Representations Of Geometric Shapes Research Articles

A survey on geometric shape representation of objects based on medial axis transform

<div><p>Geometric shape representation algorithms are key technologies in the fields of computer graphics and geometric modeling. The Medial Axis Transform (MAT) is an important geometric model description tool that provides a simplified representation of complex geometric shapes while ensuring accurate descriptions of geometric shape and topology. Therefore, it can meet the requirements of many modern research fields, including geometric modeling, pattern recognition, model segmentation, model deformation, physical simulation, path planning, and more. This paper first introduces the basic concept of the medial axis transform, including the definition of the medial axis transform and the concept of medial axis primitives. It then describes the extraction algorithms for the medial axis transform, specific research on the medial axis transform in computer vision and computer graphics, potential applications of the medial axis transform, and medial axis transform datasets. Finally, the disadvantages and advantages of the medial axis transform are discussed, and some suggestions on possible future research directions are presented.</p></div>

Some Conceptual Metaphors for Rational Numbers as Fractions in Swedish Mathematics Textbooks for Elementary Education

ABSTRACT This study examines the presence of three conceptual metaphors for fractions – The measuring stick metaphor, Arithmetic is motion along a path, and Arithmetic is object construction – in four common and popular Swedish mathematics textbook series for years 1–3. I analyse the introduction of fractions and the kinds of tasks students are given in these books. The results show an abundance of labelling exercises related to Arithmetic is object construction, with representations of geometric shapes, in three of the book series. One book series reversely introduces fractions with The measuring stick metaphor and Arithmetic is motion along a path, with a focus on number line representations. The consequences of these variations in fraction introduction and treatment are discussed in relation to previous research and to process-object theories.

An equivalent parameter geometric shape representation using independent coordinates of cubic Bézier control points

Bézier surface has been commonly applied to represent the complex geometric shape. Generally, all control points are dealt with by the same blending functions, regardless of the effect of independent coordinate. It causes to lack the modeling flexibility. Therefore, this paper proposes an equivalent parameter geometric shape representation method using the independent coordinates of control points. Since the coordinate components of control points are independent, the geometric modeling becomes more flexible. Firstly, a general Bézier curve is described in detail. Related expression is brought out in the form of independent coordinates by introducing two parameters. Then, their geometric meanings are analyzed in detail. Since both parameters are independent to parametric variables u and v, Bézier curve possess the same interval in the discrete parametric space, namely equivalent parameter. Next, a bicubic Bézier subsurface patch representation is discussed, including regular and non-regular subsurface patch. A general surface expression is given out in the form of independent coordinates, as well as the parameter structure and the geometric transformations. Finally, an example of ‘Bézier tree branch’ is constructed by using the proposed method. Results shows that the proposed method is feasible and reasonable.

Διερεύνηση γεωμετρικών γνώσεων μαθητών με ήπιες εκπαιδευτικές ανάγκες

Geometry is a structural component of mathematics, with increased spatial and design requirements that cannot be easily met by students with mild disabilities. Systematic investigation of the difficulties encountered by students with mild disabilities in their effort to learn Geometry is a prerequisite for the implementation of effective intervention programs. However, research on this issue is relatively scarce. The aim of the present study was to assess the geometric knowledge of 54 students with mild disabilities (learning disabilities or ADHD) who attended the two last classes of elementary school. Participants were asked to recognize, describe and categorize geometric shapes and solid bodies that were presented in tactile mode and through pictorial representations. Semi-structured clinical interviews were used for gathering the data in the context of Curriculum Based Assessment and the Van Hiele’s model of geometrical thinking. Participants of both categories of mild disabilities presented difficulties in distinguishing shapes and bodies, properly using the terminology, and formatting inductive geometrical reasoning. Participants with learning disabilities had higher achievement when dealing with haptic relative to pictorial representations of geometric shapes and bodies. Sixth graders performed better than fifth graders. Results are discussed in terms of the differences between the two categories of mild disabilities as well as with regard to the implementation of intervention programs.

An innovative determination approach to tooth compliance for spiral bevel and hypoid gears by using double-curved shell model and Rayleigh–Ritz approach

In high-performance sophisticated flank design for the spiral bevel and hypoid gears, tooth compliance is always an important input for identifying the tooth mechanical properties. Distinguished with the conventional modeling, a new accurate finite element structure model is established to determine tooth compliance by using elasticity-based deformation solution. In full consideration of the flank flexural behavior characteristic, a double-curved shell model with varying thickness is established to get a more accurate representation of geometric shape for face-milling or face-hobbing gear than a beam or plate model. In determination of tooth compliance, the high order shear theory of Bhimaraddi shell is used to set the displacement assumption, and the Rayleigh–Ritz approach having algebraic polynomial trail functions is performed to obtain an expression for the transverse deflection and shear rotation by considering the geometric boundary conditions. A given numerical result for the face-milling spiral bevel and hypoid gear is provided to verify the proposed methodology by comparing with the loaded tooth contact analysis (LTCA) using well-known finite element method (FEM).

Conformal-Based Surface Morphing and Multi-Scale Representation

This paper presents two algorithms, based on conformal geometry, for the multi-scale representations of geometric shapes and surface morphing. A multi-scale surface representation aims to describe a 3D shape at different levels of geometric detail, which allows analyzing or editing surfaces at the global or local scales effectively. Surface morphing refers to the process of interpolating between two geometric shapes, which has been widely applied to estimate or analyze deformations in computer graphics, computer vision and medical imaging. In this work, we propose two geometric models for surface morphing and multi-scale representation for 3D surfaces. The basic idea is to represent a 3D surface by its mean curvature function, H, and conformal factor function λ, which uniquely determine the geometry of the surface according to Riemann surface theory. Once we have the (λ, H) parameterization of the surface, post-processing of the surface can be done directly on the conformal parameter domain. In particular, the problem of multi-scale representations of shapes can be reduced to the signal filtering on the λ and H parameters. On the other hand, the surface morphing problem can be transformed to an interpolation process of two sets of (λ, H) parameters. We test the proposed algorithms on 3D human face data and MRI-derived brain surfaces. Experimental results show that our proposed methods can effectively obtain multi-scale surface representations and give natural surface morphing results.

Shape Error Concealment Based on a Shape-Preserving Boundary Approximation

In objectbased video representation, video scenes are composed of several arbitrarily shaped video objects (VOs), defined by their texture, shape and motion. In errorprone communications, packet loss results in missing information at the decoder. The impact of transmission errors is minimised through error concealment. In this paper, we propose a spatial error concealment technique for recovering lost shape data. We consider a geometric shape representation consisting of the object boundary, which can be extracted from the -plane. Missing macroblocks result in a broken boundary. A Bspline curve is constructed to replace a missing boundary segment, based on a T spline representation of the received boundary. We use Tsplines because they produce shapepreserving approximations and do not change the characteristics of the original boundary. The representation ensures a good estimation of the first derivatives at the points touching the missing segment. Applying smoothing conditions, we manage to construct a new spline that joins smoothly with the received boundary, leading to successful concealment results. Experimental results on object shapes with different concealment difficulty demonstrate the performance of the proposed method. Comparisons with prior proposed methods are also presented.

A putative model of multisensory object representation.

This review surveys the recent literature on visuo-haptic convergence in the perception of object form, with particular reference to the lateral occipital complex (LOC) and the intraparietal sulcus (IPS) and discusses how visual imagery or multisensory representations might underlie this convergence. Drawing on a recent distinction between object- and spatially-based visual imagery, we propose a putative model in which LOtv, a subregion of LOC, contains a modality-independent representation of geometric shape that can be accessed either bottom-up from direct sensory inputs or top-down from frontoparietal regions. We suggest that such access is modulated by object familiarity: spatial imagery may be more important for unfamiliar objects and involve IPS foci in facilitating somatosensory inputs to the LOC; by contrast, object imagery may be more critical for familiar objects, being reflected in prefrontal drive to the LOC.

Adaptive Shape-Sensitive Meshing of the Medial Axis

The medial axis transform provides an alternative representation of geometric shape that has many useful properties for analysis modeling. Applications include decomposition of general solids into subregions for mapped meshing, identification of slender regions for dimensional reduction and recognition of small features for suppression. To serve these purposes effectively, it is important to be able to mesh the medial axis so that its geometry is adequately approximated. This paper describes a general idea, which is based on equal distance criteria, for adaptive mesh refinement on the medial axis, assuming its topology has been defined. The completed set of theories and examples for 2D planar objects and 3D solid objects is presented.

Creativity in Drawings of Geometric Shapes

This study examines whether European Americans and Chinese differ in their creation and evaluation of drawings of geometric shapes. Two hundred ninety-four drawings created by 50 European American and 48 Chinese college students were selected from a larger study of culture and creativity. Drawings were judged by eight Chinese and six European Americans following the Consensual Assessment Technique. The drawings were coded by content to examine what the judges considered creative. Results showed high consensus between European American and Chinese judges and great similarity in the creativity of drawings generated by the two groups. Judges liked best those drawings they judged more creative. The most creative drawings typically involved representations of geometric shapes in contexts (either concrete or abstract). Results run counter to the belief that there are wide cultural variations in the evaluation of and attitudes toward creativity, demonstrate the feasibility of cross-cultural comparisons with the Consensual Assessment Technique, and provide a basis for further cross-cultural research on creativity.

Directed Edges—A Scalable Representation for Triangle Meshes

Abstract In a broad range of computer graphics applications, the representation of geometric shape is based on triangle meshes. General purpose data structures for polygonal meshes typically provide fast access to geometric objects (e.g., points) and topological entities (e.g., neighborhood relationships) but the memory requirements are rather high due to the many special configurations. In this paper, we present a new data structure which is specifically designed for triangle meshes. The data structure enables the programmer to trade memory for access time by either storing internal references explicitly, or by reconstructing them locally on demand. The trade-off can be hidden from the programmer by an object-oriented API and can automatically adapt to the available hardware resources or the complexity of the mesh (scalability).

Some Approaches to Machine Parts Modelling Using CAD Software

CAD software is a universal design tool for geometric shape representation, variation of formed shapes and their positions, computation, analysis, and production of such objects. The main objective of this paper is to suggest more effective methods for machine parts modelling using CAD software. This analysis refers to the design phase in which abstract functions have to be transformed into concrete machine parts shapes. Machine parts are classified according to the modelling capability of each part. All discussions and representations refer to three-dimensional models, which are obtained by using existing commercial CAD programs. The main characteristics of the applied software are described in general.

Hyperquadrics: Smoothly deformable shapes with convex polyhedral bounds

We present a new approach to the problem of modeling smoothly deformable shapes with convex polyhedral bounds. Our hyperquadric modeling primitives, which include superquadrics as a special case, can be viewed as hyperplanar slices of deformed hyperspheres. As the original hypersphere is deformed to its bounding hypercube, the slices undergo corresponding smooth deformations to convex polytopes. The possible shape classes include arbitrary convex polygons and polyhedra, as well as taperings and distortions that are not naturally included within the conventional superquadric framework. By generalizing Blinn's “blobby” approach to modeling complex objects, we construct single equations for nonconvex, composite shapes starting with our basic convex primitives. Hyperquadrics are of potential interest for the generation of synthetic images, for automated image interpretation and for psychological models of geometric shape representation, manipulation, and perception.