Computer Aided Geometric Design最新文献

{"title":"On discrete constant principal curvature surfaces","authors":"Yutaro Kabata ,&nbsp;Shigeki Matsutani ,&nbsp;Yuta Ogata","doi":"10.1016/j.cagd.2024.102289","DOIUrl":"https://doi.org/10.1016/j.cagd.2024.102289","url":null,"abstract":"<div><p>It has been recently discovered that a certain class of nanocarbon materials has geometrical properties related to the geometry of discrete surfaces with a pre-constant discrete curvature, based on a discrete surface theory for trivalent graphs proposed in 2017 by Kotani et al. In this paper, with the aim of an application to the nanocarbon materials, we will study discrete constant principal curvature (CPC) surfaces. Firstly, we develop the discrete surface theory on a full 3-ary oriented tree so that we define a discrete analogue of principal directions on them and investigate it. We also construct some interesting examples of discrete constant principal curvature surfaces, including discrete CPC tori.</p></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"111 ","pages":"Article 102289"},"PeriodicalIF":1.5,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140641084","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":"De Casteljau's algorithm in geometric data analysis: Theory and application","authors":"Martin Hanik ,&nbsp;Esfandiar Nava-Yazdani ,&nbsp;Christoph von Tycowicz","doi":"10.1016/j.cagd.2024.102288","DOIUrl":"https://doi.org/10.1016/j.cagd.2024.102288","url":null,"abstract":"<div><p>For decades, de Casteljau's algorithm has been used as a fundamental building block in curve and surface design and has found a wide range of applications in fields such as scientific computing and discrete geometry, to name but a few. With increasing interest in nonlinear data science, its constructive approach has been shown to provide a principled way to generalize parametric smooth curves to manifolds. These curves have found remarkable new applications in the analysis of parameter-dependent, geometric data. This article provides a survey of the recent theoretical developments in this exciting area as well as its applications in fields such as geometric morphometrics and longitudinal data analysis in medicine, archaeology, and meteorology.</p></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"110 ","pages":"Article 102288"},"PeriodicalIF":1.5,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0167839624000220/pdfft?md5=7e786d084d94b2bb29966ee6b63e857e&pid=1-s2.0-S0167839624000220-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140555016","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":"Quadratic surface preserving parameterization of unorganized point data","authors":"Dany Ríos,&nbsp;Felix Scholz,&nbsp;Bert Jüttler","doi":"10.1016/j.cagd.2024.102287","DOIUrl":"https://doi.org/10.1016/j.cagd.2024.102287","url":null,"abstract":"<div><p>Finding parameterizations of spatial point data is a fundamental step for surface reconstruction in Computer Aided Geometric Design. Especially the case of unstructured point clouds is challenging and not widely studied. In this work, we show how to parameterize a point cloud by using barycentric coordinates in the parameter domain, with the aim of reproducing the parameterizations provided by quadratic triangular Bézier surfaces. To this end, we train an artificial neural network that predicts suitable barycentric parameters for a fixed number of data points. In a subsequent step we improve the parameterization using non-linear optimization methods. We then use a number of local parameterizations to obtain a global parameterization using a new overdetermined barycentric parameterization approach. We study the behavior of our method numerically in the zero-residual case (i.e., data sampled from quadratic polynomial surfaces) and in the non-zero residual case and observe an improvement of the accuracy in comparison to standard methods. We also compare different approaches for non-linear surface fitting such as tangent distance minimization, squared distance minimization and the Levenberg Marquardt algorithm.</p></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"110 ","pages":"Article 102287"},"PeriodicalIF":1.5,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0167839624000219/pdfft?md5=865078c5e76334d3c4faef097f7aefe9&pid=1-s2.0-S0167839624000219-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140540812","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":"Genuine multi-sided parametric surface patches – A survey","authors":"Tamás Várady,&nbsp;Péter Salvi,&nbsp;Márton Vaitkus","doi":"10.1016/j.cagd.2024.102286","DOIUrl":"https://doi.org/10.1016/j.cagd.2024.102286","url":null,"abstract":"<div><p>A state-of-the-art survey is presented on various formulations of multi-sided parametric surface patches, with a focus on methods that interpolate positional and cross-derivative information along boundaries.</p></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"110 ","pages":"Article 102286"},"PeriodicalIF":1.5,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0167839624000207/pdfft?md5=1ae76dd042d095b1b1386551dabe55db&pid=1-s2.0-S0167839624000207-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140347995","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":"2D Bézier curves with monotone curvature based on Class A matrices","authors":"Aizeng Wang ,&nbsp;Chuan He ,&nbsp;Yang Song ,&nbsp;Gang Zhao","doi":"10.1016/j.cagd.2024.102279","DOIUrl":"10.1016/j.cagd.2024.102279","url":null,"abstract":"<div><p>In this paper, we construct 2D Bézier curves with monotone curvature using Class A matrices. A new sufficient condition for Class A matrices based on its singular values, is provided and proved, generalizing the 2D typical curves proposed by Mineur et al. (1998). An algorithm is provided utilizing the condition for easier Class A curve creation. Several 2D aesthetic curve examples are constructed to demonstrate the effectiveness of our approach.</p></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"110 ","pages":"Article 102279"},"PeriodicalIF":1.5,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140125542","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":"Paul de Casteljau: The story of my adventure","authors":"Andreas Müller","doi":"10.1016/j.cagd.2024.102278","DOIUrl":"https://doi.org/10.1016/j.cagd.2024.102278","url":null,"abstract":"<div><p>Paul de Faget de Casteljau (19.11.1930 - 24.3.2022) has left us an extensive autobiography, written in 1997. In 19 sections, he takes us through his eventful life which he describes with wit and humor. We read about his youth in occupied France and his education at the <em>Ecole Normale Supérieure</em>. He describes in detail various episodes from his time at Citroën, the situation during and after the discovery of his now famous algorithm, the takeover by Peugeot, his ban from working on CAD and his corporate rehabilitation thanks to his advances in polar forms and quaternions. His memoirs end with his departure from Citroën and his first invited talks at academic conferences.</p><p>The paper contains the transcribed French original, its English translation and numerous notes and annotations. The handwritten text is available as a digital supplement.</p></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"110 ","pages":"Article 102278"},"PeriodicalIF":1.5,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0167839624000128/pdfft?md5=5e7e0e7644924e2af69fdc54d4dca611&pid=1-s2.0-S0167839624000128-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140123111","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":"Fast evaluation of derivatives of Bézier curves","authors":"Filip Chudy ,&nbsp;Paweł Woźny","doi":"10.1016/j.cagd.2024.102277","DOIUrl":"https://doi.org/10.1016/j.cagd.2024.102277","url":null,"abstract":"<div><p>New geometric methods for fast evaluation of derivatives of polynomial and rational Bézier curves are proposed. They apply an algorithm for evaluating polynomial or rational Bézier curves, which was recently given by the authors. Numerical tests show that the new approach is more efficient than the methods which use the famous de Casteljau algorithm. The algorithms work well even for high-order derivatives of rational Bézier curves of high degrees.</p></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"109 ","pages":"Article 102277"},"PeriodicalIF":1.5,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139999361","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":"Shape-preserving interpolation on surfaces via variable-degree splines","authors":"P.D. Kaklis ,&nbsp;S. Stamatelopoulos ,&nbsp;A.-A.I. Ginnis","doi":"10.1016/j.cagd.2024.102276","DOIUrl":"https://doi.org/10.1016/j.cagd.2024.102276","url":null,"abstract":"<div><p>This paper proposes two, geodesic-curvature based, criteria for shape-preserving interpolation on smooth surfaces, the first criterion being of non-local nature, while the second criterion is a local (weaker) version of the first one. These criteria are tested against a family of on-surface <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> splines obtained by composing the parametric representation of the supporting surface with variable-degree (≥3) splines amended with the preimages of the shortest-path geodesic arcs connecting each pair of consecutive interpolation points. After securing that the interpolation problem is well posed, we proceed to investigate the asymptotic behaviour of the proposed on-surface splines as degrees increase. Firstly, it is shown that the local-convexity sub-criterion of the local criterion is satisfied. Second, moving to non-local asymptotics, we prove that, as degrees increase, the interpolant tends uniformly to the spline curve consisting of the shortest-path geodesic arcs. Then, focusing on isometrically parametrized developable surfaces, sufficient conditions are derived, which secure that all criteria of the first (strong) criterion for shape-preserving interpolation are met. Finally, it is proved that, for adequately large degrees, the aforementioned sufficient conditions are satisfied. This permits to build an algorithm that, after a finite number of iterations, provides a <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> shape-preserving interpolant for a given data set on a developable surface.</p></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"109 ","pages":"Article 102276"},"PeriodicalIF":1.5,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0167839624000104/pdfft?md5=7c851d71becf38df6d9c14d8e96c3615&pid=1-s2.0-S0167839624000104-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139999360","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":"Computing the intersection between a rational parametric curve and a rational parametric surface","authors":"Bingwei Zhang ,&nbsp;Xi Wu ,&nbsp;Jin-San Cheng ,&nbsp;Kexin Ding","doi":"10.1016/j.cagd.2024.102275","DOIUrl":"10.1016/j.cagd.2024.102275","url":null,"abstract":"<div><p>In this paper, we present an algorithm to compute the intersection between a rational curve and a rational surface. Evaluating the parametric curve into the matrix representation of the parametric surface for implicitization, we get a matrix with one variable. We find the intersection from the matrix with the theory of real root isolation of univariate functions without computing its determinant as we have done in <span>Jia et al. (2022)</span>.</p><p>We compare our method with the state-of-the-art methods in <span>Gershon (2022)</span>; <span>Luu Ba (2014)</span>. The given examples show that our algorithms are efficient.</p></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"109 ","pages":"Article 102275"},"PeriodicalIF":1.5,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139823658","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":"Subdivision algorithms with modular arithmetic","authors":"Ron Goldman","doi":"10.1016/j.cagd.2024.102267","DOIUrl":"10.1016/j.cagd.2024.102267","url":null,"abstract":"<div><p><span>We study the de Casteljau subdivision algorithm<span> for Bezier curves and the Lane-Riesenfeld algorithm for uniform B-spline curves over the integers </span></span><em>mod m</em>, where <span><math><mrow><mi>m</mi><mo>&gt;</mo><mn>2</mn></mrow></math></span> is an odd integer. We place the integers <em>mod m</em> evenly spaced around a unit circle so that the integer <em>k mod m</em> is located at the position on the unit circle at<span><span><span><math><mrow><msup><mrow><mi>e</mi></mrow><mrow><mn>2</mn><mi>π</mi><mi>ki</mi><mo>/</mo><mi>m</mi></mrow></msup><mo>=</mo><mi>cos</mi><mrow><mo>(</mo><mn>2</mn><mi>k</mi><mi>π</mi><mo>/</mo><mi>m</mi><mo>)</mo></mrow><mo>+</mo><mi>i</mi><mi>sin</mi><mrow><mo>(</mo><mn>2</mn><mi>k</mi><mi>π</mi><mo>/</mo><mi>m</mi><mo>)</mo></mrow><mo>↔</mo><mrow><mo>(</mo><mi>cos</mi><mrow><mo>(</mo><mn>2</mn><mi>k</mi><mi>π</mi><mo>/</mo><mi>m</mi><mo>)</mo></mrow><mo>,</mo><mspace></mspace><mi>sin</mi><mrow><mo>(</mo><mn>2</mn><mi>k</mi><mi>π</mi><mo>/</mo><mi>m</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>.</mo></mrow></math></span></span></span>Given a sequence of integers <span><math><mrow><mo>(</mo><mrow><msub><mi>s</mi><mn>0</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>s</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow></math></span> <em>mod m</em>, we connect consecutive values <span><math><mrow><msub><mi>s</mi><mi>j</mi></msub><msub><mi>s</mi><mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></math></span> on the unit circle with straight line segments to form a <span><em>control polygon</em></span>. We show that if we start these subdivision procedures with the sequence <span><math><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>m</mi></mrow><mo>)</mo></mrow></math></span> <em>mod m</em>, then the sequences generated by these recursive subdivision algorithms spawn control polygons consisting of the regular <em>m</em>-sided polygon and regular <em>m</em>-pointed stars that repeat with a period equal to the minimal integer <em>k</em> such that <span><math><mrow><msup><mrow><mn>2</mn></mrow><mi>k</mi></msup><mo>=</mo><mo>±</mo><mn>1</mn><mspace></mspace><mi>m</mi><mi>o</mi><mi>d</mi><mspace></mspace><mi>m</mi></mrow></math></span><span>. Moreover, these control polygons represent the eigenvectors of the associated subdivision matrices corresponding to the eigenvalue </span><span><math><mrow><msup><mrow><mn>2</mn></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mspace></mspace><mi>m</mi><mi>o</mi><mi>d</mi><mspace></mspace><mi>m</mi><mo>.</mo></mrow></math></span> We go on to study the effects of these subdivision procedures on more general initial control polygons, and we show in particular that certain control polygons, including the orbits of regular <em>m</em>-sided polygons and the complete graphs of <em>m</em>-sided polygons, are fixed points of these subdivision procedures.</p></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"108 ","pages":"Article 102267"},"PeriodicalIF":1.5,"publicationDate":"2024-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139410521","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}