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We review a body of literature related to the drawing and recognition of geometrical two-dimensional linear drawings including letters. Handwritten letters are viewed not as two-dimensional geometrical objects but as one dimensional trajectories of the tip of the implement. Handwritten letters are viewed as composed of a small set of kinematic primitives. Recognition of objects is mediated by processes of their creation (actual or imagined)-the imitation principle, a particular example of action-perception coupling. The concept of spatial directional field guiding the trajectories is introduced and linked to neuronal population vectors. Further. we link the kinematic description to the theory of control with spatial referent coordinates. This framework allows interpreting a number of experimental observations and clinical cases of agnosia. It also allows formulating predictions for new experimental studies of writing. [ABSTRACT FROM AUTHOR]

In this paper, based on video recordings of Orientation and Mobility (O&M) lessons for visually-disabled students, I will examine how occasioned maps (Psathas, 1979; Garfinkel, 2002), drawn in the student's palm are interactionally traced, felt, and noticed in order to represent the shape of a crossing for all practical purposes. Touching will be examined from the perspective of the live production of "trails" on a specific region of the body, the palm of the hand. We will begin to question how such hand-drawing map episodes occur during O&M courses, stressing how the coparticipants establish a participation framework that facilitates the making of the drawing, hand-map drawings are based on lines that are neither evanescent nor permanent. Their "persistence" is not an intrinsic feature but a systematic multimodal accomplishment. We will show how the drawn lines become depictions of the streets and contribute to producing two contrasting geometrical representations of the layout of a junction. [ABSTRACT FROM AUTHOR]

This paper focusses on the relationship between representation, computation and drawing measurability of geometrical shapes: a question that both architects and mathematicians address during their work and research. The case-study of the regular polyhedra is examined, acting as a support for a broader reflexion on the many aspects of scientific representation. [ABSTRACT FROM AUTHOR]

Background: In a recent study, we reported beam quality correction factors, fQ, in carbon ion beams using Monte Carlo (MC) methods for a cylindrical and a parallel-plate ionization chamber (IC). A non-negligible perturbation effect was observed;however, the magnitude of the perturbation correction due to the specific IC subcomponents was not included. Furthermore, the stopping power data presented in the International Commission on Radiation Units and Measurements (ICRU) report 73 were used, whereas the latest stopping power data have been reported in the ICRU report 90. Purpose: The aim of this study was to extend our previous work by computing fQ correction factors using the ICRU 90 stopping power data and by reporting ICspecific perturbation correction factors. Possible energy or linear energy transfer (LET) dependence of the fQ correction factor was investigated by simulating both pristine beams and spread-out Bragg peaks (SOBPs). Methods: The TOol for PArticle Simulation (TOPAS)/GEANT4 MC code was used in this study. A 30 × 30 × 50 cm3 water phantom was simulated with a uniform 10 × 10 cm2 parallel beam incident on the surface. A Farmer-type cylindrical IC (Exradin A12) and two parallel-plate ICs (Exradin P11 and A11) were simulated in TOPAS using the manufacturer-provided geometrical drawings. The fQ correction factor was calculated in pristine carbon ion beams in the 150-450 MeV/u energy range at 2 cm depth and in the middle of the flat region of four SOBPs. The kQ correction factor was calculated by simulating the fQo correction factor in a 60Co beam at 5 cm depth. The perturbation correction factors due to the presence of the individual IC subcomponents, such as the displacement effect in the air cavity, collecting electrode, chamber wall, and chamber stem, were calculated at 2 cm depth for monoenergetic beams only. Additionally, the mean dose-averaged and track-averaged LET was calculated at the depths at which the fQ was calculated. Results: The ICRU 90 fQ correction factors were reported. The pdis correction factor was found to be significant for the cylindrical IC with magnitudes up to 1.70%. The individual perturbation corrections for the parallel-plate ICs were <1.0% except for the A11 pcel correction at the lowest energy. The fQ correction for the P11 IC exhibited an energy dependence of >1.00% and displayed differences up to 0.87% between pristine beams and SOBPs. Conversely, the fQ for A11 and A12 displayed a minimal energy dependence of <0.50%. The energy dependence was found to manifest in the LET dependence for the P11 IC. A statistically significant LET dependence was found only for the P11 IC in pristine beams only with a magnitude of <1.10%. Conclusions: The perturbation and kQ correction factor should be calculated for the specific IC to be used in carbon ion beam reference dosimetry as a function of beam quality. [ABSTRACT FROM AUTHOR]

Copyright of Architectural Journal / Jian Zhu Xue Bao is the property of Architectural Journal Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Geometry is one of the most important foundations of Iranian art and architecture, which can be seen in the magnificent buildings left from the past that are important in architecture. One of the most important buildings of Iran from the Zandiyeh Era (250 years ago) is Vakil Mosque, which is located in Shiraz. Vakil Mosque is a sacred and Islamic building in Iran, which is the aim of this article to investigate and find geometry in the architectural body of this building. In the current research, descriptive-analytical methods were used to collect data. Then I studied the geometrical drawings of this mosque, which some of the most important books of Muslim scholars in this field have been studied to obtain the geometry of the plan and elevations of Vakil Mosque, and after many tests and finding acceptable answers, options were proposed as the physical geometry of this sacred building. Finally, I came to the conclusion that the architect or architects of this magnificent building have used their utmost efforts in designing its physical to create golden proportions and regular geometry on it, and this order and geometry is the reason for the glory and strength of this building. The findings of this article can be used along with other similar studies to recognize Iran's architectural works and restore its original Islamic-Iranian identity. [ABSTRACT FROM AUTHOR]

In order to achieve aesthetic and harmonious smile results, the use of anterior upper teeth golden proportion concepts represents reliable and scientific based guidelines. However, measuring, recording and analysing teeth and smiles biometric values proves to be a clinical and laboratory routine chalenge, once it is time consuming and demands additional especific math calculus or formulas. The aim of this paper is present an invention, "anterior upper teeth golden proportion millimetric templates," a set of instruments fabricated in order to achieve precise and fast millimetric measures, once they present predefined geometrical drawings and diagrams. Currently, when planning aesthetic tooth size modifications treatments, tools are used as pachymeters, dry edge compass, or some softwares. Most of times this strategy relies on professional expertise and intuiton, which shows to be limitating factors with the need of trial-and-error training and an accurate critical and artistic sense. Unfortunately, this is not inherent to all professionals, especially undergraduate dental school students. Thus, the templates bring differentials and advantages, being versatile and convenient, allowing countless clinical and laboratory uses, even over a cellphone or a computer screen. An excellent diagnostic aid providing ideal teeth proportions and positioning, increasing the chances of success in dental treatment planning. [ABSTRACT FROM AUTHOR]

Art, architecture, music and literature, which are thought to be far from mathematics, contain complex mathematics. The fact that Anatolian lands hosted many civilizations throughout history provided cultural richness including the relationship between mathematics and art and created Anatolian culture. Geometric patterns are frequently encountered in artistic and architectural works in Anatolian culture. Geometric patterns existing in Anatolian culture vary according to their fold. In this research, five-fold patterns have been explained mathematically by introducing the 'Key Length' concept. We developed a system that reconstructs an existing Islamic Geometric Pattern in the context of the girih mode that we presented verbally and visually in eleven steps. In addition, we introduced to the literature the concept of 'Key Length' which is used in the drawing process. [ABSTRACT FROM AUTHOR]

Making beneficial decisions is a vital part of running an efficacious business. Factors affecting decision-making in business contain alignment with important themes, internal and external information. The multi-attribute decision-making tool is the sub-part of the strategic decision-making technique used for evaluating awkward and unreliable problems that occurred in genuine life troubles. The major contribution of this analysis is to analyze the latest principle of complex Pythagorean fuzzy soft setting and their algebraic operations. Additionally, interaction aggregation operators are a massive part of the operators to use for aggregating the group of attributes into a singleton set. Moreover, by using the interaction aggregation operators and complex Pythagorean fuzzy soft sets, we initiated the principle of complex Pythagorean fuzzy soft interaction weighted averaging operator, complex Pythagorean fuzzy soft interaction weighted geometric operator and their well-known properties. To investigate the supremacy of the elaborated operators, a multi-attribute decision-making procedure is illustrated under the complex Pythagorean fuzzy soft information. Finally, we discussed the advantages and sensitive analysis of the presented works and drew their geometrical expressions to explore the consistency and dominancy of the initiated work. [ABSTRACT FROM AUTHOR]

Step-feature lines are one of the important geometrical elements for drawing the status quo maps of open-pit mines, and the efficient and accurate automatic extraction and updating of step-feature lines is of great significance for open-pit-mine stripping planning and analysis. In this study, an automatic extraction method of step-feature lines in an open-pit mine based on unmanned-aerial-vehicle (UAV) point-cloud data is proposed. The method is mainly used to solve the key problems, such as low accuracy, local-feature-line loss, and the discontinuity of the step-feature-line extraction method. The method first performs the regular raster resampling of the open-pit-mine cloud based on the MLS algorithm, then extracts the step-feature point set by detecting the elevation-gradient change in the resampled point cloud, further traces the step-feature control nodes by the seed-growth tracking algorithm, and finally generates smooth step-feature lines by fitting the space curve to the step-feature control nodes. The results show that the method effectively improves the accuracy of step-feature-line extraction and solves the problems of local-feature-line loss and discontinuity. [ABSTRACT FROM AUTHOR]

Copyright of Actas de Diseño is the property of Facultad de Diseno y Comunicacion, Fundacion Universidad de Palermo and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This research reveals significant and previously unknown details on the auxiliary lines and geometrical drawing of the Charles V Palace in the Alhambra, preserved at the National Historical Archive. A precise digital reproduction has made possible to formulate a hypothesis about the drawing process and discover some previous sketches of this major graphic legacy of the Early Spanish Renaissance. All this, together with the dimensional comparison with other famous drawings of the palace and data of the lost model, brings new evidence on its purpose, date and authorship, matters which still today are subject to discussion. [ABSTRACT FROM AUTHOR]

The tasks of construction of geometrical elements by the division of corners are examined in the article. A method passed the tests on employment on drawing on the fine-art faculty of the Kryvoy Rog state pedagogical university.

Measurements of form deviations of 3D elements can be conducted with various strategies. Such strategies differ in a number and distribution of sampling points located on an investigated surface. The low number of sampling points can bring about the situation that some surface irregularities cannot be detected by a measuring system. An application of higher density of sampling points, in turn, results in significant lengthening of measurement time, which is undesirable. This is why the authors propose to develop a novel adaptive measurement strategy. Proposed adaptive strategy consists of two stages: a preliminary measurement and additional measurements. During the preliminary measurement, an investigated area is scanned along a preselected trajectory. If the measurement results show that there is a significant change in sensor readings in a certain fragment of an investigated surface, then we conduct additional measurements in the area where the large change in sensor readings occurred. The paper presents the state-of-the-art on contemporary measurement strategies of 3D elements and fundamentals of the adaptive strategy proposed by the authors. [ABSTRACT FROM AUTHOR]

This paper is a historical account of the chords theorem, for conic sections from Apollonius to Boscovich. We comment the most significant proofs and applications, focusing on Newton's solution of the Pappus four lines problem. Newton's geometrical achievements drew L'Hospital's attention to the chords theorem as a fundamental one, and led him to search for a simple and direct proof, that he finally obtained by the method of projection. Stirling gave a very elegant algebraic proof; then Boscovich succeeded in finding an almost immediate geometrical proof, and showed how to develop the elements of conic sections starting from this theorem. [ABSTRACT FROM AUTHOR]

Tra il XVIII e il XIX secolo la diffusione della Geometria descrittiva di Monge in Europa determina, con tempi ed esiti diversi nei vari paesi, un cambiamento radicale nel campo della rappresentazione. Essa modifica non solo l’approccio al disegno ma anche, e in maniera sostanziale, l’istruzione professionale. In Gran Bretagna, tuttavia, la nuova scienza, per motivi sia politici che spiccatamente culturali, approda ufficialmente con grande ritardo. La sua circolazione tra professionisti, artigiani e disegnatori, attestata però ben prima, la vede innestarsi su una serie di esperienze di ricerca autonome, alimentando altresì i tentativi, da parte dei teorici dell’isola, di definire un sistema universale di comunicazione grafica. Il presente studio, attraverso una rivisitazione di ricerche sulla storia della rappresentazione relative a quest’epoca e la raccolta di fonti documentali, intende offrire una sistematica ricostruzione della diffusione della geometria descrittiva in Gran Bretagna, contestualizzandola nel clima socio-politico del tempo e intrecciando le nuove istanze del metodo mongiano con le originali ricerche condotte oltre Manica in quegli anni.

During the first half of the 20th century the Danish geometer Johannes Hjelmslev developed what he called a geometry of reality. It was presented as an alternative to the idealized Euclidean paradigm that had recently been completed by Hilbert. Hjelmslev argued that his geometry of reality was superior to the Euclidean geometry both didactically, scientifically and in practice: Didactically, because it was closer to experience and intuition, in practice because it was in accordance with the real geometrical drawing practice of the engineer, and scientifically because it was based on a smaller axiomatic basis than Hilbertian Euclidean geometry but still included the important theorems of ordinary geometry. In this paper, I shall primarily analyze the scientific aspect of Hjelmslev's new approach to geometry that gave rise to the so-called Hjelmslev (incidence) geometry or ring geometry. [ABSTRACT FROM AUTHOR]

Mathematics education researchers frequently use task-based interviews to gain insight into mathematicians' practice. However, there are a number of factors that should prevent mathematics educators from extrapolating how individual mathematicians respond to researcher-generated tasks in laboratory conditions, to how mathematicians practice their craft in authentic settings. In this paper we critically analyze the rationality of using task-based interviews to investigate mathematical practice, focusing on how task-based interview studies have been used to inform our understanding of mathematicians' use of examples in mathematical practice. We discuss four specific generalizations about mathematical practice drawn from these studies, and suggest other types of studies that can be used to corroborate or challenge those generalizations. [ABSTRACT FROM AUTHOR]

Nowadays in the plane geometry with the advancement of technology, the geometric constructions are designed and processed using the 2D dynamic geometry software's (DGSs). In order to use such programs, users should have the necessary information and skill. In this paper, pieces of information some of the known software are given, but more proposed Euklid DynaGeo is presented, in which users can draw and recognize the function graphs more easily. For this aim, some applications are displayed and finally, clearly presented how Euklid DynaGeo serves the users to realize the constructional geometrical drawings. [ABSTRACT FROM AUTHOR]

Summary: In this article, we present a novel nodal integration scheme for meshfree Galerkin methods, which draws on the mathematical framework of the virtual element method. We adopt linear maximum‐entropy basis functions for the discretization of field variables, although the proposed scheme is applicable to any linear meshfree approximant. In our approach, the weak form integrals are nodally integrated using nodal representative cells that carry the nodal displacements and state variables such as strains and stresses. The nodal integration is performed using the virtual element decomposition, wherein the bilinear form is decomposed into a consistency part and a stability part that ensure consistency and stability of the method. The performance of the proposed nodal integration scheme is assessed through benchmark problems in linear and nonlinear analyses of solids for small displacements and small‐strain kinematics. Numerical results are presented for linear elastostatics and linear elastodynamics and viscoelasticity. We demonstrate that the proposed nodally integrated meshfree method is accurate, converges optimally, and is more reliable and robust than a standard cell‐based Gauss integrated meshfree method. [ABSTRACT FROM AUTHOR]

Development of three‐dimensional (3D) printing technology has started a new chapter for in‐classroom modeling of chemical molecules. The technology provides the opportunity to design and produce various types of personalized models. However, using classical 3D printers is time consuming, and it is hard to involve students in the modeling process during traditional class times. One solution can be using hand‐held 3D printers (3D pens) that allow users to instantly draw geometrical structures. Unfortunately, drawing directly in 3D is very difficult, and precise modeling of even small molecules is simply not possible. In this article, a new approach to 3D modeling is described. It is based on 3D templates that enable the drawing of molecular models directly in three dimensions. The modular nature of the templates allows for the creation of a wide variety of structures. The resulting models provide an accurate representation of molecules including correct bond angles and geometry. This approach makes 3D pens a powerful tool for the modeling of chemical structures. [ABSTRACT FROM AUTHOR]

Analyzing and interpreting student thinking through written work is a key, often challenging, practice of teaching. It entails noticing students' mathematical thinking and drawing on mathematical knowledge for teaching. This study investigates how pre-service secondary teachers at the beginning of their preparation reason about students' written work. In interviews, participants solved mathematical tasks and the analyzed student written work in order to make assertions about student understanding. Analysis of participants' assertions revealed three primary reasoning strategies: mathematical reasoning, pedagogical reasoning, and reasoning through self-comparison. Self-comparison involved participants comparing the student work directly with their own work on the task. This strategy showed promise, as it could be used to help make inferential assertions about student understanding. It also highlighted areas of caution around how beginning pre-service teachers might make use of evidence. This study contributes to the existing literature around noticing student thinking in written work by highlighting the significance of reasoning through self-comparison. [ABSTRACT FROM AUTHOR]

Improving financial literacy for all segments of the population is a global problem which solving is aim of many international and national projects. It is not possible to find a sufficient solution without drawing attention to mathematical methods for adequate understanding and assessment of financial situation and taking optimal decisions. This is the goal of the Olympiad, established by two Bulgarian universities: the Higher School of Insurance and Finance (VUZF) in Sofia and the Economics University in Varna. Russia participated in the 2018-2019 edition of the Olympiad for the third time. This paper presents the Olympiad task solutions and gives information about the solution difficulties faced by the students. The authors hope that this information will be helpful for Olympiad questions developers and the participants' coaches. [ABSTRACT FROM AUTHOR]

Feature-based modelling has been considered as an indispensable tool for integrating CAD/CAPP systems. One of its advantages over conventional geometric modelling is the ability to associate not only geometric and topological information, but also form features, tolerances, material properties and other information which may be used during the process planning. This paper presents a STEP-based feature modeller (STEP-FM) for prismatic parts. High-level three-dimensional solid features are used as the basic entities for part design. The modeller relies on three main steps; (1) selection of the part base shape and overall size, (2) selection of the features to be added to or subtracted from the part being designed, and (3) providing information needed to define feature size, position, orientation and other attributes such as surface finish, tolerances, etc. The designed part is then exported as a STEP XML data format (according to ISO 2000). This file can be used to integrate into CAPP/CAM systems without using a complex feature recognition process. An object-oriented approach is used in the definition and implementation of the product model. This approach offers advantages of incremental system development and reusability. An example is given to demonstrate the application of the developed modeller. [ABSTRACT FROM AUTHOR]

Various Monte Carlo techniques are used to determine the complete phase diagrams of the square-well model for the attractive ranges λ=1.15 and λ=1.25. The results for the latter case are in agreement with earlier Monte Carlo simulations for the fluid-fluid coexistence curve and yield new results for the liquidus-solidus lines. Our results for λ=1.15 are new. We find that the fluid-fluid critical point is metastable for both cases, with the case λ=1.25 being just below the threshold value for metastability. We compare our results with prior studies and with experimental results for the γII-crystallin. [ABSTRACT FROM AUTHOR]

Studies on the evolution of the urban form frequently use methods of analysis in which rigorous and precise graphic representation is revealed as an optimum means in the research process for measuring, comparing and assessing the changes and transformations experienced. The purpose of this research, using the Avenida del Puerto in Valencia as a study model, is to demonstrate the validity of the cartographic coordination method, verifying relationships between the redrawn cartographs of different periods and taking permanences or invariants as support points that ensure comparison throughout the study sequence. It also intends to underline the potential offered by Geographic Information Systems in urban evolution studies, for which the spatial component constitutes a fundamental data. The analysis begins in the 18th century, to focus in greater detail between 1944 and 2004. [ABSTRACT FROM AUTHOR]

Teachers should possess a robust knowledge of graph interpretation in a world that requires increasingly scientific citizens. This study aimed to investigate teachers' knowledge of interpreting a context-based line graph, by understanding the types of difficulties teachers have in interpreting such graphs. The study also sought to determine whether there were gender differences in teachers' graph interpretation skills and determine whether these interpretation skills were different among teachers with varying teaching experiences. Sixty-one teachers from ten districts in one Indonesian province participated in this study. Empirically derived items were developed to identify the teacher's conceptual understanding of line graphs with Curcio's (Journal for Research in Mathematics Education, 18(5), 382-393, 1987) level of interpretation applied to explain the difficulties encountered. This study revealed that most of the teachers had difficulty answering questions that required 'reading beyond the data'. Specifically, these teachers interpreted the graph as an iconic representation of a real event rather than an abstract representation of data (i.e., speed vs. time). Performance differences in teachers' understanding of the graph were dependent on the grade level they taught, with differences especially evident in the interpretation of moderate and difficult items. There were no differences in teachers' understanding by gender or years of teaching experience. The results highlight the importance of focusing on teacher professional development centering on teachers' knowledge of graph comprehension. [ABSTRACT FROM AUTHOR]

In this article, we examine how it is possible, in the teaching and learning of geometry, to bridge the gap between problems involving drawings and figures, which is essential to the learning of mathematical proof. More precisely, the way students' drawing perception has to evolve, from Iconic Visualization to Non-Iconic Visualization (Duval, Annales de Didactique et de Sciences Cognitives, 10, 5-53, 2005). We show that the Instrumental Deconstruction process is multifaceted and central in this evolution. We present a theoretical framework, in relation with an experiment based on a 3D dynamic geometry environment. Based on a case study, we show that construction tasks with specific representations make the instrumental work play a key role in the learning of geometry. [ABSTRACT FROM AUTHOR]

Objectives: To assess variations in management of severe postpartum hemorrhage: 1) between obstetricians in the same situation 2) by the same obstetrician in different situations. Study design: A link to a vignette-based survey was emailed to obstetricians of 215 maternity units; the questionnaire asked them to report how they would manage the PPH described in 2 previously validated case-vignettes of different scenarios of severe PPH. Vignette 1 described a typical immediate, severe PPH, and vignette 2 a less typical case of severe but gradual PPH. They were constructed in 3 successive steps and included multiple-choice questions proposing several types of clinical practice options at each step. Variations in PPH were assessed in a descriptive analysis; agreement about management and its timing between vignette 1 and vignette 2 was assessed with the Kappa coefficient. Results: Analysis of complete responses from 119 (43.4%) obstetricians from 53 (24.6%) maternity units showed delayed or inadequate management in both vignettes. While 82.3% and 83.2% of obstetricians (in vignettes 1 and 2, respectively) would administer oxytocin 15 minutes after PPH diagnosis, only 52.9% and 29.4% would alert other team members. Management by obstetricians of the two vignette situations was inconsistent in terms of choice of treatment and timing of almost all treatments. Conclusion: Case vignettes demonstrated inadequate management as well as variations in management between obstetricians and in different PPH situations. Protocols or procedures are necessary in all maternity units to reduce the variations in practices that may explain a part of the delay in management that leads to PPH-related maternal mortality and morbidity. [ABSTRACT FROM AUTHOR]

We study a recent model for edge exchangeable random graphs introduced by Crane and Dempsey; in particular we study asymptotic properties of the random simple graph obtained by merging multiple edges. We study a number of examples, and show that the model can produce dense, sparse and extremely sparse random graphs. One example yields a power-law degree distribution. We give some examples where the random graph is dense and converges a.s. in the sense of graph limit theory, but also an example where a.s. every graph limit is the limit of some subsequence. Another example is sparse and yields convergence to a non-integrable generalized graphon defined on (0,∞). [ABSTRACT FROM AUTHOR]

Given a bipartite graph G=(A∪B,E) with strict preference lists and given an edge e∗∈E, we ask if there exists a popular matching in G that contains e∗. We call this the popular edge problem. A matching M is popular if there is no matching M′ such that the vertices that prefer M′ to M outnumber those that prefer M to M′. It is known that every stable matching is popular; however G may have no stable matching with the edge e∗. In this paper we identify another natural subclass of popular matchings called “dominant matchings” and show that if there is a popular matching that contains the edge e∗, then there is either a stable matching that contains e∗ or a dominant matching that contains e∗. This allows us to design a linear time algorithm for identifying the set of popular edges. When preference lists are complete, we show an O(n3) algorithm to find a popular matching containing a given set of edges or report that none exists, where n=|A|+|B|. [ABSTRACT FROM AUTHOR]

For a finite set X of n points from RM, the degree of an M-element subset {x1,⋯,xM} of X is defined as the number of M-simplices that can be constructed from this M-element subset using an additional point z∈X, such that no further point of X lies in the interior of this M-simplex. Furthermore, the degree of X, denoted by deg(X), is the maximal degree of any of its M-element subsets. The purpose of this paper is to show that the moments of the degree of X satisfy E[deg(X)k]≥cnk/logn, for some constant c>0, if the elements of the set X are chosen uniformly and independently from a convex body W⊂RM. Additionally, it will be shown that deg(X) converges in probability to infinity as the number of points of the set X goes to infinity. [ABSTRACT FROM AUTHOR]

Toric surface patch is the multi‐sided generalization of classical Bézier surface patch. Geometric continuity of the parametric surface patches plays a crucial role in geometric modeling. In this paper, the necessary and sufficient conditions of curvature continuity between toric surface patches are illustrated with the theory of toric degeneration. Furthermore, some practical sufficient conditions of curvature continuity of toric surface patches are also developed. [ABSTRACT FROM AUTHOR]

We present a novel method to construct subdivision stencils near extraordinary vertices with limit surfaces having optimal bounded curvature at extraordinary positions. With the proposed method, subdivision stencils for newly inserted and updated vertices near extraordinary vertices are first constructed to ensure subdivision with G1 continuity and bounded curvature at extraordinary positions. The remaining degrees of freedom of the constructed subdivision stencils are further used to optimize the eigenbasis functions corresponding to the subsubdominant eigenvalues of the subdivision with respect to G2 continuity constraints. We demonstrate the method by replacing subdivision stencils near extraordinary vertices for Catmull‐Clark subdivision and compare the results with the original Catmull‐Clark subdivision and previous tuning schemes known with small curvature variation near extraordinary positions. The results show that the proposed method produces subdivision schemes with better or comparable curvature behavior around extraordinary vertices with comparatively simple subdivision stencils. [ABSTRACT FROM AUTHOR]

Gradient mesh design tools allow users to create detailed scalable images, traditionally through the creation and manipulation of a (dense) mesh with regular rectangular topology. Through recent advances it is now possible to allow gradient meshes to have arbitrary manifold topology, using a modified Catmull‐Clark subdivision scheme to define the resultant geometry and colour [LKSD17]. We present two novel methods to allow local and hierarchical refinement of both colour and geometry for such subdivision gradient meshes. Our methods leverage the mesh properties that the particular subdivision scheme ensures. In both methods, the artists enjoy all the standard capabilities of manipulating the mesh and the associated colour gradients at the coarsest level as well as locally at refined levels. Further novel features include interpolation of both position and colour of the vertices of the input meshes, local detail follows coarser‐level edits, and support for sharp colour transitions, all at any level in the hierarchy offered by subdivision. [ABSTRACT FROM AUTHOR]

Mesh processing algorithms depend on quick access to the local neighborhood, which requires costly memory queries. Moreover, even having access to the local neighborhood is not enough to efficiently perform many geometry processing algorithms in an automatic or semi‐automatic way. As humans, we often imagine mesh editing at the level of topological information, e.g., altering surface features, adding limbs, etc., which is not supported by current data structures. These limitations come from the widely used mesh representations because the needed information is not implicitly defined by the structure. We propose a novel model representation called Skeletex. Each 3D model is decomposed into two elements: a skeletal structure that encodes the model topology and a vector displacement map to capture fine details of the geometry. Such a co‐representation contains the topology information, as well as the information about the local vertex neighborhood at each texel. Additionally, our data structure facilitates an automatic skeleton‐based cross‐parameterization. This allows us to implement the mesh manipulation tasks in parallel, using a unified streamlined pipeline that directly maps to the GPU. We demonstrate the capabilities of our data structure by implementing surface region transfer and mesh morphing of 3D models. [ABSTRACT FROM AUTHOR]

It remains a challenge to robustly and rapidly extract high quality curve skeletons from 3D models of closed surfaces, especially when there are nearby surface sheets. In this paper, we address this challenge by improving the use of LOP (Locally Optimal Projection) to adaptively contract medial surfaces of 3D models. LOP was originally designed to optimize a raw scanned point cloud to its corresponding geometry surface. It has the effect of contraction, and the contraction amplitude is controlled by a support radius. Our improvements are twofold. First, we constrain the LOP operator applied in the 2D medial surface instead of in the 3D space and take a local region growing strategy to find neighborhoods for implementing LOP. Thus, we avoid interference between disconnected surface parts and accelerate the process due to the reduced search space. Second, we adaptively adjust the support radii to have different parts of the medial surface contracted adaptively and synchronously for generating connected skeletal curves. In this paper, we demonstrate that our method allows for each part of the medial surface to be contracted symmetrically to its center line and is insensitive to surface noises. Thus, with our method, centered and connected high quality curve skeletons can be extracted robustly and rapidly, even for models with nearby surface sheets. Experimental results highlight the effectiveness and high efficiency of the method, even for noisy and topologically complex models, making it superior to other state‐of‐the‐art methods. [ABSTRACT FROM AUTHOR]

This paper studies two approaches to constructing control charts for linguistic data. Linguistic data can provide more information than the binary classification used in control charts by attributes. A generic approach for constructing attribute control charts using linguistic data is described. Several methods for calculating the values representing sample means, and for determining the centre line and the control limits are presented. [ABSTRACT FROM AUTHOR]