Article
Mathematics, Applied
Michal Bizzarri, Miroslav Lavicka
Summary: This paper discusses the problem of Hermite interpolation using clamped Minkowski Pythagorean hodograph (MPH) B-spline curves, with a focus on using MPH curves of degrees lower than previously designed algorithms. Special attention is given to C-1/C-2 Hermite interpolation by MPH B-spline cubics/quintics, resulting in interpolants obtained through exploiting properties of B-spline basis functions and solving equations in Clifford algebra Cl-2,Cl-1. The presented algorithms are purely symbolic and confirmed by various applications for high order approximation, efficient conversion, and skinning of circles in plane.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Computer Science, Information Systems
Kinga Kruppa
Summary: The article discusses the importance of special curves in Minkowski space and their applications in computer-aided geometric design. A new application area for RE curves is proposed, along with a novel method for skinning a discrete set of input circles, overcoming previous challenges. A significant advantage of the proposed method is the efficiency of trimming offsets of boundaries, particularly beneficial in computer numerical control machining.
FRONTIERS OF INFORMATION TECHNOLOGY & ELECTRONIC ENGINEERING
(2021)
Article
Mathematics, Applied
Marjeta Knez, Francesca Pelosi, Maria Lucia Sampoli
Summary: In this paper, we focus on the construction of G(2) planar Pythagorean-hodograph (PH) spline curves that interpolate points, tangent directions, and curvatures, while also having a prescribed arc-length. The interpolation method used is completely local, and each spline segment is defined as a PH biarc curve of degree 7. By fixing two free parameters to zero, it is shown that the length constraint can be satisfied for any data and any chosen ratio between boundary tangents. The bending energy is then used to select the best solution, and numerical examples are provided to illustrate the theoretical results and confirm the approximation order of 5.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Mathematics, Applied
Carlotta Giannelli, Lorenzo Sacco, Alessandra Sestini
Summary: This paper proposes a method for constructing smooth spatial paths using PH splines, introducing a local data stream interpolation algorithm for real-time computations and ensuring C-2 smooth connections between adjacent spline segments. The method is applicable to arbitrary Hermite data configurations and demonstrates high accuracy in numerical examples.
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Marjeta Knez, Francesca Pelosi, Maria Lucia Sampoli
Summary: This paper addresses the problem of constructing spatial G2 continuous Pythagorean-hodograph (PH) spline curves that interpolate points and frame data with the prescribed arc length. The proposed interpolation scheme is completely local and suitable for motion design applications. The paper presents a direct generalization of the construction done for planar curves to spatial ones by using an automatic procedure for computing the frame and velocity quaternions. Several numerical examples are provided to demonstrate the effectiveness of the proposed method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Computer Science, Software Engineering
Michal Bizzarri, Miroslav Lavicka, Jan Vrsek
Summary: This paper thoroughly investigates the problem of Hermite interpolation by clamped spatial Pythagorean hodograph (PH) B-spline curves, using symbolic computations and solving special algebraic equations. The main contribution lies in the unifying approach to the formulated problem, with results confirmed by multiple computed examples.
COMPUTER AIDED GEOMETRIC DESIGN
(2021)
Article
Mathematics, Applied
Rida T. Farouki, Francesca Pelosi, Maria Lucia Sampoli
Summary: The study investigates the use of planar Pythagorean-hodograph (PH) curves as polynomial approximants to clothoid segments, based on geometric Hermite interpolation of end points, tangents, and curvatures, with precise matching of arc length.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Thierry Bay, Isabelle Cattiaux-Huillard, Lucia Romani, Laura Saini
Summary: In this paper, we focus on a class of curves called Algebraic-Trigonometric Pythagorean Hodograph curves (ATPH) characterized by a polynomial parametric speed. We first construct interpolants for spatial G1 Hermite data, including curvature values. Compared to the solutions proposed in [24], the G1 Hermite ATPH interpolants we propose here have continuous curvature plots in terms of C0 and C1 continuity. Secondly, we investigate the existence of ATPH interpolants to spatial G2 Hermite data and prove that solutions exist under certain restrictions on the Hermite input data.
APPLIED MATHEMATICS AND COMPUTATION
(2023)
Article
Computer Science, Software Engineering
Marjeta Knez, Maria Lucia Sampoli
Summary: The construction of a curve that interpolates given initial/final positions along with orientational frames is addressed using PH curves of degree 7 and Euler-Rodrigues frames. G(1) continuity between frames is enforced, and conditions for achieving general geometric continuity are investigated using quaternion polynomials. The approach shows that G(k) continuity of ER frames implies G(k+1) continuity of the corresponding PH curves, leading to the potential definition of spline motions.
COMPUTER AIDED GEOMETRIC DESIGN
(2021)
Article
Mathematics, Applied
Rida T. Farouki, Marjeta Knez, Vito Vitrih, Emil Zagar
Summary: This paper presents a novel approach to constructing polynomial minimal surfaces using Pythagorean triples of complex polynomials, which are shown to be Pythagorean normal surfaces. The construction generalizes a previous approach based on Pythagorean triples of real polynomials and provides more free shape parameters. Additionally, the minimal surfaces obtained have the property of preserving Pythagorean-hodograph curves when one of the complex polynomials is a constant. Examples of cubic, quartic, and quintic minimal PN surfaces are provided, including solutions to the Plateau problem. The construction is also extended to minimal surfaces with non-isothermal parameterizations.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Computer Science, Software Engineering
Rida T. Farouki
Summary: This paper focuses on the problem of identifying the planar Pythagorean-hodograph curve that is closest to a given Bezier curve, while sharing the same end points (or end points and tangents). The methodology reduces the problem to the minimization of a quartic penalty function in certain real variables, subject to quadratic constraints, and can be efficiently solved using the Lagrange multiplier method.
COMPUTER-AIDED DESIGN
(2022)
Article
Mathematics, Applied
Emil Zagar
Summary: This paper focuses on the interpolation problem of two points, two corresponding tangent directions, curvatures, and arc length sampled from a circular arc (circular arc data). A general approach using Planar Pythagorean-hodograph (PH) curves of degree seven is presented, and the strong dependence of the solution on the general data is demonstrated. For circular arc data, a complicated system of nonlinear equations is reduced to a numerical solution of only one algebraic equation of degree 6, and the existence of admissible solutions is analyzed. Criteria for selecting the most appropriate solution in the case of multiple solutions are described, and an asymptotic analysis is provided. Numerical examples are used to validate the theoretical results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Computer Science, Software Engineering
Song-Hwa Kwon, Chang Yong Han
Summary: The paper presents a scheme to find spatial quintic Pythagorean-hodograph (PH) curves that interpolate given first-order Hermite data and Frenet frames. The approach determines two free parameters in general quintic PH interpolants to adjust the orientation of binormal vectors. By using a cubic interpolant as a reference, a quintic PH interpolant is produced that shares the first-order Hermite data and Frenet frames with the cubic counterpart.
COMPUTER AIDED GEOMETRIC DESIGN
(2021)
Article
Mathematics, Applied
Hans-Peter Schroecker, Zbynek Sir
Summary: All rational parametric curves with prescribed polynomial tangent direction form a vector space, including the important case of rational Pythagorean hodograph curves. The vector subspaces defined by fixing the denominator polynomial are studied, and the construction of canonical bases for them is described. It is also shown that any element of the vector space can be obtained as a finite sum of curves with single roots at the denominator, analogous to the fraction decomposition of rational functions. These results provide insight into the structure of these spaces, clarify the role of polynomial and non-polynomial curves, and suggest applications to interpolation problems.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Computer Science, Software Engineering
Miroslav Lavicka, Jan Vrsek
Summary: This article studies the geometric conditions under which a spatial curve is projected to a PH curve. The authors introduce suitable geometric characteristics of the curves with PH property, including intersection multiplicity. The results show that generic polynomial curves do not have parallel projections to PH curves, but there are finitely many ways to project spatial cubic and quintic curves.
COMPUTER AIDED GEOMETRIC DESIGN
(2022)
Article
Computer Science, Information Systems
Kinga Kruppa
Summary: The article discusses the importance of special curves in Minkowski space and their applications in computer-aided geometric design. A new application area for RE curves is proposed, along with a novel method for skinning a discrete set of input circles, overcoming previous challenges. A significant advantage of the proposed method is the efficiency of trimming offsets of boundaries, particularly beneficial in computer numerical control machining.
FRONTIERS OF INFORMATION TECHNOLOGY & ELECTRONIC ENGINEERING
(2021)
Article
Computer Science, Software Engineering
Gyorgy Papp, Miklos Hoffmann, Ildiko Papp
Summary: This paper presents a method of embedding QR codes onto surfaces using 3D printing technology. Compared to traditional methods, an improved approach is proposed to project QR codes onto highly curved surfaces robustly, avoiding deformations. Further validation and comparison with previous research are conducted.
COMPUTER-AIDED DESIGN
(2021)
Article
Computer Science, Software Engineering
Istvan Csoba, Roland Kunkli
Summary: This paper introduces a new simulation method for visual aberrations that runs at interactive, near real-time performance on commodity hardware and supports arbitrary types of aberrations. By utilizing a custom parametric eye model and parameter estimation method, along with a GPU-based interpolation scheme and convolution approach, the method achieves its goal effectively.
COMPUTER GRAPHICS FORUM
(2021)
Article
Computer Science, Software Engineering
Gyorgy Papp, Miklos Hoffmann, Ildiko Papp
Summary: QR codes are widely used for encoding information through images, which can be easily decoded using smartphones. One method to minimize deformation is to affix the label containing the QR code onto a developable surface patch of a 3D model. An alternative approach is embedding QR codes onto B-spline surfaces of CAD models.
COMPUTER GRAPHICS FORUM
(2022)
Article
Computer Science, Information Systems
Miklos Hoffmann, Imre Juhasz, Ede Troll
Summary: This paper investigates the use of developable surfaces as mirrors and presents an algorithm for computing caustic surfaces. The potential application of these surfaces in contemporary free-form architecture design is also discussed.
FRONTIERS OF INFORMATION TECHNOLOGY & ELECTRONIC ENGINEERING
(2022)
Article
History & Philosophy Of Science
Miklos Hoffmann
Summary: This paper discusses the emergence of digital virtual worlds in the animation industry and its implications for the role and authority of mathematics. It argues that while the application of mathematics to digital virtual worlds is similar to its application in describing real-world phenomena, there are significant differences. The main thesis is that mathematics in the animation industry can have a different ontological role, going beyond being just a modelling tool. Through the study of phenomena like gravity, the paper explores the creative role of mathematics in digital virtual worlds. Overall, the animation industry opens up a new chapter in the relationship between sciences and mathematics.
Article
Education & Educational Research
Csaba Csapodi, Miklos Hoffmann
Summary: The new National Core Curriculum in Hungarian schools, influenced by the COVID-19 pandemic, emphasizes the importance of skills in displaying, understanding and processing information in mathematics education. Improvements and proposals were made regarding the representation, interpretation, and critical evaluation of data and information, as well as the introduction of a new type of task for the matriculation exam: a complex essay task, to develop cross-cutting competencies in students.
EDUCATION SCIENCES
(2021)
Article
Education & Educational Research
Miklos Hoffmann, Laszlo Nemeth
Summary: In this experimental study, it was found that most people, including first-year students, have a common visual understanding of seeing a cube as opposed to a general cuboid. This common sense is surprisingly close to the conventions applied in axonometric drawings and the theoretical geometric solution in three-point perspective drawings.
EDUCATION SCIENCES
(2021)
Article
History & Philosophy Of Science
Miklos Hoffmann, Attila Egri-Nagy
Summary: The paper suggests flattening curricula by developing self-contained micro topics and providing multiple entry points to knowledge with sparse dependency graphs, in order to improve the effectiveness of teaching mathematics. A less strictly hierarchical schedule in mathematics education can reduce mathematics anxiety and prevent students from 'losing the thread', but a radical re-evaluation of standard teaching methods is required.
Article
Mathematics
Andrea Bodonyi, Gyozo Kurucz, Gabor Hollo, Roland Kunkli
Summary: Many research projects require visualization systems to manage large amounts of research-specific data with a specific structure. The growing data volume may become substantially transparent visually, necessitating custom visualization. A tool was designed to process simulation data, address issues from indirect analysis, and support data analysis through built-in tools.
ANNALES MATHEMATICAE ET INFORMATICAE
(2021)
Article
Computer Science, Interdisciplinary Applications
Andrea Bodonyi, Roland Kunkli
VISUAL COMPUTING FOR INDUSTRY BIOMEDICINE AND ART
(2020)
Article
Mathematics, Interdisciplinary Applications
Miklos Hoffmann
SYMMETRY-CULTURE AND SCIENCE
(2020)
