Computer Aided Geometric Design最新文献

{"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}

{"title":"Conics in rational cubic Bézier form made simple","authors":"Javier Sánchez-Reyes","doi":"10.1016/j.cagd.2023.102266","DOIUrl":"https://doi.org/10.1016/j.cagd.2023.102266","url":null,"abstract":"<div><p>We revisit the rational cubic Bézier representation of conics, simplifying and expanding previous works, elucidating their connection, and making them more accessible. The key ingredient is the concept of conic associated with a given (planar) cubic Bézier polygon, resulting from an intuitive geometric construction: Take a cubic semicircle, whose control polygon forms a square, and apply the perspective that maps this square to the given polygon. Since cubic conics come from a quadratic version by inserting a base point, this conic admitting the polygon turns out to be unique. Therefore, detecting whether a cubic is a conic boils down to checking out whether it coincides with the conic associated with its control polygon. These two curves coincide if they have the same shape factors (aka, shape invariants) or, equivalently, the same oriented curvatures at the endpoints. Our results hold for any cubic polygon (with no three points collinear), irrespective of its convexity. However, only polygons forming a strictly convex quadrilateral define conics whose cubic form admits positive weights. Also, we provide a geometric interpretation for the added expressive power (over quadratics) that such cubics with positive weights offer. In addition to semiellipses, they encompass elliptical segments with rho-values over the negative unit interval.</p></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"108 ","pages":"Article 102266"},"PeriodicalIF":1.5,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0167839623000985/pdfft?md5=c9318c50b5c8b9e909de36ce8c83727a&pid=1-s2.0-S0167839623000985-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138656693","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":"New algebraic and geometric characterizations of planar quintic Pythagorean-hodograph curves","authors":"Kai Hormann ,&nbsp;Lucia Romani ,&nbsp;Alberto Viscardi","doi":"10.1016/j.cagd.2023.102256","DOIUrl":"10.1016/j.cagd.2023.102256","url":null,"abstract":"<div><p>The aim of this work is to provide new characterizations of planar quintic Pythagorean-hodograph curves. The first two are algebraic and consist of two and three equations, respectively, in terms of the edges of the Bézier control polygon as complex numbers. These equations are symmetric with respect to the edge indices and cover curves with generic as well as degenerate control polygons. The last two characterizations are geometric and rely both on just two auxiliary points outside the control polygon. One requires two (possibly degenerate) quadrilaterals to be similar, and the other highlights two families of three similar triangles. All characterizations are a step forward with respect to the state of the art, and they can be linked to the well-established counterparts for planar cubic Pythagorean-hodograph curves. The key ingredient for proving the aforementioned results is a novel general expression for the hodograph of the curve.</p></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"108 ","pages":"Article 102256"},"PeriodicalIF":1.5,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0167839623000882/pdfft?md5=acc7917849022594427d74d299d56548&pid=1-s2.0-S0167839623000882-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138562097","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":"Detecting and parametrizing polynomial surfaces without base points","authors":"Sonia Pérez-Díaz ,&nbsp;Marian Fernández de Sevilla ,&nbsp;Rafael Magdalena Benedicto ,&nbsp;Li-Yong Shen","doi":"10.1016/j.cagd.2023.102255","DOIUrl":"https://doi.org/10.1016/j.cagd.2023.102255","url":null,"abstract":"<div><p>Given an algebraic surface implicitly defined by an irreducible polynomial, we present a method that decides whether or not this surface can be parametrized by a polynomial parametrization without base points and, in the affirmative case, we show how to compute this parametrization.</p></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"107 ","pages":"Article 102255"},"PeriodicalIF":1.5,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0167839623000870/pdfft?md5=e4c621695cd77f86295b60c5b8f9398c&pid=1-s2.0-S0167839623000870-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138397003","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":"Apollonian de Casteljau–type algorithms for complex rational Bézier curves","authors":"Bert Jüttler ,&nbsp;Josef Schicho ,&nbsp;Zbyněk Šír","doi":"10.1016/j.cagd.2023.102254","DOIUrl":"https://doi.org/10.1016/j.cagd.2023.102254","url":null,"abstract":"<div><p>We describe a new de Casteljau–type algorithm for complex rational Bézier curves. After proving that these curves exhibit the maximal possible circularity, we construct their points via a de Casteljau–type algorithm over complex numbers. Consequently, the line segments that correspond to convex linear combinations in affine spaces are replaced by circular arcs. In difference to the algorithm of <span>Sánchez-Reyes (2009)</span>, the construction of all the points is governed by (generically complex) roots of the denominator, using one of them for each level. Moreover, one of the bi-polar coordinates is fixed at each level, independently of the parameter value. A rational curve of the complex degree <em>n</em> admits generically <em>n</em>! distinct de Casteljau–type algorithms, corresponding to the different orderings of the denominator's roots.</p></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"107 ","pages":"Article 102254"},"PeriodicalIF":1.5,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49707548","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":"Efficient computation of moving planes for rational parametric surfaces with base points using Dixon resultants","authors":"Kai Li ,&nbsp;Xiaohong Jia ,&nbsp;Falai Chen","doi":"10.1016/j.cagd.2023.102253","DOIUrl":"https://doi.org/10.1016/j.cagd.2023.102253","url":null,"abstract":"<div><p>Moving planes have been widely recognized as a potent algebraic tool in various fundamental problems of geometric modeling, including implicitization, intersection computation, singularity calculation, and point inversion of parametric surfaces. For instance, a matrix representation that inherits the key properties of a parametric surface is constructed from a set of moving planes. In this paper, we present an efficient approach to computing such a set of moving planes that follow the given rational parametric surface. Our method is based on the calculation of Dixon resultant matrices, which allows for the computation of moving planes with simpler coefficients, improved efficiency and superior numerical stability when compared to the direct way of solving a linear system of equations for the same purpose. We also demonstrate the performance of our algorithm through experimental examples when applied to implicitization, surface intersection, singularity computation as well as inversion formula computation.</p></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"107 ","pages":"Article 102253"},"PeriodicalIF":1.5,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49707524","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":"On the uniqueness of the multirational blossom","authors":"O. Oğulcan Tuncer ,&nbsp;Plamen Simeonov ,&nbsp;Ron Goldman","doi":"10.1016/j.cagd.2023.102252","DOIUrl":"https://doi.org/10.1016/j.cagd.2023.102252","url":null,"abstract":"<div><p>The multirational blossom of order <em>k</em> and degree −<em>n</em> of a <em>k</em><span>-times differentiable function </span><span><math><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span><span> is defined as a multivariate function </span><span><math><mi>f</mi><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>/</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>k</mi><mo>+</mo><mi>n</mi></mrow></msub><mo>)</mo></math></span> characterized by four axioms: bisymmetry in the <em>u</em> and <em>v</em> parameters, multiaffine in the <em>u</em> parameters, satisfies a cancellation property and reduces to <span><math><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> along the diagonal. The existence of a multirational blossom was established in <span>Goldman (1999a)</span> by providing an explicit formula for this blossom in terms of divided differences. Here we show that these four axioms uniquely characterize the multirational blossom. We go on to introduce a homogeneous version of the multirational blossom. We then show that for differentiable functions derivatives can be computed in terms of this homogeneous multirational blossom. We also use the homogeneous multirational blossom to convert between the Taylor bases and the negative degree Bernstein bases.</p></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"107 ","pages":"Article 102252"},"PeriodicalIF":1.5,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49759702","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":"Topological classification of the intersection curves of two quadrics using a set of discriminants","authors":"Wenbing Shao,&nbsp;Falai Chen","doi":"10.1016/j.cagd.2023.102244","DOIUrl":"https://doi.org/10.1016/j.cagd.2023.102244","url":null,"abstract":"<div><p>Computing the intersection curves<span> of two quadrics (QSIC) is a fundamental problem in computer aided design<span> system, where the topology analysis of the QSIC is an essential task to realize the robust computation. In this paper, a new method is proposed to classify the topology of the QSIC in three dimensional projective space by using a set of discriminants associated with the two quadrics. The new topology classification method presents explicit representations and can be applied even when the coefficients of the two quadrics contain symbols. Some examples are provided to illustrate the usage of the new classification method.</span></span></p></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"107 ","pages":"Article 102244"},"PeriodicalIF":1.5,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49703445","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":"On C0 and C1 continuity of envelopes of rotational solids and its application to 5-axis CNC machining","authors":"Felipe Ponce-Vanegas ,&nbsp;Michal Bizzarri ,&nbsp;Michael Bartoň","doi":"10.1016/j.cagd.2023.102245","DOIUrl":"https://doi.org/10.1016/j.cagd.2023.102245","url":null,"abstract":"<div><p>We study the smoothness of envelopes generated by motions of rotational rigid bodies in the context of 5-axis Computer Numerically Controlled (CNC) machining. A moving cutting tool, conceptualized as a rotational solid, forms a surface, called envelope, that delimits a part of 3D space where the tool engages the material block. The smoothness of the resulting envelope depends both on the smoothness of the motion and smoothness of the tool. While the motions of the tool are typically required to be at least <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, the tools are frequently only <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msup></math></span> continuous, which results in discontinuous envelopes. In this work, we classify a family of instantaneous motions that, in spite of only <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msup></math></span> continuous shape of the tool, result in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msup></math></span> continuous envelopes. We show that such motions are flexible enough to follow a free-form surface, preserving tangential contact between the tool and surface along two points, therefore having applications in shape slot milling or in a semi-finishing stage of 5-axis flank machining. We also show that <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> tools and motions still can generate smooth envelopes.</p></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"107 ","pages":"Article 102245"},"PeriodicalIF":1.5,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49703472","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":"An efficient analytical model for the swept volume generation of a flat-end mill in 5-axis CNC milling","authors":"Ahmet Dogrusadik","doi":"10.1016/j.cagd.2023.102241","DOIUrl":"10.1016/j.cagd.2023.102241","url":null,"abstract":"<div><p>5-axis CNC (Computer Numerical Control) milling is widely used to create complex part geometries in the industrial area. The cutting tool creates the swept volume as it moves along the defined path. The swept volume is subtracted from the initial stock for the machining simulation. Although a swept volume consists of three parts such as ingress, egress, and swept envelope, the number of faces of the swept volume is higher than three. In this work, parametric equations<span> of the faces of the swept volume were obtained for a flat-end mill in four steps. The model is based on the decomposition of the tool into circles along the tool orientation vector<span> since a flat-end mill can be modeled as a cylinder. The fundamental principle is that each circle is tangent to the swept envelope of the cylinder. Locations of the grazing points with respect to a local coordinate system were determined by applying the Envelope Theory to the parameterized circle of the cylinder. Then, each face of the swept volume was represented based on the equation of the parameterized circle. The model was verified by using an alternative analytical model. As a result, boundaries of the swept volume were represented fully analytically for a flat-end mill in 5-axis CNC milling in an efficient way.</span></span></p></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"106 ","pages":"Article 102241"},"PeriodicalIF":1.5,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49041671","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}