GEOMETRIC MODELING OF ADAPTIVE ALGEBRAIC CURVES PASSING THROUGH PREDETERMINED POINTS

2021 ◽  
pp. 26-34
Author(s):  
E. V. Konopatskiy ◽  
I. V. Seleznev ◽  
O. A. Chernysheva ◽  
M. V. Lagunova ◽  
A. A. Bezditnyi
Keyword(s):  
Initial Data ◽  
Geometric Modeling ◽  
Algebraic Curves ◽  
Geometric Theory ◽  
Multidimensional Space ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Multidimensional Interpolation ◽  
Geometric Objects ◽  
Parameter Values

In this paper, the geometric theory of multidimensional interpolation was further developed in terms of modeling and using adaptive curves passing through predetermined points. A feature of the proposed approach to modeling curved lines is the ability to adapt to any initial data for high-quality interpolation, which excludes unplanned oscillations, due to the uneven distribution of parameter values, the source of which are the initial data. This is the improvement of the previously proposed method for constructing and analytically describing arcs of algebraic curves passing through predetermined points, obtained on the basis of Bezier curves, which are compiled taking into account the expansion coefficients of the Newton binomial. The paper gives an example of using adaptive algebraic curves passing through predetermined points for geometric modeling of the stress-strain state of membrane coatings cylindrical shells using two-dimensional interpolation. The given example an illustrative showed the advantages of the proposed adaptation of algebraic curves passing through predetermined points and obtained on the basis of Bezier curves for geometric modeling of multifactor processes and phenomena. The use of such adaptation allows not only to avoid unplanned oscillations, but also self-intersection of geometric objects when generalized to a multidimensional space. Adaptive algebraic curves can also be effectively used as formative elements for constructing geometric objects of multidimensional space, both as guide lines and as generatrix’s.

GEOMETRIC MODELING OF MULTI-FACTOR PROCESSES BASED ON VARIABLE POINT ALGORITHMS

2021 ◽  
pp. 29-38
Author(s):  
E. V. Konopatskiy ◽  
I. V. Seleznev ◽  
M. V. Lagunova ◽  
A. A. Bezditniy
Keyword(s):  
Geometric Modeling ◽  
Geometric Theory ◽  
Physical And Mechanical Properties ◽  
Multidimensional Space ◽  
Fine Grained ◽  
Variable Approach ◽  
Multidimensional Interpolation ◽  
Geometric Objects ◽  
Variable Point ◽  
Current Point

In this paper, the geometric theory of multidimensional interpolation was further developed. It has been established that the geometric models of multivariate processes obtained using multidimensional interpolation are characterized by variability, which is a consequence of the multiplicity of choice of reference lines in the process of developing a geometric modeling scheme. At the same time, all possible variations of geometric interpolants fully satisfy the initial experimental and statistical data, but have different curvature between the node points of the interpolation. As the dimension of the space increases, the number of variations increases significantly. The variable approach to geometric modeling of multifactorial processes generates a number of scientific problems that require further research, such as: comparison of geometric objects of multidimensional space, development of criteria for choosing the best solutions, construction of averaged geometric objects as one of the tools for optimizing the results of modeling, etc. The article also presents the results of a computational experiment on geometric modeling of the dependence of the physical and mechanical properties of fine-grained concrete on the composition of the combined aggregate based on variable point algorithms with the subsequent construction of an averaged response surface, the current point of which is the center of gravity of a multidimensional tetrahedron, for which the dimension of space depends on the amount possible interpolation options.

GEOMETRIC THEORY OF MULTIDIMENSIONAL INTERPOLATION

2020 ◽  
Vol 2020 (1) ◽  
pp. 9-16
Author(s):  
Evgeniy Konopatskiy
Keyword(s):  
Response Function ◽  
Algebraic Curves ◽  
Geometric Theory ◽  
Analytical Description ◽  
Geometric Interpretation ◽  
Affine Geometry ◽  
Mathematical Apparatus ◽  
Multidimensional Interpolation ◽  
Pass Through

The paper presents a geometric theory of multidimensional interpolation based on invariants of affine geometry. The analytical description of geometric interpolants is performed within the framework of the mathematical apparatus BN-calculation using algebraic curves that pass through preset points. A geometric interpretation of the interaction of parameters, factors, and the response function is presented, which makes it possible to generalize the geometric theory of multidimensional interpolation in the direction of increasing the dimension of space. The conceptual principles of forming the tree of the geometric interpolant model as a geometric basis for modeling multi-factor processes and phenomena are described.

scholarly journals Geometric Modeling of the Valencia Orange (Citrus sinensis L.) by Applying Bézier Curves and an Image-Based CAD Approach

Agriculture ◽  
2020 ◽  
Vol 10 (8) ◽  
pp. 313
Author(s):  
Hector A. Tinoco ◽  
Daniel R. Barco ◽  
Olga Ocampo ◽  
Jaime Buitrago-Osorio
Keyword(s):  
Computer Aided Design ◽  
Citrus Sinensis ◽  
Geometric Modeling ◽  
Prediction Models ◽  
Two Dimensions ◽  
Radius Of Curvature ◽  
Computer Design ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Valencia Orange

The computer-aided design of fruits are used for different purposes, e.g., to determine mechanical properties by applying engineering simulations, to design postharvest equipment, and to study the natural changes related to the topology. This paper developed a methodology to model Valencia orange (Citrus sinensis), applying Bézier curves and an image-based CAD approach; the orange geometry was designed for different ripening stages. In the modeling process, a 3D construction was carried out using third-order Bézier curves, adjusted to the images taken in orthogonal planes. Four control points defined each profile to compose the geometric pattern of the orange, with geometric errors lower than 3%. Two prediction models were proposed to relate the orthogonal dimensions with a factor size; this means that two dimensions out of three can be predicted. The results showed that the shape ratios kept constant in any ripening stage; however, the radius of curvature evidenced differences in the analyzed shape profiles. The methodological framework presented in the paper might be used to draw other types of citrus fruits. This contribution is a tool to model fruits in 3D, instead of using expensive technological equipment, since it is only necessary to apply computer design tools.

scholarly journals Geometric Modeling of Novel Generalized Hybrid Trigonometric Bézier-Like Curve with Shape Parameters and Its Applications

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 967 ◽  
Author(s):  
Samia BiBi ◽  
Muhammad Abbas ◽  
Kenjiro T. Miura ◽  
Md Yushalify Misro
Keyword(s):  
Geometric Modeling ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Geometric Continuity ◽  
Shape Parameters ◽  
Bernstein Basis ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Curvature Continuity ◽  
Bernstein Basis Functions

The main objective of this paper is to construct the various shapes and font designing of curves and to describe the curvature by using parametric and geometric continuity constraints of generalized hybrid trigonometric Bézier (GHT-Bézier) curves. The GHT-Bernstein basis functions and Bézier curve with shape parameters are presented. The parametric and geometric continuity constraints for GHT-Bézier curves are constructed. The curvature continuity provides a guarantee of smoothness geometrically between curve segments. Furthermore, we present the curvature junction of complex figures and also compare it with the curvature of the classical Bézier curve and some other applications by using the proposed GHT-Bézier curves. This approach is one of the pivotal parts of construction, which is basically due to the existence of continuity conditions and different shape parameters that permit the curve to change easily and be more flexible without altering its control points. Therefore, by adjusting the values of shape parameters, the curve still preserve its characteristics and geometrical configuration. These modeling examples illustrate that our method can be easily performed, and it can also provide us an alternative strong strategy for the modeling of complex figures.

scholarly journals Approximation the geometric objects of multidimensional space using arcs of curves passing through the given points

2019 ◽  
Author(s):  
Евгений Конопацкий ◽  
Evgeniy Konopatskiy ◽  
Сергей Ротков ◽  
Sergey Rotkov
Keyword(s):  
Response Function ◽  
Algebraic Curves ◽  
Influence Factors ◽  
Geometric Object ◽  
Multidimensional Space ◽  
Global Coordinate System ◽  
Factors Affecting ◽  
Geometric Objects ◽  
Basic Ideas ◽  
The Given

The paper presents the basic ideas of geometric objects approximation in multidimensional space by means the arcs of algebraic curves passing through given points, which is as follows. A special network of points with a dimension one less than the dimension of the space in which the simulated geometric object is located is formed. Taking into account the special properties the arcs of algebraic curves passing through the given points, a linear relationship between the parameters of the geometric object and the influence factors corresponding to the axes of the global coordinate system is established. Next, the nodes of the network are calculated such values of the response function, which provide the minimum value of the quadratic residual function. The proposed method allows to perform the generalization the method of least squares in the direction of increasing space dimension and, consequently, the number of investigated factors affecting the response function, which is especially important for modeling and optimization of multifactorial processes and phenomena.

scholarly journals MODELISASI KURSI DENGAN PENGGABUNGAN HASIL DEFORMASI BENDA-BENDA RUANG MENGGUNAKAN KURVA BEZIER

2021 ◽  
Vol 21 (2) ◽  
pp. 107
Author(s):  
Annisa Ayu Nadzira ◽  
Bagus Juliyanto ◽  
Ahmad Kamsyakawuni
Keyword(s):  
Rectangular Prism ◽  
Office Workers ◽  
Regular Hexagon ◽  
Bezier Curves ◽  
Bézier Curves ◽  
The Third ◽  
Geometric Objects ◽  
Hexagon Prism ◽  
Prism Model

Chairs are needed by humans to do some work, especially students and office workers. The parts contained in the chair are the chair legs, chair legs eats and chair backs. The purpose of this study is to obtain variations in the shape of office chairs using Bezier curves and incorporate the results of deformation of space geometric objects. In modeling this chair, it is divided into several stages, namely first, building the chair leg components. This chair leg component consists of chair wheels, connecting two wheels with tube deformation, modeling the chair leg branch components and modeling chair leg supports. Second, namely the model of the chair leg seat. Chair leg seat consists of regular hexagon prism deformation and regular quadrangle prism deformation. The third is the modelization of the back of the chair by using a rectangular prism model. The result of combining several components of the chair using one modeling axis produces 36 chair models, with special provisions, namely that the seat support parts can only be joined using a tube.

scholarly journals Geometric modeling and applications of generalized blended trigonometric Bézier curves with shape parameters

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Sidra Maqsood ◽  
Muhammad Abbas ◽  
Kenjiro T. Miura ◽  
Abdul Majeed ◽  
Azhar Iqbal
Keyword(s):  
Geometric Modeling ◽  
Recursive Formula ◽  
Basis Functions ◽  
Shape Parameters ◽  
Shape Modification ◽  
Research Topics ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Computer Aided Geometric Design ◽  
Curve Modeling

Abstract A Bézier model with shape parameters is one of the momentous research topics in geometric modeling and computer-aided geometric design. In this study, a new recursive formula in explicit expression is constructed that produces the generalized blended trigonometric Bernstein (or GBT-Bernstein, for short) polynomial functions of degree m. Using these basis functions, generalized blended trigonometric Bézier (or GBT-Bézier, for short) curves with two shape parameters are also constructed, and their geometric features and applications to curve modeling are discussed. The newly created curves share all geometric properties of Bézier curves except the shape modification property, which is superior to the classical Bézier. The $C^{3}$ C 3 and $G^{2}$ G 2 continuity conditions of two pieces of GBT-Bézier curves are also part of this study. Moreover, in contrast with Bézier curves, our generalization gives more shape adjustability in curve designing. Several examples are presented to show that the proposed method has high applied values in geometric modeling.

scholarly journals TO THE QUESTION OF GEOMETRIC MODELING USING BEZIER CURVES

2020 ◽  
Vol 0 (98) ◽  
pp. 29-34
Author(s):  
Volodymyr Vanin ◽  
Gennadii Virchenko ◽  
Petro Yablonskyi
Keyword(s):  
Geometric Modeling ◽  
Bezier Curves ◽  
Bézier Curves

L-SYSTEMS IN GEOMETRIC MODELING

2012 ◽  
Vol 23 (01) ◽  
pp. 133-146 ◽  
Author(s):  
PRZEMYSLAW PRUSINKIEWICZ ◽  
MITRA SHIRMOHAMMADI ◽  
FARAMARZ SAMAVATI
Keyword(s):  
Geometric Modeling ◽  
Rational Curves ◽  
Affine Geometry ◽  
Bezier Curves ◽  
Bézier Curves ◽  
De Casteljau Algorithm ◽  
B Splines ◽  
Context Sensitive ◽  
L Systems ◽  
Subdivision Curves

We show that parametric context-sensitive L -systems with affine geometry interpretation provide a succinct description of some of the most fundamental algorithms of geometric modeling of curves. Examples include the Lane-Riesenfeld algorithm for generating B -splines, the de Casteljau algorithm for generating Bézier curves, and their extensions to rational curves. Our results generalize the previously reported geometric-modeling applications of L -systems, which were limited to subdivision curves.

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