it is not easy to notice the rectangular form of the vault.

CONSTRUCTION OF ANY ANGLE WITHOUT PROTRACTOR

Geometry is one of the foremost significant chapters in mathematics. Construction of angles is one of the important aspects of geometry. The term construction in geometry refers to drawing shapes, lines, or angles accurately with the assistance of mathematical instruments. You will simply need a compass, a ruler (straightedge), and a pencil to construct angles, and a protractor to measure the constructed angle. This is the pure form of geometric construction as no numbers are involved in it. We can only draw some special angles without employing a protractor, such as 30, 60, 90, 120, 180, etc. and non-special angles such as 83, 72, 68, etc. with 1or 2 degrees deviation from the required angles. The method of constructing special angles is generally taught in our lower secondary school. However, using the strategy we devised, we will draw any angle between 0 and 360 degrees without any deviation from the required angle. Constructing angles is important in maths as this method can be extended for the construction of other geometric figures as well, primarily triangles. Keywords: Angle, Degree, Intersection Points, Reference Arc, Primary arc

Snakes Isogeometric Analysis (SIGA): Towards accurate and flexible geometrical models of the respiratory airways

The construction of accurate patient-specific geometries from medical images is critical to achieving predictive numerical simulations of the respiratory system. However, the generation of surface representations of the airway tree is very challenging due to their complex morphology. In this work, we present a novel framework for creating Non-Uniform Rational B-Splines surface models of the respiratory airways. Our method relies on solving the variational formulation of the Snakes segmentation problem using Isogeometric Analysis (SIGA). We validate the SIGA method by comparing the resulting surface mesh against those delivered by two benchmark surface-fitting methods based on control-point projection. Our results confirm that SIGA outperforms the benchmark methods in creating surface models from computed-tomography images of a normal and a diseased lung. Further, we show that SIGA meshes adapt well to pathological airways with non-convex cross-sections where traditional surface-fitting methods fail. We envision that the geometrical flexibility and accuracy of SIGA will enhance the creation of computational models of the respiratory system under healthy and diseased conditions.

A comparative study on environmental performance of 3D printing and conventional casting of concrete products with industrial wastes

3D printing construction techniques are believed to have potential sustainability benefits, including improved resource efficiency, increased construction productivity, and construction of complex geometries without supporting structures. 3D printable concrete materials, when introducing industrial wastes such as fly ash, silica fume, and slag, may also bring additional sustainability benefits. These advantages need to be verified quantitatively. This study investigated the environmental impact of 3D printable concrete materials using industrial wastes compared with the conventional ones via life cycle assessment (LCA). Two types of concrete materials applied in concrete casting or 3D printing were compared, that is, cement-based concrete and geopolymer concrete. The results indicate that using waste materials as cement replacement could bring environmental benefits; however, such environmental benefits might be diminished with increasing activator content in geopolymer concrete for 3D concrete printing. Based on the material-level LCA results, this study further conducted an LCA study at the component level, which investigated the life-cycle environmental impact of concrete components of different shapes constructed by Contour Crafting method. Results show that the potential environmental benefit of 3D concrete printing increases with the level of building complexity while decreases with the reuse times of formwork, which leads to the conclusion that 3D concrete printing method is more desirable for constructing non-repetitive freeform concrete structures.

Modeling of geometric correspondences and transformations via geometry constructions language

Geometric correspondences and transformations can be applied in solutions to numerous problems of geometric modeling, mainly in order to create models of curves and surfaces. Among the tools of CAD systems, the geometric transformations are presented in the form of a limited set of editing tools, array tools, tools providing formation of 3D solids and surfaces through motion. In practice, this set of tools restricts application of the wide capabilities of geometric transformations and correspondences. In this regard, it is proposed to model the geometric transformations and correspondences via the geometry construction language. The elements of the geometric transformation theory are analyzed. The geometric transformations are proposed to be created on par with other objects of the geometric model. The commands of the geometry constructions language enabling construction of the transformation objects are considered, their application is discussed. Solutions to practical problems of surface modeling via the geometry constructions language with the use of geometric transformation are demonstrated on examples. The proposed approach allows us to facilitate modeling of curves and surfaces in CAD systems, if said curves and surfaces can be acquired through geometric transformation or correspondence that can be defined via a certain constructive algorithm.

Coagulation Bath‐Assisted 3D Printing of PEDOT:PSS with High Resolution and Strong Substrate Adhesion for Bioelectronic Devices

AbstractPoly(3,4‐ethylenedioxythiophene):poly(styrenesulfonate) (PEDOT:PSS) is a promising material because of its favorable electrical and mechanical properties, stability in ambient environments, and biocompatibility. It finds broad application in energy storage, flexible electronics, and bioelectronics. Additive manufacturing opens a plethora of new avenues to form and shape PEDOT:PSS, allowing for the rapid construction of customized geometries. However, there are difficulties in printing PEDOT:PSS while maintaining its attractive properties. A 3D printing method for PEDOT:PSS using a room‐temperature coagulation bath‐based direct ink writing technique is reported. This technique enables fabrication of PEDOT:PSS into parts that are of high resolution and high conductivity, while maintaining stable electrochemical properties. The coagulation bath can be further modified to improve the mechanical properties of the resultant printed part via a one‐step reaction. Furthermore, it is demonstrated that a simple post‐processing step allows the printed electrodes to strongly adhere to several substrates under aqueous conditions, broadening their use in bioelectronics. Employing 3D printing of PEDOT:PSS, a cortex‐wide neural interface is fabricated, and intracranial electrical stimulation and simultaneous optical monitoring of mice brain activity with wide field calcium imaging are demonstrated. This reported 3D‐printing technique eliminates the need for cumbersome experimental setups while offering desired material properties.

Seismic Assessment of Roman Concrete Groin Vaults through UAV, NDT and 3D Analyses

In Roman Baths, the Romans employed barrel and groin vaults of great dimensions, with maximum span more than 20 m; simple tools of structural analysis of ancient wide span vaulted halls are still lacking, due to geometrical and material complexity. In this paper, we study the collapse behavior, under horizontal static action, of a corner cross vault of the Baths of Diocletian in Rome (Hall I). Two methods of analysis are here used: non-linear incremental finite element and limit analysis. In both cases, 3D models have been developed by means of UAV inspection, NDT measures, and AVT monitoring. The construction of the overall 3D geometry has been here afforded with a specific pre-processing approach. Midas commercial software has been employed for FEM analysis, assuming a constitutive law specifically developed for Roman concrete. In limit analysis, masonry is discretized as a system of interacting rigid bodies in no-tension and frictional contact. The computational code consists in a linear approach, which makes use of a series of optimization packages via lower and upper bound techniques. Finally, a strategy based on FEM analysis including discontinuities was implemented, and the results were compared with the two previous approaches.

Content Areas and Performance of Students in the Old Mutual Mathematics Olympiad in Malawi

The Old Mutual Mathematics Olympiad (OMMO) in Malawi is a national mathematics competition open to all final year secondary school students. In this paper, we analyse the results of the preliminary round of the competition from 2015 to 2020. We developed seven content areas for the OMMO based on core elements of the Malawi secondary school mathematics curriculum. We found that the OMMO is dominated by questions on the core element Number and numeration, particularly topics like number systems, indices and logarithms, linear equations and number patterns. We also analysed the general performance of students in these areas. We found that students performed well in topics like inequalities, trigonometry and mensuration and did very poorly on proportions, sets, coordinate geometry and geometric constructions, among other topics. These findings suggest that mathematics Olympiads can help diagnose areas that are difficult for students and hence enhance the teaching and learning of Mathematics, especially in Malawi and other countries with a similar mathematics curriculum. To take care of under-represented areas, future OMMO examinations need to be structured to adequately cover all the core elements.

Fast Identification, and Construction of Adsorbate-Adsorbent Geometries for High Throughput Computational Applications: The Automatic Surface Adsorbate Structure Provider (Asap) Algorithm

Fast Identification, and Construction of Adsorbate-Adsorbent Geometries for High Throughput Computational Applications: The Automatic Surface Adsorbate Structure Provider (Asap) Algorithm