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 Curve and surface construction based on the generalized toric-Bernstein basis functions

2020 ◽  
Vol 18 (1) ◽  
pp. 36-56 ◽  
Author(s):  
Jing-Gai Li ◽  
Chun-Gang Zhu
Keyword(s):  
Computer Aided Design ◽  
Geometric Modeling ◽  
Basis Functions ◽  
Bernstein Basis ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Parametric Curve ◽  
Curves And Surfaces ◽  
Computer Aided ◽  
Bernstein Basis Functions

Abstract The construction of parametric curve and surface plays an important role in computer aided geometric design (CAGD), computer aided design (CAD), and geometric modeling. In this paper, we define a new kind of blending functions associated with a real points set, called generalized toric-Bernstein (GT-Bernstein) basis functions. Then, the generalized toric-Bézier (GT-Bézier) curves and surfaces are constructed based on the GT-Bernstein basis functions, which are the projections of the (irrational) toric varieties in fact and the generalizations of the classical rational Bézier curves/surfaces and toric surface patches. Furthermore, we also study the properties of the presented curves and surfaces, including the limiting properties of weights and knots. Some representative examples verify the properties and results.

Approximation of Physical Spline with Large Deflections

2021 ◽  
pp. 3-18
Author(s):  
Viktor Korotkiy ◽  
Igor' Vitovtov
Keyword(s):  
Mathematical Problem ◽  
Minimum Energy ◽  
Internal Stresses ◽  
Radius Of Curvature ◽  
Large Deflections ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Cross Sectional ◽  
Elastic Line ◽  
Average Curvature

Physical spline is a resilient element whose cross-sectional dimensions are very small compared to its axis’s length and radius of curvature. Such a resilient element, passing through given points, acquires a "nature-like" form, having a minimum energy of internal stresses, and, as a consequence, a minimum of average curvature. For example, a flexible metal ruler, previously used to construct smooth curves passing through given coplanar points, can be considered as a physical spline. The theoretical search for the equation of physical spline’s axis is a complex mathematical problem with no elementary solution. However, the form of a physical spline passing through given points can be obtained experimentally without much difficulty. In this paper polynomial and parametric methods for approximation of experimentally produced physical spline with large deflections are considered. As known, in the case of small deflections it is possible to obtain a good approximation to a real elastic line by a set of cubic polynomials ("cubic spline"). But as deflections increase, the polynomial model begins to differ markedly from the experimental physical spline, that limits the application of polynomial approximation. High precision approximation of an elastic line with large deflections is achieved by using a parameterized description based on Ferguson or Bézier curves. At the same time, not only the basic points, but also the tangents to the elastic line of the real physical spline should be given as boundary conditions. In such a case it has been shown that standard cubic Bézier curves have a significant computational advantage over Ferguson ones. Examples for modelling of physical splines with free and clamped ends have been considered. For a free spline an error of parametric approximation is equal to 0.4 %. For a spline with clamped ends an error of less than 1.5 % has been obtained. The calculations have been performed with SMath Studio computer graphics system.

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.

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.

3D-Olympics And Computer Design Competition in Technical Universities Programs

2015 ◽  
Vol 3 (2) ◽  
pp. 52-59 ◽  
Author(s):  
Вельтищев ◽  
Vitaliy Weltishev
Keyword(s):  
Computer Aided Design ◽  
Geometric Modeling ◽  
Surface Modeling ◽  
Practical Training ◽  
Training Methods ◽  
Significant Activity ◽  
Computer Design ◽  
Self Study ◽  
Design Competition ◽  
Design Solutions

The author shares his experience of training 3D geometric design competition for technical objects using modern of CAD-systems. It proposed the idea of development of creative abilities of students in the process of learning the basics of geometric modeling by incorporating elements of the structure of the training methods of self-development. It is noted that the principle of competition among the students in the group, encourages the pursuit of the study of modern packages and the acquisition of new skills. Created aimed at the development of creative abilities formation technique Olympiad learning objectives based on the real experience of designing in engineering offices and proposed a new form of training for a competition at the expense of an additional self-study students. As the experience of the first 3D-competitions, the task of a typical section of the course of the IG does not attract active and creative students who want to learn to work at a high level and modern packages. Systematic preparation for the 3D-competition is a necessary and very useful tool to stimulate and self-study of the subject as "Fundamentals of geometric modeling." This subject is not in the programs of universities, but it is indispensable. The tasks of the Olympiad are encouraged to include practical design solutions, both in the solid state, and in surface modeling. In more complex tasks with surface modeling is particularly interesting to students. They seek to independently study the theoretical and practical side of the work in computer-aided design packages, so the competition celebrated its significant activity. In tasks you are offered the use of standard methods of practical surface modeling to create the technical and design form the projections and conceptual sketches. The article gives examples of practical solutions, which can be seen a high level of theoretical and practical training of participants in 3D geometric design competition, there is speed of execution of tasks and the ability to solve professional design solutions.

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.

Approximation of Physical Spline with Large Deflections

2021 ◽  
Vol 9 (1) ◽  
pp. 3-19
Author(s):  
Viktor Korotkiy ◽  
Igor' Vitovtov
Keyword(s):  
Mathematical Problem ◽  
Minimum Energy ◽  
Internal Stresses ◽  
Radius Of Curvature ◽  
Large Deflections ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Cross Sectional ◽  
Elastic Line ◽  
Average Curvature

Physical spline is a resilient element whose cross-sectional dimensions are very small compared to its axis’s length and radius of curvature. Such a resilient element, passing through given points, acquires a "nature-like" form, having a minimum energy of internal stresses, and, as a consequence, a minimum of average curvature. For example, a flexible metal ruler, previously used to construct smooth curves passing through given coplanar points, can be considered as a physical spline. The theoretical search for the equation of physical spline’s axis is a complex mathematical problem with no elementary solution. However, the form of a physical spline passing through given points can be obtained experimentally without much difficulty. In this paper polynomial and parametric methods for approximation of experimentally produced physical spline with large deflections are considered. As known, in the case of small deflections it is possible to obtain a good approximation to a real elastic line by a set of cubic polynomials ("cubic spline"). But as deflections increase, the polynomial model begins to differ markedly from the experimental physical spline, that limits the application of polynomial approximation. High precision approximation of an elastic line with large deflections is achieved by using a parameterized description based on Ferguson or Bézier curves. At the same time, not only the basic points, but also the tangents to the elastic line of the real physical spline should be given as boundary conditions. In such a case it has been shown that standard cubic Bézier curves have a significant computational advantage over Ferguson ones. Examples for modelling of physical splines with free and clamped ends have been considered. For a free spline an error of parametric approximation is equal to 0.4 %. For a spline with clamped ends an error of less than 1.5 % has been obtained. The calculations have been performed with SMath Studio computer graphics system.

scholarly journals Approximate Multi-Degree Reduction of SG-Bézier Curves Using the Grey Wolf Optimizer Algorithm

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1242
Author(s):  
Hu ◽  
Qiao ◽  
Qin ◽  
Wei
Keyword(s):  
Computer Aided Design ◽  
Fitness Function ◽  
Grey Wolf Optimizer ◽  
Grey Wolf ◽  
Degree Reduction ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Geometric Properties ◽  
Reduction Effect ◽  
Aided Design

SG-Bézier curves have become a useful tool for shape design and geometric representation in computer aided design (CAD), owed to their good geometric properties, e.g., symmetry and convex hull property. Aiming at the problem of approximate degree reduction of SG-Bézier curves, a method is proposed to reduce the n-th SG-Bézier curves to m-th (m < n) SG-Bézier curves. Starting from the idea of grey wolf optimizer (GWO) and combining the geometric properties of SG-Bézier curves, this method converts the problem of multi-degree reduction of SG-Bézier curves into solving an optimization problem. By choosing the fitness function, the approximate multi-degree reduction of SG-Bézier curves with adjustable shape parameters is realized under unrestricted and corner interpolation constraints. At the same time, some concrete examples of degree reduction and its errors are given. The results show that this method not only achieves good degree reduction effect, but is also easy to implement and has high accuracy.

Regularity of Bézier Curves

2011 ◽  
Vol 48-49 ◽  
pp. 877-880
Author(s):  
Qing Xian Meng ◽  
Hui Li Liu
Keyword(s):  
Computer Aided Design ◽  
Practical Method ◽  
Singular Points ◽  
Algebraic Equations ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Computer Aided ◽  
Aided Design ◽  
Zero Points ◽  
Isolation Theorem

Regularity is one of important properties of curves in computer aided design. In this paper, we convert the problem of determining regularity of Bézier curves to that of detecting existence of zero points of polynomials. Based on the properties of algebraic equations and isolation theorem of roots, a simple and practical method is presented. Regularity of Bézier curves and number of singular points can be determined by easier computation.

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
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