3D Object Representation Research Articles

2D Representations of 3D Point Clouds Via the Stereographic Projection with Encryption Applications

Using the topological equivalence between the Riemann sphere S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {S}$$\\end{document} and the extended complex plane C¯=C∪{∞}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{\\mathbb {C}} = \\mathbb {C} \\cup \\{\\infty \\}$$\\end{document}, where C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {C}$$\\end{document} is the field of complex numbers, we establish 2D-bijective representations of 3D point clouds. Points of 3D point clouds are mapped into the Riemann sphere S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {S}$$\\end{document}, and a stereographic projection is implemented to map the points into the complex plane C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {C}$$\\end{document}. The way the 3D objects are mapped into S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {S}$$\\end{document} may be varied for various applications. To prove the accuracy and efficiency of the proposed 2D representation of 3D objects, we apply this correspondence to 3D point cloud encryption. We utilize chaotic permutations, chaotic circuits, and Latin cubes in addition to the stereographic projection representation to construct our scheme. The permutation steps using chaotic maps and Latin cubes are carried out on the object data points in both S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {S}$$\\end{document} and C¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{\\mathbb {C}}$$\\end{document}, while the chaotic circuits are applied to 2D projections of the 3D objects. To the best of our knowledge, no earlier work employed stereographic projections for 3D object encryption. Experimental simulations of this method show high encryption strength and strong confusion and diffusion properties based on quantitative and statistical measures.

Integration and evaluation of the high-precision MotionCam-3D into a 3D thermography system

Abstract. Infrared thermal imaging enables fast, accurate and non-contact measurement of temperature distributions. However, 2D representations of 3D objects often require several images to provide significant information. For such cases, 3D thermograms allow a quick temporal and spatial analysis. In this paper, the integration of an industrial high-precision 3D sensor into a 3D thermography system is presented. The performances of the existing and new systems are assessed and compared by analyzing 3D thermograms of an industry-related test object. The geometry of the obtained point cloud is evaluated by means of a non-referenced point cloud quality assessment approach. It is shown that, in the presence of the spatial resolution and the local curvature, the proposed system performs significantly better than the existing one.

The impact of presentation modes on mental rotation processing: a comparative analysis of eye movements and performance

Mental rotation is the ability to rotate mental representations of objects in space. Shepard and Metzler’s shape-matching tasks, frequently used to test mental rotation, involve presenting pictorial representations of 3D objects. This stimulus material has raised questions regarding the ecological validity of the test for mental rotation with actual visual 3D objects. To systematically investigate differences in mental rotation with pictorial and visual stimuli, we compared data of N=54\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N=54$$\\end{document} university students from a virtual reality experiment. Comparing both conditions within subjects, we found higher accuracy and faster reaction times for 3D visual figures. We expected eye tracking to reveal differences in participants’ stimulus processing and mental rotation strategies induced by the visual differences. We statistically compared fixations (locations), saccades (directions), pupil changes, and head movements. Supplementary Shapley values of a Gradient Boosting Decision Tree algorithm were analyzed, which correctly classified the two conditions using eye and head movements. The results indicated that with visual 3D figures, the encoding of spatial information was less demanding, and participants may have used egocentric transformations and perspective changes. Moreover, participants showed eye movements associated with more holistic processing for visual 3D figures and more piecemeal processing for pictorial 2D figures.

SCA-PVNet: Self-and-cross attention based aggregation of point cloud and multi-view for 3D object retrieval

To address 3D object retrieval, substantial efforts have been made to generate highly discriminative descriptors for 3D objects represented by a single modality, such as voxels, point clouds, or multiview images. It is promising to leverage complementary information from multimodal representations of 3D objects to further improve retrieval performance. However, multimodal 3D object retrieval has rarely been developed or analyzed for large-scale datasets. In this paper, we propose a self-and-cross-attention-based aggregation of point clouds and multiview images (SCA-PVNet) for 3D object retrieval. With deep features extracted from point clouds and multi-view images, we design two types of feature aggregation modules, namely the in-modality aggregation module (IMAM) and the cross-modality aggregation module (CMAM ), for effective feature fusion. IMAM leverages a self-attention mechanism to aggregate multiview features, whereas CMAM exploits a cross-attention mechanism to interact with point-cloud and multiview features. The final descriptor of a 3D object for object retrieval can be obtained by concatenating the aggregated feature outputs of both modules. Extensive experiments and analyses were conducted on four datasets ranging from small to large scales to demonstrate the superiority of the proposed SCA-PVNet over state-of-the-art methods. In addition to achieving state-of-the-art retrieval performance, our method is more robust in challenging scenarios when views or points are missing during inference.

Hypergraph-Based Multi-Modal Representation for Open-Set 3D Object Retrieval.

The traditional 3D object retrieval (3DOR) task is under the close-set setting, which assumes the categories of objects in the retrieval stage are all seen in the training stage. Existing methods under this setting may tend to only lazily discriminate their categories, while not learning a generalized 3D object embedding. Under such circumstances, it is still a challenging and open problem in real-world applications due to the existence of various unseen categories. In this paper, we first introduce the open-set 3DOR task to expand the applications of the traditional 3DOR task. Then, we propose the Hypergraph-Based Multi-Modal Representation (HGM 2 R) framework to learn 3D object embeddings from multi-modal representations under the open-set setting. The proposed framework is composed of two modules, i.e., the Multi-Modal 3D Object Embedding (MM3DOE) module and the Structure-Aware and Invariant Knowledge Learning (SAIKL) module. By utilizing the collaborative information of modalities derived from the same 3D object, the MM3DOE module is able to overcome the distinction across different modality representations and generate unified 3D object embeddings. Then, the SAIKL module utilizes the constructed hypergraph structure to model the high-order correlation among 3D objects from both seen and unseen categories. The SAIKL module also includes a memory bank that stores typical representations of 3D objects. By aligning with those memory anchors in the memory bank, the aligned embeddings can integrate the invariant knowledge to exhibit a powerful generalized capacity toward unseen categories. We formally prove that hypergraph modeling has better representative capability on data correlation than graph modeling. We generate four multi-modal datasets for the open-set 3DOR task, i.e., OS-ESB-core, OS-NTU-core, OS-MN40-core, and OS-ABO-core, in which each 3D object contains three modality representations: multi-view, point clouds, and voxel. Experiments on these four datasets show that the proposed method can significantly outperform existing methods. In particular, the proposed method outperforms the state-of-the-art by 12.12%/12.88% in terms of mAP on the OS-MN40-core/OS-ABO-core dataset, respectively. Results and visualizations demonstrate that the proposed method can effectively extract the generalized 3D object embeddings on the open-set 3DOR task and achieve satisfactory performance.

Three-dimensional shape generation via variational autoencoder generative adversarial network with signed distance function

<span lang="EN-US">Mesh-based 3-dimensional (3D) shape generation from a 2-dimensional (2D) image using a convolution neural network (CNN) framework is an open problem in the computer graphics and vision domains. Most existing CNN-based frameworks lack robust algorithms that can scale well without combining different shape parts. Also, most CNN-based algorithms lack suitable 3D data representations that can fit into CNN without modification(s) to produce high-quality 3D shapes. This paper presents an approach that integrates a variational autoencoder (VAE) and a generative adversarial network (GAN) called 3 dimensional variational autoencoder signed distance function generative adversarial network (3D-VAE-SDFGAN) to create a 3D shape from a 2D image that considerably improves scalability and visual quality. The proposed method only feeds a single 2D image into the network to produce a mesh-based 3D shape. The network encodes a 2D image of the 3D object into the latent representations, and implicit surface representations of 3D objects corresponding to those 2D images are subsequently generated. Hence, a signed distance function (SDF) is proposed to maintain object inside-outside information in the implicit surface representation. Polygon mesh surfaces are then produced using the marching cubes algorithm. The ShapeNet dataset was used in the experiments to evaluate the proposed 3D-VAE-SDFGAN. The experimental results show that 3D-VAE-SDFGAN outperforms other state-of-the-art models.</span>

Using Mixed Reality for Iterative Sketching and Prototyping in Product Design

Abstract: Mixed reality (MR) integrates virtual objects and environments with the real-world using sensors, cameras, and displays. Unlike virtual reality, which is completely immersive, MR allows users to interact with virtual objects while still being aware of the physical environment. This technology can be applied to education, healthcare, entertainment, and product design. MR can simulate user experiences, allowing designers to test the usability and functionality of their designs before they are manufactured. In education, MR provides immersive learning experiences that allow students to explore and interact with virtual objects in a real-world context. For example, MR can create interactive educational simulations that enable students to explore complex scientific concepts or historical events. In healthcare, MR simulates medical procedures, allowing medical professionals to practice and improve their skills in a safe and controlled environment. MR can also be used for patient education, allowing patients to visualize and better understand medical conditions and procedures. Overall, MR offers a unique opportunity to integrate virtual and real-world environments, providing a powerful tool for various industries. A visual interface is a computerized representation of a user interface that may or may not correspond to the physical interface controller. Virtual interface designs are static in nature, representing signals used to connect design modules and physical products. With the advent of the internet and its technologies, newer possibilities for remote access to technology have emerged, such as the concept of virtual reality. Virtual reality uses graphical displays to represent objects or scenes in three dimensions, making it easier for users to visualize real-world objects, scenes, or processes. This technology combines visual representations of 3D elements and objects, such as polygons and surfaces, to create a computer-generated scene that resembles a real environment. This scene is known as virtual reality, and it is often projected as what is popularly known as a "virtual world.

MixFormer: A Self-Attentive Convolutional Network for 3D Mesh Object Recognition

3D mesh as a complex data structure can provide effective shape representation for 3D objects, but due to the irregularity and disorder of the mesh data, it is difficult for convolutional neural networks to be directly applied to 3D mesh data processing. At the same time, the extensive use of convolutional kernels and pooling layers focusing on local features can cause the loss of spatial information and dependencies of low-level features. In this paper, we propose a self-attentive convolutional network MixFormer applied to 3D mesh models. By defining 3D convolutional kernels and vector self-attention mechanisms applicable to 3D mesh models, our neural network is able to learn 3D mesh model features. Combining the features of convolutional networks and transformer networks, the network can focus on both local detail features and long-range dependencies between features, thus achieving good learning results without stacking multiple layers and saving arithmetic overhead compared to pure transformer architectures. We conduct classification and semantic segmentation experiments on SHREC15, SCAPE, FAUST, MIT, and Adobe Fuse datasets. Experimental results show that the network can achieve 96.7% classification and better segmentation results by using fewer parameters and network layers.

Disentangling Geometric Deformation Spaces in Generative Latent Shape Models

A complete representation of 3D objects requires characterizing the space of deformations in an interpretable manner, from articulations of a single instance to changes in shape across categories. In this work, we improve on a prior generative model of geometric disentanglement for 3D shapes, wherein the space of object geometry is factorized into rigid orientation, non-rigid pose, and intrinsic shape. The resulting model can be trained from raw 3D shapes, without correspondences, labels, or even rigid alignment, using a combination of classical spectral geometry and probabilistic disentanglement of a structured latent representation space. Our improvements include more sophisticated handling of rotational invariance and the use of a diffeomorphic flow network to bridge latent and spectral space. The geometric structuring of the latent space imparts an interpretable characterization of the deformation space of an object. Furthermore, it enables tasks like pose transfer and pose-aware retrieval without requiring supervision. We evaluate our model on its generative modelling, representation learning, and disentanglement performance, showing improved rotation invariance and intrinsic-extrinsic factorization quality over the prior model.

Assessment of 3D Geoportals of Cities According to CityGML Standard Guidelines

Along with the development of three-dimensional computer graphics, methods of collecting and making 3D spatial data available became a significant issue covering the interoperability of data derived from multiple sources. Between 2006 and 2008, the Open Geospatial Consortium designed a CityGML model as a proposal for a uniform classification, graphic representation, construction, and storage of 3D objects. A considerable part of three-dimensional visualisations, now gaining popularity, make use of solutions based on the CityGML standard, with which they are compatible to various degrees. The survey involved a comprehensive analysis of sixteen generally accessible 3D geoportals of cities in Europe, Asia, and North America in terms of their broad-sense functionality as well as technical and thematic compatibility with the assumptions of CityGML standards. The level of realisation of various features related to the provided spatial data services was evaluated, taking into account elements that the present-day world literature deems to be particularly desirable. The analysis resulted in an elaborate ranking of websites according to 21 criteria. The most common objects and features of the analysed geoportals were also detailed. In addition, the authors presented several solutions to improve the quality of three-dimensional geoportals of cities by implementing external data from various sources.

3D Model Retrieval Algorithm Based on DSP-SIFT Descriptor and Codebook Combination

Recently, extensive research efforts have been dedicated to view-based 3D object retrieval, owing to its advantage of using a set of 2D images to represent 3D objects. Some existing image processing technologies can be employed. In this paper, we adopt Bag-of-Words for view-based 3D object retrieval. Instead of SIFT, DSP-SIFT is extracted from all images as object features. Moreover, two codebooks of the same size are generated by approximate k-means. Then, we combine two codebooks to correct the quantization artifacts and improve recall. Bayes merging is applied to address the codebook correlation (overlapping among different vocabularies) and to provide the benefit of high recall. Moreover, Approximate Nearest Neighbor (ANN) is used to quantization. Experimental results on ETH-80 datasets show that our method improves the performance significantly compared with the state-of-the-art approaches.

Variable Rate Point Cloud Attribute Compression with Non-Local Attention Optimization

Point clouds are widely used as representations of 3D objects and scenes in a number of applications, including virtual and mixed reality, autonomous driving, antiques reconstruction. To reduce the cost for transmitting and storing such data, this paper proposes an end-to-end learning-based point cloud attribute compression (PCAC) approach. The proposed network adopts a sparse convolution-based variational autoencoder (VAE) structure to compress the color attribute of point clouds. Considering the difficulty of stacked convolution operations in capturing long range dependencies, the attention mechanism is incorporated in which a non-local attention module is developed to capture the local and global correlations in both spatial and channel dimensions. Towards the practical application, an additional modulation network is offered to achieve the variable rate compression purpose in a single network, avoiding the memory cost of storing multiple networks for multiple bitrates. Our proposed method achieves state-of-the-art compression performance compared to other existing learning-based methods and further reduces the gap with the latest MPEG G-PCC reference software TMC13 version 14.

Visibility Detection of 3D Objects and Visual K-Nearest Neighbor Query Based on Convex Hull Model

With the rapid development of the technologies of virtual reality (VR) and augmented reality (AR), the accurate visual query of 3D objects which are closer to real world has become a hotspot of study. There are a lot of redundancy problems in the existing axis-aligned bounding box model (AABB) and oriented bounding box model (OBB), which are used to represent 3D objects mostly at present, and cannot be used for the accurate visual query. For example, the results of the visual query must be accurate, and without any error in automatic driving technology in the automotive industry, otherwise, serious safety problems would be caused. The convex hull model could be used to solve those problems. However, there are few studies on visual query based on the convex hull model. A visual query method for 3D obstacle space based on the convex hull model is proposed in this paper. Firstly, the definitions of “query point-object visual body” and “combined obstacle” are proposed. Secondly, the visibility detection algorithm of the convex hull model is given in detail. The collision detection method of “query point-object visible body” combined with other obstacles and line-plane geometric calculation operations are used to detect the visibility of the convex hull model. Then, a 3D visual k-nearest neighbor algorithm based on the convex hull model is given in the paper. Finally, the algorithm is verified by experiments and compared with the traditional visibility detection algorithm, and the analysis of experimental results shows that the algorithm has a great performance and higher accuracy. The growth rate of the query time is smaller, and the speed is faster; especially, when there are fewer query points, the query speed can be increased by more than 50%.

Visual working memory for connected 3D objects: effects of stimulus complexity, dimensionality and connectivity

Visual working memory (VWM) is typically measured using arrays of two-dimensional isolated stimuli with simple visual identities (e.g., color or shape), and these studies typically find strong capacity limits. Science, technology, engineering and mathematics (STEM) experts are tasked with reasoning with representations of three-dimensional (3D) connected objects, raising questions about whether those stimuli would be subject to the same limits. Here, we use a color change detection task to examine working memory capacity for 3D objects made up of differently colored cubes. Experiment 1a shows that increasing the number of parts of an object leads to less sensitivity to color changes, while change-irrelevant structural dimensionality (the number of dimensions into which parts of the structure extend) does not. Experiment 1b shows that sensitivity to color changes decreases similarly with increased complexity for multipart 3D connected objects and disconnected 2D squares, while sensitivity is slightly higher with 3D objects. Experiments 2a and 2b find that when other stimulus characteristics, such as size and visual angle, are controlled, change-irrelevant dimensionality and connectivity have no effect on performance. These results suggest that detecting color changes on 3D connected objects and on displays of isolated 2D stimuli are subject to similar set size effects and are not affected by dimensionality and connectivity when these properties are change-irrelevant, ruling out one possible explanation for scientists’ advantages in storing and manipulating representations of complex 3D objects.

Holographic augmented reality display with conical holographic optical element for wide viewing zone

In this study, we propose a holographic augmented reality (AR) display with a wide viewing zone realized by using a special-designed reflective optical element. A conical holographic optical element (HOE) is used as such a reflective optical element. This conical HOE was implemented to reconstruct a diverging spherical wave with a wide spread angle. It has a sharp wavelength selectivity by recording it as a volume hologram, enabling augmented reality (AR) representation of real and virtual 3D objects. The quality of the generated spherical wave and the spectral reflectivity of the fabricated conical HOE were investigated. An optical superimposition between real and virtual 3D objects was demonstrated, thereby enhancing the validity of our proposed method. A horizontal viewing zone of 140° and a vertical viewing zone of 30° were experimentally confirmed. The fabrication procedure for the conical HOE is presented, and the calculation method of the computer-generated hologram (CGH) based on Fermat’s principle is explained in detail.

Bias-Induced Point Auto-Encoder and Comparative Analysis of Point Encoder-Decoder Combinations

As a combination of a point encoder and a decoder, a point auto-encoder (AE) facilitates the reconstruction and segmentation of the point cloud representation of 3D objects. In this paper, first, we present a novel approach to point decoding, referred to here as bias-induced point decoding, which learns pointwise decoding of the same global feature by the layer-wise control of mapping biases while sharing mapping weights, instead of pointwise differentiating the input by concatenating local features or grids to the global feature as is done conventionally. Then, in-depth comparative analyses of 12 possible encoder-decoder combinations are conducted to identify the strengths and weaknesses in terms of accuracy and robustness in reconstruction and segmentation. We verified that a bias-induced point AE could be a viable alternative to conventional point AEs with its own strength and weakness. Finally, the effectiveness of the proposed bias-induced point AE is demonstrated based on its application to the automatic transformation from partial-to-full 3D point cloud representations.

SIEV-Net: A Structure-Information Enhanced Voxel Network for 3D Object Detection From LiDAR Point Clouds

As one of the fundamental tasks in scene understanding, 3D object detection from LiDAR point clouds has drawn extensive attention in the past few years. Although the existing voxel-based methods have achieved remarkable performance, how to effectively exploit geometric structure information of the point clouds to boost the detection performance remains to be explored. In this paper, we propose a novel structure-information enhanced voxel network (SIEV-Net) for 3D object detection from LiDAR point clouds. The proposed SIEV-Net learns feature representations of 3D objects by jointly considering uneven spatial distribution and height information of the point clouds. Specifically, considering the uneven spatial distribution characteristics of point clouds, a hierarchical-voxel feature encoding module is proposed to effectively extract features of voxels in both sparse and dense regions. Besides, by utilizing the Bird’s Eye View (BEV) map of point clouds, a height information complement module is designed to minimize the height information lost in the process of point feature aggregation in a voxel network. Experimental results on the widely used KITTI benchmark dataset have demonstrated the efficacy of the proposed SIEV-Net.

3D TOPOLOGICAL SUPPORT IN SPATIAL DATABASES: AN OVERVIEW

Abstract. The storage of spatial data that consists of spatial and non-spatial properties requires a database management system that possesses spatial functions that can cater to the spatial characteristics of data. These characteristics include the geometrical shape, topological and positional information. Parallel to how geometries describe the shape of an object, topological information is also an important spatial property which describes how the geometries in a space are related to each other. This information describes the connectivity, containment and adjacencies of spatial objects which are the foundation for more complex analysis such as navigation, data reconstruction, spatial queries and others. However, the topological support provided by spatial databases varies. This paper provided an overview on the current implementations of topological support in spatial databases such as ArcGIS, QGIS, PostgreSQL and others. The native topology in most spatial databases was found to be 2D topology maintained by 2D topology rules with limited representation of 3D topological relationships. Consequently, 3D objects represented by 2D topology had to be decomposed into objects of lower dimensions. Approaches to implement additional topological support for spatial databases included the use of topological data models, data structures, operators, and rules. 3D applications such as 3D cadastre required more detailed representations of topological information which required a more comprehensive 3D topological data model. Nonetheless, comprehensive preservation of topological information also mandates voluminous storage and higher computational efficiency. Thus, the appropriate 3D topological support should be provided in spatial databases to accurately represent 3D objects and meet 3D analysis requirements.

Iso-scallop tool path planning for triangular mesh surfaces in multi-axis machining

• A robust and universal isoscallop tool path planning method for multi-axis machining is proposed. • The tool path with the constant scallop height is generated from the first contour of a normalized geodesic distance field (GDF) on triangular mesh surfaces. • The proposed algorithm starts from the boundary curve and runs forward recursively to ensure that the surface is covered completely by the isoscallop tool path. • The feasibility and superiority are studied by several simulations and experiments. Triangular mesh enables the flexible construction of complex surface geometry and has become a general representation of 3D objects in computer graphics. However, the creation of a tool path with constant residual scallop height on triangular mesh surfaces in multi-axis machining is not a convenient task for current algorithms. In this study, an isoscallop tool path planning method for triangular mesh surfaces, in which the tool path is derived directly from the contours of a normalized geodesic distance field (GDF), without any post-processing is proposed. First, the GDF is built to determine the shortest geodesic distance from each vertex to the mesh boundary. Then, the normalizing process is performed on the GDF to ensure that its first contour meets the isoscallop height requirement considering the mesh curvature and effective cutter radius. To improve the computational efficiency, the GDF is only built in the mesh area related to the first contour by specifying a stop distance. Moreover, an adaptive refinement process is conducted on the mesh to improve the smoothness and accuracy of the tool path. Finally, the triangular mesh is trimmed along this first contour for a new round of tool path planning. The proposed method is organized recursively and terminated when no new paths are generated. Simulations and experiments are conducted to verify the effectiveness and superiority of the proposed tool path planning method.

Visualization of Archimedian and Platonic polyhedra using a web environment in Augmented Reality and Virtual Reality

This paper shows the development of a web environment for the construction of Archimedes and Plato polyhedra in Augmented Reality (AR) and Virtual Reality (VR). In this environment we used the geometric transformations of translation and rotation with the structure of hierarchies of HTML pages, without the use of the coordinates of each polyhedra vertex. The developed environment can be used in classroom to visualize the polyhedra in Augmented Reality, with the possibility of manipulations of the graphical representations by students in the environment created in Virtual Reality. Other studies that can be developed with the polyhedra modeled are areas, volumes and the relation of Euler. Another important content that can be developed is truncation, because seven Archimedes polyhedra are obtained by using truncation of Plato's polyhedrons. With this work, it becomes possible to develop didactic materials with a simple technology, free and with great contribution to improvement of the teaching of Geometry and other areas that use representation of 3D objects.