{"title":"A point-normal interpolatory subdivision scheme preserving conics","authors":"Niels Bügel , Lucia Romani , Jiří Kosinka","doi":"10.1016/j.cagd.2024.102347","DOIUrl":"10.1016/j.cagd.2024.102347","url":null,"abstract":"<div><p>The use of subdivision schemes in applied and real-world contexts requires the development of conceptually simple algorithms that can be converted into fast and efficient implementation procedures. In the domain of interpolatory subdivision schemes, there is a demand for developing an algorithm capable of (i) reproducing all types of conic sections whenever the input data (in our case point-normal pairs) are arbitrarily sampled from them, (ii) generating a visually pleasing limit curve without creating unwanted oscillations, and (iii) having the potential to be naturally and easily extended to the bivariate case. In this paper we focus on the construction of an interpolatory subdivision scheme that meets all these conditions simultaneously. At the centre of our construction lies a conic fitting algorithm that requires as few as four point-normal pairs for finding new edge points (and associated normals) in a subdivision step. Several numerical results are included to showcase the validity of our algorithm.</p></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"111 ","pages":"Article 102347"},"PeriodicalIF":1.5,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0167839624000815/pdfft?md5=73e6c19a1540d186507b4ba5fb902a3e&pid=1-s2.0-S0167839624000815-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141132176","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Splines on manifolds: A survey","authors":"Claudio Mancinelli, Enrico Puppo","doi":"10.1016/j.cagd.2024.102349","DOIUrl":"https://doi.org/10.1016/j.cagd.2024.102349","url":null,"abstract":"<div><p>Splines in the manifold setting have been defined as extensions from the standard Euclidean setting, but they are far more complicated. Alternative approaches, which are equivalent in the Euclidean case, lead to different results in the manifold case; the existence conditions are often quite restrictive; and the necessary computations are rather involved. All difficulties stem from the peculiar nature of the geodesic distance: in general, shortest geodesics may be not unique and the dependence on their endpoints may not be smooth; and distances cannot be computed in closed form. The former issue may impose strong limitations on the placement of control points. While the latter may greatly complicate the computations. Nevertheless, some recent results suggest that splines on surfaces may have practical impact on CAGD applications. We review the literature on this topic, accounting for both theoretical results and practical implementations.</p></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"112 ","pages":"Article 102349"},"PeriodicalIF":1.5,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0167839624000839/pdfft?md5=41f1c9fe29c029db48fc5c7dba197bf1&pid=1-s2.0-S0167839624000839-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141241867","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Singularity distance computations for 3-RPR manipulators using intrinsic metrics","authors":"Aditya Kapilavai, Georg Nawratil","doi":"10.1016/j.cagd.2024.102343","DOIUrl":"https://doi.org/10.1016/j.cagd.2024.102343","url":null,"abstract":"<div><p>Avoiding singularities is a crucial task in robotics and path planning. This paper proposes a novel algorithm for detecting the closest singularity to a given pose for nine interpretations of the 3-RPR manipulator. The algorithm utilizes intrinsic metrics based on the framework's total elastic strain energy density, employing the physical concept of Green-Lagrange strain. The constrained optimization problem for detecting the closest singular configuration with respect to these metrics is solved globally using tools from numerical algebraic geometry implemented in the software package <span>Bertini</span>. The effectiveness of the proposed algorithm is demonstrated on a 3-RPR manipulator executing a one-parametric motion. Additionally, the obtained intrinsic singularity distances are compared with extrinsic metrics. Finally, the paper illustrates the advantage of employing a well-defined metric for identifying the closest singularities in comparison with the existing methods in the literature, highlighting its application in design optimization.</p></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"111 ","pages":"Article 102343"},"PeriodicalIF":1.5,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0167839624000773/pdfft?md5=7aaa62823b27cbe61f490d7566678c2e&pid=1-s2.0-S0167839624000773-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141095077","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Michal Bizzarri , Miroslav Lávička , Jan Vršek , Michael Bartoň , Jiří Kosinka
{"title":"On tiling spherical triangles into quadratic subpatches","authors":"Michal Bizzarri , Miroslav Lávička , Jan Vršek , Michael Bartoň , Jiří Kosinka","doi":"10.1016/j.cagd.2024.102344","DOIUrl":"https://doi.org/10.1016/j.cagd.2024.102344","url":null,"abstract":"<div><p>Various interpolation and approximation methods arising in several practical applications in geometric modeling deal, at a particular step, with the problem of computing suitable rational patches (of low degree) on the unit sphere. Therefore, we are concerned with the construction of a system of spherical triangular patches with prescribed vertices that globally meet along common boundaries. In particular, we investigate various possibilities for tiling a given spherical triangular patch into quadratically parametrizable subpatches. We revisit the condition that the existence of a quadratic parameterization of a spherical triangle is equivalent to the sum of the interior angles of the triangle being <em>π</em>, and then circumvent this limitation by studying alternative scenarios and present constructions of spherical macro-elements of the lowest possible degree. Applications of our method include algorithms relying on the construction of (interpolation) surfaces from prescribed rational normal vector fields.</p></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"111 ","pages":"Article 102344"},"PeriodicalIF":1.5,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141090078","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Michelangelo Marsala , Carla Manni , Hendrik Speleers
{"title":"Maximally smooth cubic spline quasi-interpolants on arbitrary triangulations","authors":"Michelangelo Marsala , Carla Manni , Hendrik Speleers","doi":"10.1016/j.cagd.2024.102348","DOIUrl":"10.1016/j.cagd.2024.102348","url":null,"abstract":"<div><p>We investigate the construction of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> cubic spline quasi-interpolants on a given arbitrary triangulation <span><math><mi>T</mi></math></span> to approximate a sufficiently smooth function <em>f</em>. The proposed quasi-interpolants are locally represented in terms of a simplex spline basis defined on the cubic Wang–Shi refinement of the triangulation. This basis behaves like a B-spline basis within each triangle of <span><math><mi>T</mi></math></span> and like a Bernstein basis for imposing smoothness across the edges of <span><math><mi>T</mi></math></span>. Any element of the cubic Wang–Shi spline space can be uniquely identified by considering a local Hermite interpolation problem on every triangle of <span><math><mi>T</mi></math></span>. Different <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> cubic spline quasi-interpolants are then obtained by feeding different sets of Hermite data to this Hermite interpolation problem, possibly reconstructed via local polynomial approximation. All the proposed quasi-interpolants reproduce cubic polynomials and their performance is illustrated with various numerical examples.</p></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"112 ","pages":"Article 102348"},"PeriodicalIF":1.5,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0167839624000827/pdfft?md5=3323aa8370f60e90cbc4a16e5bad56ff&pid=1-s2.0-S0167839624000827-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141133010","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Zhenyu Zhang , Mingyang Zhao , Zeyu Shen , Yuqing Wang , Xiaohong Jia , Dong-Ming Yan
{"title":"Interactive reverse engineering of CAD models","authors":"Zhenyu Zhang , Mingyang Zhao , Zeyu Shen , Yuqing Wang , Xiaohong Jia , Dong-Ming Yan","doi":"10.1016/j.cagd.2024.102339","DOIUrl":"https://doi.org/10.1016/j.cagd.2024.102339","url":null,"abstract":"<div><p>Reverse engineering <em>Computer-Aided Design</em> (CAD) models based on the original geometry is a valuable and challenging research problem that has numerous applications across various tasks. However, previous approaches have often relied on excessive manual interaction, leading to limitations in reconstruction speed. To mitigate this issue, in this study, we approach the reconstruction of a CAD model by sequentially constructing geometric primitives (such as vertices, edges, loops, and faces) and performing Boolean operations on the generated CAD modules. We address the complex reconstruction problem in four main steps. Firstly, we use a plane to cut the input mesh model and attain a loop cutting line, ensuring accurate normals. Secondly, the cutting line is automatically fitted to edges using primitive information and connected to form a primitive loop. This eliminates the need for time-consuming manual selection of each endpoint and significantly accelerates the reconstruction process. Subsequently, we construct the loop of primitives as a chunked CAD model through a series of CAD procedural operations, including <em>extruding, lofting, revolving, and sweeping</em>. Our approach incorporates an automatic height detection mechanism to minimize errors that may arise from manual designation of the extrusion height. Finally, by merging Boolean operations, these CAD models are assembled together to closely approximate the target geometry. We conduct a comprehensive evaluation of our algorithm using a diverse range of CAD models from both the Thingi10K dataset and real-world scans. The results validate that our method consistently delivers accurate, efficient, and robust reconstruction outcomes while minimizing the need for manual interactions. Furthermore, our approach demonstrates superior performance compared to competing methods, especially when applied to intricate geometries.</p></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"111 ","pages":"Article 102339"},"PeriodicalIF":1.5,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140951770","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Investigations of a functional version of a blending surface scheme for regular data interpolation","authors":"Stephen Mann","doi":"10.1016/j.cagd.2024.102345","DOIUrl":"10.1016/j.cagd.2024.102345","url":null,"abstract":"<div><p>This paper describes an implementation and tests of a blending scheme for regularly sampled data interpolation, and in particular studies the order of approximation for the method. This particular implementation is a special case of an earlier scheme by Fang for fitting a parametric surface to interpolate the vertices of a closed polyhedron with <em>n</em>-sided faces, where a surface patch is constructed for each face of the polyhedron, and neighbouring faces can meet with a user specified order of continuity. The specialization described in this paper considers functions of the form <span><math><mi>z</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> with the patches meeting with <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> continuity. This restriction allows for investigation of order of approximation, and it is shown that the functional version of Fang's scheme has polynomial precision.</p></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"111 ","pages":"Article 102345"},"PeriodicalIF":1.5,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0167839624000797/pdfft?md5=1305b2062d4430bed826c2aa0d0b8ef1&pid=1-s2.0-S0167839624000797-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141040547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Hermite subdivision schemes for manifold-valued Hermite data","authors":"Hofit Ben-Zion Vardi, Nira Dyn, Nir Sharon","doi":"10.1016/j.cagd.2024.102342","DOIUrl":"https://doi.org/10.1016/j.cagd.2024.102342","url":null,"abstract":"<div><p>This paper introduces a family of subdivision schemes that generate curves over manifolds from manifold-Hermite data. This data consists of points and tangent directions sampled from a curve over a manifold. Using a manifold-Hermite average based on the De Casteljau algorithm as our main building block, we show how to adapt a geometric approach for curve approximation over manifold-Hermite data. The paper presents the various definitions and provides several analysis methods for characterizing properties of both the average and the resulting subdivision schemes based on it. Demonstrative figures accompany the paper's presentation and analysis.</p></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"111 ","pages":"Article 102342"},"PeriodicalIF":1.5,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141067696","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Zhihao Wang , Juan Cao , Tuan Guan , Zhonggui Chen , Yongjie Jessica Zhang
{"title":"pκ-Curves: Interpolatory curves with curvature approximating a parabola","authors":"Zhihao Wang , Juan Cao , Tuan Guan , Zhonggui Chen , Yongjie Jessica Zhang","doi":"10.1016/j.cagd.2024.102330","DOIUrl":"https://doi.org/10.1016/j.cagd.2024.102330","url":null,"abstract":"<div><p>This paper introduces a novel class of fair and interpolatory planar curves called <em>pκ</em>-curves. These curves are comprised of smoothly stitched Bézier curve segments, where the curvature distribution of each segment is made to closely resemble a parabola, resulting in an aesthetically pleasing shape. Moreover, each segment passes through an interpolated point at a parameter where the parabola has an extremum, encouraging the alignment of interpolated points with curvature extrema. To achieve these properties, we tailor an energy functional that guides the optimization process to obtain the desired curve characteristics. Additionally, we develop an efficient algorithm and an initialization method, enabling interactive modeling of the <em>pκ</em>-curves without the need for global optimization. We provide various examples and comparisons with existing state-of-the-art methods to demonstrate the curve modeling capabilities and visually pleasing appearance of <em>pκ</em>-curves.</p></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"111 ","pages":"Article 102330"},"PeriodicalIF":1.5,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140951771","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jiaze Li , Luo Zhang , Jiangbei Hu , Zhebin Zhang , Hongyu Sun , Gaochao Song , Ying He
{"title":"Real-time volume rendering with octree-based implicit surface representation","authors":"Jiaze Li , Luo Zhang , Jiangbei Hu , Zhebin Zhang , Hongyu Sun , Gaochao Song , Ying He","doi":"10.1016/j.cagd.2024.102322","DOIUrl":"https://doi.org/10.1016/j.cagd.2024.102322","url":null,"abstract":"<div><p>Recent breakthroughs in neural radiance fields have significantly advanced the field of novel view synthesis and 3D reconstruction from multi-view images. However, the prevalent neural volume rendering techniques often suffer from long rendering time and require extensive network training. To address these limitations, recent initiatives have explored explicit voxel representations of scenes to expedite training. Yet, they often fall short in delivering accurate geometric reconstructions due to a lack of effective 3D representation. In this paper, we propose an octree-based approach for the reconstruction of implicit surfaces from multi-view images. Leveraging an explicit, network-free data structure, our method substantially increases rendering speed, achieving real-time performance. Moreover, our reconstruction technique yields surfaces with quality comparable to state-of-the-art network-based learning methods. The source code and data can be downloaded from <span>https://github.com/LaoChui999/Octree-VolSDF</span><svg><path></path></svg>.</p></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"111 ","pages":"Article 102322"},"PeriodicalIF":1.5,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140913891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}