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Journal articles on the topic 'Bézier triangles'

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Published: 9 March 2023

Last updated: 10 March 2023

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1

Goldman, Ronald N., and Daniel J. Filip. "Conversion from Bézier rectangles to Bézier triangles." Computer-Aided Design 19, no. 1 (January 1987): 25–27. http://dx.doi.org/10.1016/0010-4485(87)90149-7.

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2

Prautzsch, H. "On convex Bézier triangles." ESAIM: Mathematical Modelling and Numerical Analysis 26, no. 1 (1992): 23–36. http://dx.doi.org/10.1051/m2an/1992260100231.

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3

Lee, Chang-Ki, Hae-Do Hwang, and Seung-Hyun Yoon. "Bézier Triangles with G2 Continuity across Boundaries." Symmetry 8, no. 3 (March 15, 2016): 13. http://dx.doi.org/10.3390/sym8030013.

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4

Yan, Lanlan. "Construction Method of Shape Adjustable Bézier Triangles." Chinese Journal of Electronics 28, no. 3 (May 1, 2019): 610–17. http://dx.doi.org/10.1049/cje.2019.03.016.

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5

Gregory, John A., and Jianwei Zhou. "Convexity of Bézier nets on sub-triangles." Computer Aided Geometric Design 8, no. 3 (August 1991): 207–11. http://dx.doi.org/10.1016/0167-8396(91)90003-t.

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6

Feng, Yu-Yu. "Rates of convergence of Bézier net over triangles." Computer Aided Geometric Design 4, no. 3 (November 1987): 245–49. http://dx.doi.org/10.1016/0167-8396(87)90016-1.

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7

Belbis, Bertrand, Lionel Garnier, and Sebti Foufou. "Construction of 3D Triangles on Dupin Cyclides." International Journal of Computer Vision and Image Processing 1, no. 2 (April 2011): 42–57. http://dx.doi.org/10.4018/ijcvip.2011040104.

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This paper considers the conversion of the parametric Bézier surfaces, classically used in CAD-CAM, into patched of a class of non-spherical degree 4 algebraic surfaces called Dupin cyclides, and the definition of 3D triangle with circular edges on Dupin cyclides. Dupin cyclides was discovered by the French mathematician Pierre-Charles Dupin at the beginning of the 19th century. A Dupin cyclide has one parametric equation, two implicit equations, and a set of circular lines of curvature. The authors use the properties of these surfaces to prove that three families of circles (meridian arcs, parallel arcs, and Villarceau circles) can be computed on every Dupin cyclide. A geometric algorithm to compute these circles so that they define the edges of a 3D triangle on the Dupin cyclide is presented. Examples of conversions and 3D triangles are also presented to illustrate the proposed algorithms.

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8

Walz, Guido. "Trigonometric Bézier and Stancu polynomials over intervals and triangles." Computer Aided Geometric Design 14, no. 4 (May 1997): 393–97. http://dx.doi.org/10.1016/s0167-8396(96)00061-1.

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9

Filip, Daniel J. "Adaptive subdivision algorithms for a set of Bézier triangles." Computer-Aided Design 18, no. 2 (March 1986): 74–78. http://dx.doi.org/10.1016/0010-4485(86)90153-3.

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10

Hermes, Danny. "Helper for Bézier Curves, Triangles, and Higher Order Objects." Journal of Open Source Software 2, no. 16 (August 2, 2017): 267. http://dx.doi.org/10.21105/joss.00267.

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11

Liu, Zhi, Jie-qing Tan, Xiao-yan Chen, and Li Zhang. "The conditions of convexity for Bernstein–Bézier surfaces over triangles." Computer Aided Geometric Design 27, no. 6 (August 2010): 421–27. http://dx.doi.org/10.1016/j.cagd.2010.05.004.

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12

Chang, Hanjiang, Cheng Liu, Qiang Tian, Haiyan Hu, and Aki Mikkola. "Three new triangular shell elements of ANCF represented by Bézier triangles." Multibody System Dynamics 35, no. 4 (June 17, 2015): 321–51. http://dx.doi.org/10.1007/s11044-015-9462-y.

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13

Chan, E. S., and B. H. Ong. "Range restricted scattered data interpolation using convex combination of cubic Bézier triangles." Journal of Computational and Applied Mathematics 136, no. 1-2 (November 2001): 135–47. http://dx.doi.org/10.1016/s0377-0427(00)00580-x.

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14

Lorente-Pardo, J., P. Sablonnière, and M. C. Serrano-Pérez. "Subharmonicity and convexity properties of Bernstein polynomials and Bézier nets on triangles." Computer Aided Geometric Design 16, no. 4 (May 1999): 287–300. http://dx.doi.org/10.1016/s0167-8396(98)00050-8.

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15

López, Jorge, Cosmin Anitescu, Navid Valizadeh, Timon Rabczuk, and Naif Alajlan. "Structural shape optimization using Bézier triangles and a CAD-compatible boundary representation." Engineering with Computers 36, no. 4 (May 31, 2019): 1657–72. http://dx.doi.org/10.1007/s00366-019-00788-z.

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16

Bez, H. E. "The invariant functions and invariant-image conditions of the rational Bézier triangles." Applicable Algebra in Engineering, Communication and Computing 23, no. 3-4 (September 21, 2012): 195–205. http://dx.doi.org/10.1007/s00200-012-0174-8.

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17

Bastl, Bohumír, Bert Jüttler, Miroslav Lávička, Josef Schicho, and Zbyněk Šír. "Spherical quadratic Bézier triangles with chord length parameterization and tripolar coordinates in space." Computer Aided Geometric Design 28, no. 2 (February 2011): 127–34. http://dx.doi.org/10.1016/j.cagd.2010.11.001.

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18

Abdul Karim, Samsul Ariffin Abdul, Azizan Saaban, and Van Thien Nguyen. "Scattered Data Interpolation Using Quartic Triangular Patch for Shape-Preserving Interpolation and Comparison with Mesh-Free Methods." Symmetry 12, no. 7 (June 30, 2020): 1071. http://dx.doi.org/10.3390/sym12071071.

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Scattered data interpolation is important in sciences, engineering, and medical-based problems. Quartic Bézier triangular patches with 15 control points (ordinates) can also be used for scattered data interpolation. However, this method has a weakness; that is, in order to achieve C 1 continuity, the three inner points can only be determined using an optimization method. Thus, we cannot obtain the exact Bézier ordinates, and the quartic scheme is global and not local. Therefore, the quartic Bézier triangular has received less attention. In this work, we use Zhu and Han’s quartic spline with ten control points (ordinates). Since there are only ten control points (as for cubic Bézier triangular cases), all control points can be determined exactly, and the optimization problem can be avoided. This will improve the presentation of the surface, and the process to construct the scattered surface is local. We also apply the proposed scheme for the purpose of positivity-preserving scattered data interpolation. The sufficient conditions for the positivity of the quartic triangular patches are derived on seven ordinates. We obtain nonlinear equations that can be solved using the regula-falsi method. To produce the interpolated surface for scattered data, we employ four stages of an algorithm: (a) triangulate the scattered data using Delaunay triangulation; (b) assign the first derivative at the respective data; (c) form a triangular surface via convex combination from three local schemes with C 1 continuity along all adjacent triangles; and (d) construct the scattered data surface using the proposed quartic spline. Numerical results, including some comparisons with some existing mesh-free schemes, are presented in detail. Overall, the proposed quartic triangular spline scheme gives good results in terms of a higher coefficient of determination (R2) and smaller maximum error (Max Error), requires about 12.5% of the CPU time of the quartic Bézier triangular, and is on par with Shepard triangular-based schemes. Therefore, the proposed scheme is significant for use in visualizing large and irregular scattered data sets. Finally, we tested the proposed positivity-preserving interpolation scheme to visualize coronavirus disease 2019 (COVID-19) cases in Malaysia.

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19

Liu, Ning, and Ann E. Jeffers. "Feature-preserving rational Bézier triangles for isogeometric analysis of higher-order gradient damage models." Computer Methods in Applied Mechanics and Engineering 357 (December 2019): 112585. http://dx.doi.org/10.1016/j.cma.2019.112585.

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20

Wang, Zheng-bin, and Qi-ming Liu. "An improved condition for the convexity and positivity of Bernstein-Bézier surfaces over triangles." Computer Aided Geometric Design 5, no. 4 (November 1988): 269–75. http://dx.doi.org/10.1016/0167-8396(88)90008-8.

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21

Yu, Kai-Ming, Yu Wang, and Charlie C. L. Wang. "Smooth geometry generation in additive manufacturing file format: problem study and new formulation." Rapid Prototyping Journal 23, no. 1 (January 16, 2017): 34–43. http://dx.doi.org/10.1108/rpj-06-2015-0067.

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Purpose In the newly released ASTM standard specification for additive manufacturing file (AMF) format – version 1.1 – Hermite curve-based interpolation is used to refine input triangles to generate denser mesh with smoother geometry. This paper aims to study the problems of constructing smooth geometry based on Hermite interpolation on curves and proposes a solution to overcome these problems. Design/methodology/approach A formulation using triangular Bézier patch is proposed to generate smooth geometry from input polygonal models. Different configurations on the boundary curves in the formulation are analyzed to further enrich this formulation. Findings The study shows that the formulation given in the AMF format (version 1.1) can lead to the problems of inconsistent normals and undefined end-tangents. Research limitations/implications The scheme has requirements on the input normals of a model, only C0 interpolation can be generated on those cases with less-proper input. Originality/value To overcome the problems of smooth geometry generation in the AMF format, the authors propose an enriched scheme for computing smooth geometry by using triangular Bézier patch. For the configurations with less-proper input, the authors adopt the Boolean sum and the Nielson’s point-opposite edge interpolation for triangular Coons patch to generate the smooth geometry as a C0 interpolant.

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22

Chau, Hau Hing, Alison McKay, Christopher F. Earl, Amar Kumar Behera, and Alan de Pennington. "Exploiting lattice structures in shape grammar implementations." Artificial Intelligence for Engineering Design, Analysis and Manufacturing 32, no. 2 (May 2018): 147–61. http://dx.doi.org/10.1017/s0890060417000282.

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AbstractThe ability to work with ambiguity and compute new designs based on both defined and emergent shapes are unique advantages of shape grammars. Realizing these benefits in design practice requires the implementation of general purpose shape grammar interpreters that support: (a) the detection of arbitrary subshapes in arbitrary shapes and (b) the application of shape rules that use these subshapes to create new shapes. The complexity of currently available interpreters results from their combination of shape computation (for subshape detection and the application of rules) with computational geometry (for the geometric operations need to generate new shapes). This paper proposes a shape grammar implementation method for three-dimensional circular arcs represented as rational quadratic Bézier curves based on lattice theory that reduces this complexity by separating steps in a shape computation process from the geometrical operations associated with specific grammars and shapes. The method is demonstrated through application to two well-known shape grammars: Stiny's triangles grammar and Jowers and Earl's trefoil grammar. A prototype computer implementation of an interpreter kernel has been built and its application to both grammars is presented. The use of Bézier curves in three dimensions opens the possibility to extend shape grammar implementations to cover the wider range of applications that are needed before practical implementations for use in real life product design and development processes become feasible.

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23

Riso, Marzia, Giacomo Nazzaro, Enrico Puppo, Alec Jacobson, Qingnan Zhou, and Fabio Pellacini. "BoolSurf." ACM Transactions on Graphics 41, no. 6 (November 30, 2022): 1–13. http://dx.doi.org/10.1145/3550454.3555466.

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We port Boolean set operations between 2D shapes to surfaces of any genus, with any number of open boundaries. We combine shapes bounded by sets of freely intersecting loops, consisting of geodesic lines and cubic Bézier splines lying on a surface. We compute the arrangement of shapes directly on the surface and assign integer labels to the cells of such arrangement. Differently from the Euclidean case, some arrangements on a manifold may be inconsistent. We detect inconsistent arrangements and help the user to resolve them. Also, we extend to the manifold setting recent work on Boundary-Sampled Halfspaces, thus supporting operations more general than standard Booleans, which are well defined on inconsistent arrangements, too. Our implementation discretizes the input shapes into polylines at an arbitrary resolution, independent of the level of resolution of the underlying mesh. We resolve the arrangement inside each triangle of the mesh independently and combine the results to reconstruct both the boundaries and the interior of each cell in the arrangement. We reconstruct the control points of curves bounding cells, in order to free the result from discretization and provide an output in vector format. We support interactive usage, editing shapes consisting up to 100k line segments on meshes of up to 1M triangles.

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24

Albrecht, Gudrun, and Wendelin L. F. Degen. "Construction of Bézier rectangles and triangles on the symmetric Dupin horn cyclide by means of inversion." Computer Aided Geometric Design 14, no. 4 (May 1997): 349–75. http://dx.doi.org/10.1016/s0167-8396(97)00002-2.

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25

Šimák, Jan. "A software tool for blade design." EPJ Web of Conferences 269 (2022): 01055. http://dx.doi.org/10.1051/epjconf/202226901055.

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An interactive software tool for blade design of axial flow machines was created. It is written as an extension module to the open-source software FreeCAD. In its graphical interface, the user can modify the blade profiles, stack them to create the whole blade and generate end walls and other stage features. Or everything can be controlled by a simple Python script. Results can be saved as STEP file or STL mesh and export to a mesh generator and CFD solver. Blade profiles are given by a set of parameters describing Bézier curves, the blade is represented by b-spline surfaces. Up to now, the tool helps the designer to find the right shapes by a 1D method, which evaluates the velocity triangles and the state variables through the machine stage on a mean line (mean radius) from the given design parameters (mass flow, total states in front of the stage, rotational speed etc.) and distribute them along the radius from the hub to the tip. The CFD results of a sample designed stage are included.

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26

Lasser, Dieter. "Tensor product Bézier surfaces on triangle Bézier surfaces." Computer Aided Geometric Design 19, no. 8 (October 2002): 625–43. http://dx.doi.org/10.1016/s0167-8396(02)00145-0.

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27

Garnier, Lionel, Lucie Druoton, Jean-Paul Bécar, Laurent Fuchs, and Géraldine Morin. "Subdivisions of Horned or Spindle Dupin Cyclides Using Bézier Curves with Mass Points." WSEAS TRANSACTIONS ON MATHEMATICS 20 (December 31, 2021): 756–76. http://dx.doi.org/10.37394/23206.2021.20.80.

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This paper shows the same algorithm is used for subdivisions of Dupin cyclides with singular points and quadratic Bézier curves passing through infinity. The mass points are usefull for any quadratic Bézier representation of a parabola or an hyperbola arc. The mass points are mixing weighted points and pure vectors. Any Dupin cyclide is considered in the Minkowski-Lorentz space. In that space, the Dupin cyclide is defined by the union of two conics laying on the unit pseudo-hypersphere, called the space of spheres. The subdivision of any Dupin cyclide, is equivalent to subdivide two Bézier curves of degree 2 with mass points, independently. The use of these two curves eases the subdivision of a Dupin cyclide patch or triangle.

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28

Razdan, Anshuman, and MyungSoo Bae. "Curvature estimation scheme for triangle meshes using biquadratic Bézier patches." Computer-Aided Design 37, no. 14 (December 2005): 1481–91. http://dx.doi.org/10.1016/j.cad.2005.03.003.

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29

Kobayashi, Ken, Naoki Hamada, Akiyoshi Sannai, Akinori Tanaka, Kenichi Bannai, and Masashi Sugiyama. "Bézier Simplex Fitting: Describing Pareto Fronts of´ Simplicial Problems with Small Samples in Multi-Objective Optimization." Proceedings of the AAAI Conference on Artificial Intelligence 33 (July 17, 2019): 2304–13. http://dx.doi.org/10.1609/aaai.v33i01.33012304.

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Multi-objective optimization problems require simultaneously optimizing two or more objective functions. Many studies have reported that the solution set of an M-objective optimization problem often forms an (M − 1)-dimensional topological simplex (a curved line for M = 2, a curved triangle for M = 3, a curved tetrahedron for M = 4, etc.). Since the dimensionality of the solution set increases as the number of objectives grows, an exponentially large sample size is needed to cover the solution set. To reduce the required sample size, this paper proposes a Bézier simplex model and its fitting algorithm. These techniques can exploit the simplex structure of the solution set and decompose a high-dimensional surface fitting task into a sequence of low-dimensional ones. An approximation theorem of Bézier simplices is proven. Numerical experiments with synthetic and real-world optimization problems demonstrate that the proposed method achieves an accurate approximation of high-dimensional solution sets with small samples. In practice, such an approximation will be conducted in the postoptimization process and enable a better trade-off analysis.

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30

Hongyi, Wu. "Dual functionals of said-bézier type generalized ball bases over triangle domain and their application." Applied Mathematics-A Journal of Chinese Universities 21, no. 1 (March 2006): 96–106. http://dx.doi.org/10.1007/s11766-996-0028-x.

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31

Shafieipour, Mohammad, Jonatan Aronsson, Ian Jeffrey, Chen Nui, and Vladimir I. Okhmatovski. "On New Triangle Quadrature Rules for the Locally Corrected Nyström Method Formulated on NURBS-Generated Bézier Surfaces in 3-D." IEEE Transactions on Antennas and Propagation 64, no. 7 (July 2016): 3027–38. http://dx.doi.org/10.1109/tap.2016.2560958.

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32

Barroso, Elias Saraiva, John Andrew Evans, Joaquim Bento Cavalcante-Neto, Creto Augusto Vidal, and Evandro Parente. "An efficient automatic mesh generation algorithm for planar isogeometric analysis using high-order rational Bézier triangles." Engineering with Computers, February 9, 2022. http://dx.doi.org/10.1007/s00366-022-01613-w.

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33

Peters, Jörg, Kyle Shih-Huang Lo, and Kȩstutis Karčiauskas. "Algorithm ⋆: Bi-cubic splines for polyhedral control nets." ACM Transactions on Mathematical Software, October 31, 2022. http://dx.doi.org/10.1145/3570158.

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For control nets outlining a large class of topological polyhedra, not just tensor-product grids, bi-cubic polyhedral spline s form a piecewise polynomial, first-order differentiable space that associates one function with each vertex. Akin to tensor-product splines, the resulting smooth surface approximates the polyhedron. Admissible polyhedral control net s consist of quadrilateral faces in a grid-like layout, star-configuration where n ≠ 4 quadrilateral faces join around an interior vertex, n -gon configurations, where 2 n quadrilaterals surround an n -gon, polar configurations where a cone of n triangles meeting at a vertex is surrounded by a ribbon of n quadrilaterals, and three types of T-junctions where two quad-strips merge into one. The bi-cubic pieces of a polyhedral spline have matching derivatives along their break lines, possibly after a known change of variables. The pieces are represented in Bernstein-Bézier form with coefficients depending linearly on the polyhedral control net, so that evaluation, differentiation, integration, moments, etc. are no more costly than for standard tensor-product splines. Bi-cubic polyhedral spline s can be used both to model geometry and for computing functions on the geometry. Although polyhedral spline s do not offer nested refinement by refinement of the control net, polyhedral spline s support engineering analysis of curved smooth objects. Coarse nets typically suffice since the splines efficiently model curved features. Algorithm ⋆ is a C++ library with input-output example pairs and an iges output choice.

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34

Kapl, Mario, Giancarlo Sangalli, and Thomas Takacs. "A family of C1 quadrilateral finite elements." Advances in Computational Mathematics 47, no. 6 (November 3, 2021). http://dx.doi.org/10.1007/s10444-021-09878-3.

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AbstractWe present a novel family of C1 quadrilateral finite elements, which define global C1 spaces over a general quadrilateral mesh with vertices of arbitrary valency. The elements extend the construction by Brenner and Sung (J. Sci. Comput. 22(1-3), 83-118, 2005), which is based on polynomial elements of tensor-product degree p ≥ 6, to all degrees p ≥ 3. The proposed C1 quadrilateral is based upon the construction of multi-patch C1 isogeometric spaces developed in Kapl et al. (Comput. Aided Geometr. Des. 69, 55–75 2019). The quadrilateral elements possess similar degrees of freedom as the classical Argyris triangles, developed in Argyris et al. (Aeronaut. J. 72(692), 701–709 1968). Just as for the Argyris triangle, we additionally impose C2 continuity at the vertices. In contrast to Kapl et al. (Comput. Aided Geometr. Des. 69, 55–75 2019), in this paper, we concentrate on quadrilateral finite elements, which significantly simplifies the construction. We present macro-element constructions, extending the elements in Brenner and Sung (J. Sci. Comput. 22(1–3), 83–118 2005), for polynomial degrees p = 3 and p = 4 by employing a splitting into 3 × 3 or 2 × 2 polynomial pieces, respectively. We moreover provide approximation error bounds in $L^{\infty }$ L ∞ , L2, H1 and H2 for the piecewise-polynomial macro-element constructions of degree p ∈{3,4} and polynomial elements of degree p ≥ 5. Since the elements locally reproduce polynomials of total degree p, the approximation orders are optimal with respect to the mesh size. Note that the proposed construction combines the possibility for spline refinement (equivalent to a regular splitting of quadrilateral finite elements) as in Kapl et al. (Comput. Aided Geometr. Des. 69, 55–75 30) with the purely local description of the finite element space and basis as in Brenner and Sung (J. Sci. Comput. 22(1–3), 83–118 2005). In addition, we describe the construction of a simple, local basis and give for p ∈{3,4,5} explicit formulas for the Bézier or B-spline coefficients of the basis functions. Numerical experiments by solving the biharmonic equation demonstrate the potential of the proposed C1 quadrilateral finite element for the numerical analysis of fourth order problems, also indicating that (for p = 5) the proposed element performs comparable or in general even better than the Argyris triangle with respect to the number of degrees of freedom.

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