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Journal articles on the topic 'Bézier triangles'

Author: Grafiati

Published: 9 March 2023

Last updated: 31 July 2025

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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 (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 (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 (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 (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 (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 (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, pa

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8

Walz, Guido. "Trigonometric Bézier and Stancu polynomials over intervals and triangles." Computer Aided Geometric Design 14, no. 4 (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 (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 (2017): 267. http://dx.doi.org/10.21105/joss.00267.

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11

de Cássia Jerônimo da Silva, Rita, Thiago de Aguiar Leal Domingues, Marlos Antônio Pinheiro Rolim, Suzan Diniz, and Silvio de Barros Melo. "Efficient Slicing of rational Bézier triangles for additive manufacturing." Additive Manufacturing 78 (September 2023): 103876. http://dx.doi.org/10.1016/j.addma.2023.103876.

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12

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 (2010): 421–27. http://dx.doi.org/10.1016/j.cagd.2010.05.004.

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13

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 (2015): 321–51. http://dx.doi.org/10.1007/s11044-015-9462-y.

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14

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 (2001): 135–47. http://dx.doi.org/10.1016/s0377-0427(00)00580-x.

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15

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 (1999): 287–300. http://dx.doi.org/10.1016/s0167-8396(98)00050-8.

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16

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 (2019): 1657–72. http://dx.doi.org/10.1007/s00366-019-00788-z.

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17

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 (2012): 195–205. http://dx.doi.org/10.1007/s00200-012-0174-8.

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18

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 (2011): 127–34. http://dx.doi.org/10.1016/j.cagd.2010.11.001.

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19

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 (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 te

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20

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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21

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 (1988): 269–75. http://dx.doi.org/10.1016/0167-8396(88)90008-8.

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22

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 (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 formul

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23

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 (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 computati

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24

Riso, Marzia, Giacomo Nazzaro, Enrico Puppo, Alec Jacobson, Qingnan Zhou, and Fabio Pellacini. "BoolSurf." ACM Transactions on Graphics 41, no. 6 (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-Sampl

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25

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 (1997): 349–75. http://dx.doi.org/10.1016/s0167-8396(97)00002-2.

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26

Liu, Shibo, Yang Ji, Jia-Peng Guo, Ligang Liu, and Xiao-Ming Fu. "Smooth Bijective Projection in a High-order Shell." ACM Transactions on Graphics 43, no. 4 (2024): 1–13. http://dx.doi.org/10.1145/3658207.

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We propose a new structure called a higher-order shell, which is composed of a set of triangular prisms. Each triangular prism is enveloped by three Bézier triangles (top, middle, and bottom) and three side surfaces, each of which is trimmed from a bilinear surface. Moreover, we define a continuous vector field to smoothly and bijectively transfer attributes between two surfaces inside the shell. Since the higher-order shell has several hard construction constraints, we apply an interior-point strategy to robustly and automatically construct a high-order shell for an input mesh. Specifically,

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27

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 to

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28

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

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29

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 cur

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30

Fernández-Jambrina, Leonardo. "Inverse Relations for Blossoms and Parametrisations of Rational Curves and Surfaces." Mathematics 12, no. 18 (2024): 2841. http://dx.doi.org/10.3390/math12182841.

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In this paper, we make use of an inverse formula that relates the blossom of a NURBS curve, surface or Bézier triangle with its parametrisation, with no explicit reference to the control points and weights of the parametrisation. We make use of this inverse formula to raise and lower the degree elevation and reduction.

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31

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

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32

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 a

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33

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 (2006): 96–106. http://dx.doi.org/10.1007/s11766-996-0028-x.

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34

Zhang, Ziyi, Nicolas Roussel, and Wenzel Jakob. "Projective Sampling for Differentiable Rendering of Geometry." ACM Transactions on Graphics 42, no. 6 (2023): 1–14. http://dx.doi.org/10.1145/3618385.

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Discontinuous visibility changes at object boundaries remain a persistent source of difficulty in the area of differentiable rendering. Left untreated, they bias computed gradients so severely that even basic optimization tasks fail. Prior path-space methods addressed this bias by decoupling boundaries from the interior, allowing each part to be handled using specialized Monte Carlo sampling strategies. While conceptually powerful, the full potential of this idea remains unrealized since existing methods often fail to adequately sample the boundary proportional to its contribution. This paper

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35

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 (2016): 3027–38. http://dx.doi.org/10.1109/tap.2016.2560958.

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36

Wang, Wanyi, Zhonggui Chen, Lincong Fang, and Juan Cao. "Curved Image Triangulation Based on Differentiable Rendering." Computer Graphics Forum, October 24, 2024. http://dx.doi.org/10.1111/cgf.15232.

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AbstractImage triangulation methods, which decompose an image into a series of triangles, are fundamental in artistic creation and image processing. This paper introduces a novel framework that integrates cubic Bézier curves into image triangulation, enabling the precise reconstruction of curved image features. Our developed framework constructs a well‐structured curved triangle mesh, effectively preventing overlaps between curves. A refined energy function, grounded in differentiable rendering, establishes a direct link between mesh geometry and rendering effects and is instrumental in guidin

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37

Silva, Francisco Davyd Pereira, Elias Saraiva Barroso, Gabriel Braga Alves de Matos, Evandro Parente, and João Batista Marques de Sousa. "Isogeometric analysis of functionally graded panels using Bézier triangles." Composite Structures, June 2024, 118310. http://dx.doi.org/10.1016/j.compstruct.2024.118310.

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38

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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39

Sichetti, Federico, Zizhou Huang, Marco Attene, Denis Zorin, Enrico Puppo, and Daniele Panozzo. "High-Order Continuous Geometrical Validity." ACM Transactions on Graphics, June 26, 2025. https://doi.org/10.1145/3745763.

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We propose a conservative algorithm to test the geometrical validity of simplicial (triangles, tetrahedra), tensor product (quadrilaterals, hexahedra), and mixed (prisms) elements of arbitrary polynomial order as they deform linearly within a time interval. Our algorithm uses a combination of adaptive Bézier refinement and bisection search to determine if, when, and where the Jacobian determinant of an element’s polynomial geometric map becomes negative in the transition from one configuration to another. In elastodynamic simulation, our algorithm guarantees that the system remains physically

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40

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

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41

Kapl, Mario, Giancarlo Sangalli, and Thomas Takacs. "A family of C1 quadrilateral finite elements." Advances in Computational Mathematics 47, no. 6 (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

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