scholarly journals Subdivisions of Ring Dupin Cyclides Using Bézier Curves with Mass Points

2021 ◽  
Vol 20 ◽  
pp. 581-596
Author(s):  
Lionel Garnier ◽  
Lucie Druoton ◽  
Jean-Paul Bécar ◽  
Laurent Fuchs ◽  
Géraldine Morin
Keyword(s):  
Lorentz Space ◽  
Geometric Modeling ◽  
Linear Equations ◽  
Algebraic Surfaces ◽  
Bezier Curves ◽  
Bézier Curves ◽  
French Mathematician ◽  
Cad Cam ◽  
Dupin Cyclide ◽  
Dupin Cyclides

Dupin cyclides are algebraic surfaces introduced for the first time in 1822 by the French mathematician Pierre-Charles Dupin. A Dupin cyclide can be defined as the envelope of a one-parameter family of oriented spheres, in two different ways. R. Martin is the first author who thought to use these surfaces in CAD/CAM and geometric modeling. The Minkowski-Lorentz space is a generalization of the space-time used in Einstein’s theory, equipped of the non-degenerate indefinite quadratic form QM(u) = x^2 + y^2 + z^2 - c^2 t^2where (x, y, z) are the spacial components of the vector u and t is the time component of u and c is the constant of the speed of light. In this Minkowski-Lorentz space, a Dupin cyclide is the union of two conics on the unit pseudo-hypersphere, called the space of spheres, and a singular point of a Dupin cyclide is represented by an isotropic vector. Then, we model Dupin cyclides using rational quadratic Bézier curves with mass points. The subdivisions of a surface i.e. a Dupin cyclide, is equivalent to subdivide two curves of degree 2, independently, whereas in the 3D Euclidean space ε3, the same work implies the subdivision of a rational quadratic Bézier surface and resolutions of systems of three linear equations. The first part of this work is to consider ring Dupin cyclides because the conics are circles which look like ellipses.

Construction of 3D Triangles on Dupin Cyclides

2011 ◽  
Vol 1 (2) ◽  
pp. 42-57 ◽  
Author(s):  
Bertrand Belbis ◽  
Lionel Garnier ◽  
Sebti Foufou
Keyword(s):  
Algebraic Surfaces ◽  
Parametric Equation ◽  
French Mathematician ◽  
Lines Of Curvature ◽  
Implicit Equations ◽  
Cad Cam ◽  
Dupin Cyclide ◽  
Dupin Cyclides ◽  
Definition Of ◽  
The 19Th Century

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, parallel arcs, and Villarceau circles) can be computed on every Dupin cyclide. A geometric algorithm to compute these circles so that they define the edges of a 3D triangle on the Dupin cyclide is presented. Examples of conversions and 3D triangles are also presented to illustrate the proposed algorithms.

Construction of 3D Triangles on Dupin Cyclides

2013 ◽  
pp. 113-127
Author(s):  
Bertrand Belbis ◽  
Lionel Garnier ◽  
Sebti Foufou
Keyword(s):  
Algebraic Surfaces ◽  
Parametric Equation ◽  
French Mathematician ◽  
Lines Of Curvature ◽  
Implicit Equations ◽  
Cad Cam ◽  
Dupin Cyclide ◽  
Dupin Cyclides ◽  
Definition Of ◽  
The 19Th Century

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, parallel arcs, and Villarceau circles) can be computed on every Dupin cyclide. A geometric algorithm to compute these circles so that they define the edges of a 3D triangle on the Dupin cyclide is presented. Examples of conversions and 3D triangles are also presented to illustrate the proposed algorithms.

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 Approximate Multidegree Reduction ofλ-Bézier Curves

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Gang Hu ◽  
Huanxin Cao ◽  
Suxia Zhang
Keyword(s):  
Shape Control ◽  
Control Parameter ◽  
Linear Equations ◽  
Least Square ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Control Points ◽  
Degree Reduction ◽  
Bezier Curves ◽  
Bézier Curves

Besides inheriting the properties of classical Bézier curves of degreen, the correspondingλ-Bézier curves have a good performance in adjusting their shapes by changing shape control parameter. In this paper, we derive an approximation algorithm for multidegree reduction ofλ-Bézier curves in theL2-norm. By analysing the properties ofλ-Bézier curves of degreen, a method which can deal with approximatingλ-Bézier curve of degreen+1byλ-Bézier curve of degreem  (m≤n)is presented. Then, in unrestricted andC0,C1constraint conditions, the new control points of approximatingλ-Bézier curve can be obtained by solving linear equations, which can minimize the least square error between the approximating curves and the original ones. Finally, several numerical examples of degree reduction are given and the errors are computed in three conditions. The results indicate that the proposed method is effective and easy to implement.

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.

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.

Blending and Joining Using Cyclides

1994 ◽  
Vol 116 (4) ◽  
pp. 1034-1041 ◽  
Author(s):  
Y. L. Srinivas ◽  
D. Dutta
Keyword(s):  
Geometric Modeling ◽  
Algebraic Surfaces ◽  
Variable Radius ◽  
Dupin Cyclides ◽  
Definition Of ◽  
General Method

We consider the use of Dupin cyclides in geometric modeling. These are algebraic surfaces of degree four with interesting properties such as rational parametric forms and closure under offsets. We have been focusing on methods for expanding the geometric coverage of solid modelers, and the cyclide as a new primitive offers promise. In this paper, we briefly describe the cyclide and discuss in details two applications. The first is the modeling of blending surfaces, an important problem in geometric modeling. We outline methods to use various forms of the cyclide in variable-radius blending. Next, we consider the problem of automatic joining of pipes, and describe a general method rooted in the definition of a cyclide for its solution. Implemented examples are provided for both applications.

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

scholarly journals Offset Approximation of Rational Trigonometric Bezier ´ Curves

2021 ◽  
pp. 351-365
Author(s):  
Uzma Bashir ◽  
Aqsa Rasheed
Keyword(s):  
Root Function ◽  
Square Root ◽  
Control Points ◽  
Bezier Curves ◽  
Bézier Curves ◽  
End Points ◽  
Control Polygon ◽  
Cad Cam ◽  
Offset Curves ◽  
Definition Of

Offset curves are one of the crucial curves, but the presence of square root function in the representation is main hindrance towards their applications in CAD/CAM. The presented technique is based on offset approximation using rational trigonometric Bezier curves. The idea is ´ to construct a new control polygon parallel to original one. The two end points of the offset control polygon have been taken as exact offset end points, while the middle control points and weights have been computed using definition of parallel curves. As a result, offsets of rational and nonrational trigonometric Bezier curves have been approximated by rational ´ cubic trigonometric Bezier curve. An error between exact and approxi- ´ mated offset curves have also been computed to show the efficacy of the method.

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