Geometric Knowledge Research Articles

Geometry: An innate ability that nevertheless makes it difficult for students worldwide

The importance of geometry has been highlighted since antiquity. Studies show that we have geometric knowledge from infancy age. Some geometric skills are innate and are not taught in school. However, cultivating them can be achieved in many ways at a very young age and has positive academic results in the future. Unfortunately, nowadays students do not show the necessary interest in the subject and as a result they have not developed their geometric thinking and encounter difficulties both in everyday life and at school.

An automatic generation approach of process model based on feature knowledge and geometric modeling

Traditional process planning applies knowledge to convert design information within the part model into process information. This information is then displayed by manually drawing two-dimensional (2D) process cards from multiple views, illustrating the geometric design and the process details. However, this method has revealed several issues, including poor efficiency, information misinterpretation, and the potential for manual drawing errors. This traditional approach hampers the integration of manufacturing information across engineering software. It exacerbates the digital divide between computer-aided design (CAD), computer-aided process planning (CAPP), and computer-aided manufacturing (CAM). To address these issues, this study proposes an automated approach for creating three-dimensional (3D) process models based on feature knowledge and geometric modeling. The 3-axis milling features are recognized as the machining objects from the boundary representation (B-Rep) part model. Geometric and parameter knowledge for modeling is derived from the machining step. In the proposed approach, two geometric modeling methods are investigated to construct feature state volumes (FSVs). The first method utilizes FSV modeling to simulate the shape of unmachined features before rough machining, while the second method utilizes FSV modeling to construct the shape of machined features before finishing machining. The case studies illustrate that the proposed approach can construct FSVs for various machining states and autonomously generate process models. In digital manufacturing, this approach assists process planners in intuitively evaluating the reasonableness of process routes. Additionally, it provides the essential driving geometry required for the autonomous generation of tool paths.

A Comparative Study of Geometric Principles in the Sulba Sutras and the Pythagorean Theorem: Historical Context and Mathematical Applications

This research delves into the geometric principles detailed in the ancient Sulba Sutras and their parallels with the Pythagorean Theorem, traditionally attributed to the Greek mathematician Pythagoras. While the Pythagorean theorem is widely regarded as a cornerstone of Greek geometry, recent studies suggest that its formulation appeared centuries earlier in Indian mathematics, as evidenced in Baudhāyana’s Sulba Sutra (circa 800 BCE). These texts offer extensive insights into the geometric constructions used in Vedic rituals, including the use of Pythagorean triples and transformations between geometric shapes. The Sulba Sutras present a rich mathematical framework, emphasizing practical applications in altar construction, land measurements, and urban planning. This study aims to bridge the research gap by comparing the geometric developments in the Sulba Sutras with Greek formulations, particularly focusing on their mathematical methods, cultural contexts, and historical significance. The findings underscore the advanced state of Indian mathematics and its profound influence on subsequent mathematical traditions, offering a nuanced understanding of the parallel evolution of geometric knowledge across cultures.

МЕКТЕП ГЕОМЕТРИЯСЫНДАГЫ ФИГУРАЛАРДЫН КЛАССИФИКАЦИЯСЫ

This work proposes a classification of geometric figures studied in a systematic course in school geometry. Classification of geometric shapes is said to be an important aspect of teaching geometry to enable students to understand and analyze different shapes; identify similarities and differences between figures; recognize common features and differences between figures belonging to the same class or type; systematize and organize knowledge; reveal patterns and relationships between figures with similar properties, which provides the basis for subsequent discoveries; derive new properties and theorems; solve spatial geometric problems. New areas of spatial perception (virtual space, multidimensional space, digital technologies, including cloud, etc.), technologies and industries are increasingly emerging. In such changes, the relevance of the science of geometry does not disappear, but rather increases. This is due to the fact that geometric knowledge and geometric thinking give a person the basis for solving various problems in theoretical (geometric) and practical (physical) spaces. However, despite this, geometry is considered a difficult subject to perceive and understand.

Spatial guided image captioning: Guiding attention with object's spatial interaction

AbstractNowadays relational position embedding is widely used in many large multi‐modal models. It begins with relational captioning (a branch of image captioning) and contains two procedures: geometric modelling and prior attention. However, there are some problems that remain unsolved in the conventional procedures. This paper reviews the shortcomings of geometric modelling and prior attention. Then, a new framework called relational guided transformer (RGT) is proposed to verify the authors' conclusion from the origin of relational position embedding—relational captioning. Specifically, RGT has two simple but effective improvements in geometric modelling and prior attention: (1) A machine‐learned geometric modelling strategy called multi‐task geometric modelling (MTG) is used under multi‐task learning, replacing the original hand‐made geometric feature. (2) The effectiveness of multiple kinds of prior attention is discussed and preserved in a better form, which is called spatial guided attention (SGA) to integrate the geometric prior knowledge. Extensive experiments on MSCOCO and Flickr30k have been performed to investigate the effectiveness of each module and prove our argument. The superiority of the model comparing to the authors' baseline has also been proven on the offline evaluation with the “Karpathy” test split of both datasets.

Enhancing Geometry Understanding in Grade 8: Development and Evaluation of a Learning Package

This study developed and evaluated a comprehensive learning package for Grade 8 Geometry, aligned with the curriculum to address diverse student needs. The package included a teacher’s guide, modules, manipulatives, PowerPoint presentations, and lesson plans. Evaluators assessed the package based on instructional, content, and technical quality, rating it as very good. A quasi-experimental approach was employed to measure the package's effectiveness in enhancing geometric knowledge and problem-solving skills. The study involved two schools, one public and one private, each with two sections of Grade 8 students. One section in each school served as the experimental group, while the other served as the control group. Effectiveness was determined by comparing pretest and posttest scores using appropriate statistical tests. The results indicated significant improvements in student performance due to the learning package, with notable differences between public and private school students. These findings emphasize the importance of tailored educational interventions that consider the specific contexts of different school environments. It is recommended that teachers adopt this learning package to enhance student learning in geometry.

Efficient aerodynamic shape optimization by using unsupervised manifold learning to filter geometric features

Many aerodynamic shape optimization methods often focus on utilizing the end-to-end relationship between design variables and aerodynamic performance to find the optimal design, while overlooking the exploration of geometric knowledge of the shape itself. To fully use geometric knowledge to improve optimization efficiency, this paper proposes an efficient method by exploring the potential correlation between geometric features and aerodynamic performance at a low cost. We use unsupervised isometric feature mapping in manifold learning to capture geometric features that can distinguish the aerodynamic performance of different airfoils without embedding any tags. Then a filter criterion is establish based on the geometric features. During the optimization process, airfoils that deliver poor aerodynamic performance can be filtered out with a high probability before being precisely evaluated through computational fluid dynamics simulations. This helps improve samples quality to enhance the optimization efficiency. We applied the proposed method to the unconstrained and constrained optimizations of the Royal Aircraft Establishment (RAE) 2822 airfoil to validate its performance. The results demonstrate that the proposed method can improve the efficiency of optimization by over 50% compared with the original evolutionary optimization algorithm. It performs well across various optimization problems, demonstrating high engineering practical value.

KANT, FREGE, HILBERT Y LAS FUENTES DEL CONOCIMIENTO GEOMÉTRICO

1. The scientific revolution of the 18th century was based in capability of predicting, using mathematical models, the structure of the spacio-temporal world. From there, aro se the notion of experimental science whose goals are determine wich of the theorical architectures correspond to spatio-temporal facts. 2. Producing the scientific revolution and destroying the psychologism inherent in the realism of Arostotle is the question "What are the mental structures that permit the creation and development ofthe scientific model achieved by mathematico-experimental physics? This is the problem conceming the sources of geometric knowledge proposed by Kant and later by Fregge and still later by David Hilbert. 3. These lead to a new concept of science seen as creativity of the human mind in order to systematize our experience in the spatio-temporal world inhabited by Horno sapiens.

Student Teachers’ Knowledge of School-level Geometry: Implications for Teaching and Learning

<p style="text-align:justify">This study aimed to assess the geometric knowledge of student teachers from a university in the Eastern Cape province of South Africa. The study used a sample of 225 first-year student teachers who completed school mathematics baseline assessments on a computer- aided mathematics instruction (CAMI) software. The study adopted a descriptive cross-sectional research design, using quantitative data to measure student teachers’ geometry achievement level, and qualitative data to explain the challenges encountered. The results show that student teachers exhibited a low level of understanding of school-level geometry. The low achievement levels were linked to various factors, such as insufficient grasp of geometry concepts in their secondary school education, difficulty in remembering what was done years ago, low self-confidence, and lack of Information and Communications Technology (ICT) skills along with the limited time for the baseline tests. These results suggest that appropriate measures should be taken to ensure that student teachers acquire the necessary subject-matter knowledge to teach effectively in their future classrooms.</p>

Exploring pre-service primary teachers’ emotions in a geometry project with 3D design

This study explores the emotions that pre-service teachers (PSTs) experience when they interact with geometric knowledge and 3D design in a project focused on the way of Saint James pilgrimage route. The sample was made up of 101 PSTs from the University of Santiago de Compostela, the final point of the pilgrimage path. An emotional questionnaire, together with a participant observation notebook and a rubric, served as data collection instruments. The analytical frameworks included the didactical suitability criteria for mediational and affective facets, and SAMR (substitution, augmentation, modification, and redefinition) model for the effective integration of technology in schools. The results show a greater presence of pleasant emotions, highlighting the emotions of curiosity and cheerfulness. Despite this, less pleasant emotions such as brain-taster or bewilderment were notable. Pleasant emotions shown, as well, higher correlation rates. In conclusion, 3D design seems to indicate great potential for working on emotions with this group of students.

Multiscale Dynamic Curvelet Scattering Network.

The feature representation learning process greatly determines the performance of networks in classification tasks. By combining multiscale geometric tools and networks, better representation and learning can be achieved. However, relatively fixed geometric features and multiscale structures are always used. In this article, we propose a more flexible framework called the multiscale dynamic curvelet scattering network (MSDCCN). This data-driven dynamic network is based on multiscale geometric prior knowledge. First, multiresolution scattering and multiscale curvelet features are efficiently aggregated in different levels. Then, these features can be reused in networks flexibly and dynamically, depending on the multiscale intervention flag. The initial value of this flag is based on the complexity assessment, and it is updated according to feature sparsity statistics on the pretrained model. With the multiscale dynamic reuse structure, the feature representation learning process can be improved in the following training process. Also, multistage fine-tuning can be performed to further improve the classification accuracy. Furthermore, a novel multiscale dynamic curvelet scattering module, which is more flexible, is developed to be further embedded into other networks. Extensive experimental results show that better classification accuracies can be achieved by MSDCCN. In addition, necessary evaluation experiments have been performed, including convergence analysis, insight analysis, and adaptability analysis.

Advancing surgical VQA with scene graph knowledge

PurposeThe modern operating room is becoming increasingly complex, requiring innovative intra-operative support systems. While the focus of surgical data science has largely been on video analysis, integrating surgical computer vision with natural language capabilities is emerging as a necessity. Our work aims to advance visual question answering (VQA) in the surgical context with scene graph knowledge, addressing two main challenges in the current surgical VQA systems: removing question–condition bias in the surgical VQA dataset and incorporating scene-aware reasoning in the surgical VQA model design.MethodsFirst, we propose a surgical scene graph-based dataset, SSG-VQA, generated by employing segmentation and detection models on publicly available datasets. We build surgical scene graphs using spatial and action information of instruments and anatomies. These graphs are fed into a question engine, generating diverse QA pairs. We then propose SSG-VQA-Net, a novel surgical VQA model incorporating a lightweight Scene-embedded Interaction Module, which integrates geometric scene knowledge in the VQA model design by employing cross-attention between the textual and the scene features.ResultsOur comprehensive analysis shows that our SSG-VQA dataset provides a more complex, diverse, geometrically grounded, unbiased and surgical action-oriented dataset compared to existing surgical VQA datasets and SSG-VQA-Net outperforms existing methods across different question types and complexities. We highlight that the primary limitation in the current surgical VQA systems is the lack of scene knowledge to answer complex queries.ConclusionWe present a novel surgical VQA dataset and model and show that results can be significantly improved by incorporating geometric scene features in the VQA model design. We point out that the bottleneck of the current surgical visual question–answer model lies in learning the encoded representation rather than decoding the sequence. Our SSG-VQA dataset provides a diagnostic benchmark to test the scene understanding and reasoning capabilities of the model. The source code and the dataset will be made publicly available at: https://github.com/CAMMA-public/SSG-VQA.

CrossBind: Collaborative Cross-Modal Identification of Protein Nucleic-Acid-Binding Residues

Accurate identification of protein nucleic acid binding residues poses a significant challenge with important implications for various biological processes and drug design. Many typical computational methods for protein analysis rely on a single model that could ignore either the semantic context of the protein or the global 3D geometric information. Consequently, these approaches may result in incomplete or inaccurate protein analysis. To address the above issue, in this paper, we present CrossBind, a novel collaborative cross modal approach for identifying binding residues by exploiting both protein geometric structure and its sequence prior knowledge extracted from a large scale protein language model. Specifically, our multi modal approach leverages a contrastive learning technique and atom wise attention to capture the positional relationships between atoms and residues, thereby incorporating fine grained local geometric knowledge, for better binding residue prediction. Extensive experimental results demonstrate that our approach outperforms the next best state of the art methods, GraphSite and GraphBind, on DNA and RNA datasets by 10.8/17.3% in terms of the harmonic mean of precision and recall (F1 Score) and 11.9/24.8% in Matthews correlation coefficient (MCC), respectively. We release the code at https://github.com/BEAM-Labs/CrossBind.

Boosting deep neural networks with geometrical prior knowledge: a survey

Deep neural networks achieve state-of-the-art results in many different problem settings by exploiting vast amounts of training data. However, collecting, storing and—in the case of supervised learning—labelling the data is expensive and time-consuming. Additionally, assessing the networks’ generalization abilities or predicting how the inferred output changes under input transformations is complicated since the networks are usually treated as a black box. Both of these problems can be mitigated by incorporating prior knowledge into the neural network. One promising approach, inspired by the success of convolutional neural networks in computer vision tasks, is to incorporate knowledge about symmetric geometrical transformations of the problem to solve that affect the output in a predictable way. This promises an increased data efficiency and more interpretable network outputs. In this survey, we try to give a concise overview about different approaches that incorporate geometrical prior knowledge into neural networks. Additionally, we connect those methods to 3D object detection for autonomous driving, where we expect promising results when applying those methods.

Inherited Geometry: a Socio-Professional Pattern in the Low Countries, 1480-1560

At the beginning of the 16th century the Habsburg Low Countries witnessed the emergence of a highly skilled class of painters who also often actively engaged in architectural design practice. This phenomenon has often been explained as being the result of an influx of Italian and humanist thinking about the artist such via the writings of figures like Alberti and Vitruvius. In this article, I propose an alternative account, using selected case studies to argue that this development was an internal process. Essential to changing design practices was the transfer of geometrical design knowledge between guilds, a transfer that transgressed traditional boundaries. Finally, I explore the effect of this knowledge dissemination on the evolving status of the visual artist.

Retaining and Mobilising Geometric Knowledge

Abstract This paper is dedicated to the history of an extremely particular manuscript: Segre’s personal card index. Consisting of 515 handwritten cards on which he annotated bibliographic references, excerpts of readings that could be useful in his studies and teaching, comments and memoranda, Segre’s Schedario is the manuscript catalogue of a virtual library, a sort of reasoned bibliography that photographs the cultural heritage of Corrado Segre and, more generally, of an Italian geometer of the Belle Époque. In this paper, we will analyse how Segre constructed, organised and used the Schedario in order to structure his teaching and research activity.

Task Space Compliant Control and Six-Dimensional Force Regulation Toward Automated Robotic Ultrasound Imaging

<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Objective:</i> We propose a general control framework for task space compliant motion and six-dimensional (6-D) force regulation towards automated robotic ultrasound (US) imaging. The framework endows a position-controlled robotic manipulator with the capability of accurate compliant motion in free space and accurate force control in motion-constrained environment. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Methods:</i> An intuitive six degree-of-freedom (6-DoF) admittance control model expressed in an arbitrary Cartesian body frame is mathematically derived with closed-form task space error mapping. Its practical implementation on widely-used collaborative manipulators is proposed to achieve full task space compliant behaviors and accurate 6-D force control. A hybrid control law is presented to achieve good motion accuracy in free space and improved coupled stability in motion-constrained environment. The coupled model of physical human-robot interaction is established and the reason for the improved coupled stability is analyzed through simulation. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Results:</i> Evaluation experiments on the proposed control framework were performed to show the effectiveness. The mean error of compliant trajectory following was less than 0.30 mm in free space. The mean relative force and moment control accuracy in three orthogonal directions was better than 0.5% and 0.8%, respectively. The improved coupled stability under the same model parameters was also confirmed by human-robot interaction experiments. Finally, an automated robotic US imaging experiment on a human volunteer in a real clinical scenario was carried out to show the potential application of our proposed framework. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Conclusion:</i> Experimental results have shown the advantages of the control framework, including satisfied force control accuracy, high accuracy of compliant motion, improved coupled stability, and system effectiveness on a human volunteer. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This paper was motivated by the increasing needs of automated ultrasound (US) scanning for both diagnostic and interventional purpose. Clinical sonographers suffer from repeated workload when performing diagnostic US imaging, which could benefit from automated robotic scanning. Robotic US imaging involves physical interaction between the robot end-effector (i.e., US transducer) and the human body. The dynamics of the interaction is regulated by the control law to guarantee the contact of the US transducer and the safety of the procedure. Most existing works have focused on regulating in-plane contact force in terms of the position without considering the compliance in other dimensions. However, it is not a trivial work to extend the positional compliance to six degree-of-freedom (6-DoF) compliance. As the prevalence of low-cost collaborative robotic arms in medical scenarios, how to perform 6-DoF compliant trajectory following and accurate six-dimensional (6-D) force control on these robotic arms becomes increasingly important. This paper gives a complete general solution to achieve 6-DoF compliant control and 6-D force regulation with accurate kinematics on a position-controlled robotic arm. A hybrid control law is proposed to switch the government of “instantaneous model” and “theoretical model” to achieve compliant motion accuracy in free space and improved coupled stability in motion-constrained environment. No expensive torque sensors and torque control interface are required. And no prior geometric knowledge about the scanning object is needed. We have demonstrated the application for robotic US imaging in a real clinical scenario.

Expressiveness Analysis and Enhancing Framework for Geometric Knowledge Graph Embedding Models

Expressiveness Analysis and Enhancing Framework for Geometric Knowledge Graph Embedding Models

Reciprocal Learning Mathematics through Honeycomb Patterns

The purpose of this paper is to examine the mathematical ideas, problems, and solutions from the honeycomb patterns. This is a qualitative research design under an interpretive paradigm. I used interview, participant observation, small group discussions, photos, notes taken, and personal reflection for the data collection tools. The information was analyzed by participants generated text, activities, and experience of learning mathematics through the honeycomb patterns. The major themes that generated were algebra and students knowing of in-depth processes and solutions; algebraic and geometrical interpretation of 2D shapes with formulas; and geometrical knowledge of honeycomb patterns. The study showed that honeycomb patterns are the source for reciprocal learning of multidimensional area contents like algebra (quadratic and arithmetic sequence, hexagonal congruent tessellation, and geometry parallel lines, angles, and similar and congruent shapes together). What the students and the teachers need in reciprocal learning is the skill of observing patterns, redrawing, imagining, estimating, and logically intervening and shearing in them. The findings of the study can be helpful to teachers, students, and educators for reciprocal ways of learning, teaching, and training mathematics, connecting such patterns interestingly and meaningfully.

Semantic correction, enrichment and enhancement of social and transport infrastructure BIM models

The use of Building Information Modelling (BIM) models in the design, construction and operation of buildings and infrastructure is leading to a stronger focus on the quality of the models. Models may need correction, enrichment or enhancement to meet the expectations for quality and completeness, especially if models are to be taken as legal documents, for example for regulatory approval. Past work on semantic development has looked at specific scenarios such as scanned geometry or missing classification. This paper describes an innovative unified approach to the documentation of semantic expectations by actors in the AECO (Architectural, Engineering, Construction and Operations) domain and the means to put them into effect. RASE (Requirements, Applications, Selections and Exceptions) semantic mark-up is used to make both the requirements and any supporting resources both human-readable and machine-operable. Two example models from industry, a motorway bridge and a healthcare space, are used to demonstrate applying geometric, schema and classification knowledge. This knowledge is represented in a number of different styles. This extends our understanding of the nature of the knowledge found in dictionaries, classifications and development specifications, demonstrating how this knowledge can be made operable. This bridges the gap between the application of static compliance knowledge and the accurate and efficient application of correction, enrichment and enhancement knowledge.