NATURE-LIKE CURVE MODELING

2021 ◽  
Vol 2021 (6) ◽  
pp. 11-22
Author(s):  
Viktor Korotkiy ◽  
Igor' Vitovtov
Keyword(s):  
Geometric Modeling ◽  
Mathematical Problem ◽  
Minimum Energy ◽  
System Of Nonlinear Equations ◽  
Large Deflections ◽  
Elastic Rod ◽  
Elastic Line ◽  
Investigation Methods ◽  
Curve Modeling ◽  
Straight Lines

A physical spline is called an elastic rod the cross- section dimensions of which are rather small as compared with the length and radius of its axis curvature. Such a rod when passing through specified points obtains in natural way a nature-like shape characterized with minimum energy of inner stresses and minimum mean curvature. A search for the equation of elastic line is a difficult mathematical problem having no elementary solution. The work purpose: the development of the experimental-rated procedure for modeling a nature-like elastic curve passing through complanar points specified in advance. The investigation methods: methods of piece-cubic interpolation based on the application of polynomial splines and compound curves specified by parametric equations. In the paper there are considered polynomial and parametric methods of the geometric modeling of the physical spline passing through the points specified in advance. The elastic line of the physical spline is obtained experimentally. The investigation results: it is shown that unlike a polynomial model a parametrized model on the basis of Fergusson curve gives high accuracy of approximation if in basic points there are specified tangents to the elastic line of the physical spline with large deflections. Novelty: there is offered a simplified method for the computation of factors of an approximating spline allowing the substitution of the 2n system of nonlinear equations (smoothness conditions) by the successive solution of n systems of two equations. Conclusions: for the modeling of nature-like curves with large deflections there is offered the application of Fergusson cubic spline passing through specified points and touching the specified straight lines in these points. The error of the modeling of the natural elastic line with free ends at n=5 does not exceed 0.4%.

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.

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.

Applications of PNC in Numerical Methods

2017 ◽  
pp. 235-268
Keyword(s):  
Probability Distribution ◽  
Shape Representation ◽  
Mathematical Problem ◽  
Distribution Functions ◽  
Two Dimensional ◽  
Key Points ◽  
Characteristic Points ◽  
Curve Modeling ◽  
Curve Contour ◽  
Dimensional Object

Nodes are treated as characteristic points of data for modeling and analyzing. The model of data can be built by choice of probability distribution function and nodes combination. Two-dimensional object is extrapolated and interpolated via nodes combination and different functions as discrete or continuous probability distribution functions: polynomial, sine, cosine, tangent, cotangent, logarithm, exponent, arc sin, arc cos, arc tan, arc cot or power function. Curve interpolation represents one of the most important problems in mathematics and computer science: how to model the curve via discrete set of two-dimensional points? Also the matter of shape representation (as closed curve - contour) and curve parameterization is still opened. For example pattern recognition, signature verification or handwriting identification problems are based on curve modeling via the choice of key points. So interpolation is not only a pure mathematical problem but important task in computer vision and artificial intelligence.

The shape of a Möbius band

1993 ◽  
Vol 440 (1908) ◽  
pp. 149-162 ◽  
Keyword(s):  
Aspect Ratio ◽  
Asymptotic Solution ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Elastic Rods ◽  
Large Deflections ◽  
Mechanical Equilibrium ◽  
Elastic Rod ◽  
Flexible Material ◽  
Möbius Band

The shape of a Möbius band made of a flexible material, such as paper, is determined. The band is represented as a bent, twisted elastic rod with a rectangular cross-section. Its mechanical equilibrium is governed by the Kirchhoff–Love equations for the large deflections of elastic rods. These are solved numerically for various values of the aspect ratio of the cross-section, and an asymptotic solution is found for large values of this ratio. The resulting shape is shown to agree well with that of a band made from a strip of plastic.

A Quadratic Trigonometric Nu Spline with Shape Control

2017 ◽  
Vol 17 (03) ◽  
pp. 1750015 ◽  
Author(s):  
Muhammad Sarfraz ◽  
Shamaila Samreen ◽  
Malik Zawwar Hussain
Keyword(s):  
Computer Aided Design ◽  
Geometric Modeling ◽  
Shape Control ◽  
Partition Of Unity ◽  
Control Points ◽  
Geometric Continuity ◽  
Spline Method ◽  
Shape Effects ◽  
Curve Modeling ◽  
Aided Design

A significant curve modeling technique has been introduced with a view to its applications in geometric modeling, computer graphics and computer-aided design. It is a new spline method using quadratic trigonometric functions with well-controlled shape influences of parameters introduced through geometric continuity of order two. The proposed curve model owns the best possible geometric properties such as convex hull, partition of unity, affine invariance and variation diminishing. A quadratic normally has three control points giving lesser flexibility to one piece of curve. However, a cubic has four control points giving higher flexibility to one piece of curve. In the proposed scheme, we have introduced a quadratic trigonometric with four control points. Thus, the proposed quadratic trigonometric has embedded geometric features of cubic/cubic trigonometric. The proposed spline method, constrained with Nu spline like GC2 smoothness, produces a quadratic trigonometric Nu spline (QTNS) with interesting shape control locally and globally. The method is helpful for a variety of shape effects, like point tension, interval tension or global tension. It also produces a quadratic trigonometric alternative to cubic/cubic trigonometric spline because of having four control points in its piecewise description.

scholarly journals FORMATION OF OPTIMAL PARAMETRS OF THE TRAJECTORY OF THE OVERFLIGHT OF UNMANNED AERIAL VEHICLE THROUGH THE SPECIFIED POINTS OF SPACE

Doklady BGUIR ◽  
2019 ◽  
pp. 50-57
Author(s):  
A. A. Lobaty ◽  
A. Y. Bumai ◽  
Du Jun
Keyword(s):  
Unmanned Aerial Vehicle ◽  
Mathematical Problem ◽  
Minimum Energy ◽  
Optimization Criterion ◽  
Flight Path ◽  
Reference Points ◽  
Control Synthesis ◽  
Control Law ◽  
Analytical Control ◽  
Aerial Vehicle

The purpose of the scientific research, results are determinated in the article, is to analytically synthesize the control law of an unmanned aerial vehicle while guiding one along the trajectory that specified by the reference points of space in an inertial coordinate system. The analysis of various existing approaches of the formation of a given flight path of an unmanned aerial vehicle based on various mathematical formulations of the problem is carried out. To achieve the goal, the flight path is considered as separate intervals, where the control optimization problem is solved. The optimization criterion in general form is substantiated and its presentation in the form of a minimized quadratic quality functional is convenient for analytical control synthesis. As components of the functional, the parameters of the deviation of the flight path of the aircraft from the specified points of space are considered, as well as the predicted parameters of the velocity vector and the control normal acceleration. Moreover, at each specified point in space, the direction of the trajectory to the subsequent point is taken into account, that ensures optimal curvature of the trajectory by specified flight speed of the unmanned aerial vehicle. As a result of analytical synthesis, mathematical dependences are obtained to determine control acceleration, which allow us to get a specified optimal control law on board an unmanned aerial vehicle, which ultimately ensures minimum energy consumption. The validity of the proposed theoretical provisions is confirmed by a clear example, where for a simplified mathematical problem statement the optimal laws of change in control acceleration and the trajectory parameters of an unmanned aerial vehicle are calculated by computer simulation.

Flat Spring With Large Deflections

1946 ◽  
Vol 13 (3) ◽  
pp. A223-A230
Author(s):  
F. Hymans
Keyword(s):  
Single Point ◽  
The Other ◽  
Large Deflections ◽  
Elastic Line

Abstract This paper gives an analysis of a flat spring initially curved and inserted in the apparatus of which it is an element in a buckled condition. The unstressed spring is shaped in the arc of a circle, and it is shown that then, and only then, the problem may be reduced to that of an initially straight bar by adding to the actually impressed force a couple which, if acting alone, produces the initial curved shape. On this basis the paper ascertains the various possible shapes of the buckled spring and the forces associated with them. It is shown that these must have at least one point of inflection of the elastic line. More particularly, there are two shapes, each with a single point of inflection, yet differing widely in their properties. In one of them there is a range of instability while the other is stable throughout. Finally, it is shown how by a suitable variation of parameter shapes two or more points of inflection may be obtained.

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.

Second Order Curves on Computer Screen

2018 ◽  
Vol 6 (2) ◽  
pp. 100-112 ◽  
Author(s):  
Виктор Короткий ◽  
Viktor Korotkiy ◽  
Е. Усманова ◽  
E. Usmanova
Keyword(s):  
Computer Graphics ◽  
Computer Program ◽  
Geometric Modeling ◽  
Main Tool ◽  
Second Order ◽  
Modern Computer ◽  
Order Curve ◽  
Junction Points ◽  
Tools And Techniques ◽  
Straight Lines

Modern computer graphics is based on methods of computational geometry. The curves and surfaces’ description is based on apparatus of spline functions, which became the main tool for geometric modeling. Methods of projective geometry are almost not applying. One of the reasons for this is impossibility to exactly construct a second-order curve passing through given points and tangent to given straight lines. To eliminate this defect a computer program for second order curves construction has been developed. The program performs the construction of second-order curve’s metric (center, vertices, asymptotes, foci) for following combinations: • The second-order curve is given by five points; • The second-order curve is given by five tangent lines; • The second-order curve is given by a point and two tangent lines with points of contact indicated on them; • The parabola is given by four tangent lines; • The parabola is given by four points. In this paper are presented algorithms for construction a metric for each combination. After construction the metric the computer program written in AutoLISP language and using geometrically exact projective algorithms which don’t require algebraic computations draws a second-order curve. For example, to construct vertices and foci of two parabolas passing through four given points, it is only necessary to draw an arbitrary circle and several straight lines. To construct a conic metric passing through five given points, it is necessary to perform only three geometrically exact operations: to construct an involution of conjugate diameters, to find the main axes and asymptotes; to note the vertices of desired second-order curve. Has been considered the architectural appearance of a new airport in Simferopol. It has been demonstrated that a terminal facade’s wavelike form can be obtained with a curve line consisting of conic sections’ areas with common tangent lines at junction points. The developed computer program allows draw second-order curves. The program application will promote the development of computer graphics’ tools and techniques.

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