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.

Data Driven Parametrization for Flexible Airfoils and Predictive Aerodynamics

2016 ◽  
Author(s):  
P. Venkataraman
Keyword(s):  
Three Dimensional ◽  
Aircraft Design ◽  
Local Variation ◽  
Parameterization Scheme ◽  
Bottom Surface ◽  
Control Points ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Flexible Airfoils ◽  
Definition Of

NASA achieved an important milestone in aircraft design the past year by flight testing a shapeshifting wing. The design moved the rear region of the wing through large deflection to provide flap operation for takeoff and landing. The next step is inflight surface modification of the entire wing. Underlying the three dimensional wing is the two-dimensional airfoil shape that anchors the wing aerodynamic performance. Many parametric definition of airfoils have been used for optimizing airfoil and wing aerodynamics but these analysis were made for fixed wing configurations. For flexible airfoils, it is important to recognize that the lofting of shapes in flight will happen around a parent airfoil. From a practical perspective it is likely that only a narrow range of shapes will be possible because of limited actuator locations. With this in mind a new Bézier parameterization scheme is introduced that can reproduce current airfoils with the assurance that original aerodynamics is maintained if not improved. Two Bézier curves are used to define the airfoil. One for the top surface and the other for the bottom surface. It is shown that this parametrization lends itself to fixed abscissa placement of control points for all airfoils, identifying possible actuator locations. Bézier curves change globally to local variation in geometry so a few points can generate an effective flexible airfoil. Coupling these changes with a simple analysis program can easily generate aerodynamic sensitivity information to physical shape changes based on the changes in a limited set of control points. This will provide the ability to create a shape based on a new aerodynamic demand while in flight. This paper presents the development of the parameterization scheme only.

Smoothness of C-Bezier Curves

2000 ◽  
Author(s):  
Manhong Wen ◽  
Kwun-Lon Ting
Keyword(s):  
Shape Control ◽  
Control Point ◽  
Design Procedure ◽  
Design Parameters ◽  
Control Points ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Point Distribution ◽  
G1 Continuity ◽  
G2 Continuity

Abstract This paper presents G1 and G2 continuity conditions of c-Bezier curves. It shows that the collinear condition for G1 continuity of Bezier curves is generally no longer necessary for c-Bezier curves. Such a relaxation of constraints on control points is beneficial from the structure of c-Bezier curves. By using vector weights, each control point has two extra free design parameters, which offer the probability of obtaining G1 and G2 continuity by only adjusting the weights if the control points are properly distributed. The enlargement of control point distribution region greatly simplifies the design procedure to and enhances the shape control on constructing composite curves.

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.

Style Design System Based on Class A Bézier Curves for High-Quality Aesthetic Shapes

2012 ◽  
Author(s):  
Tetsuo Oya ◽  
Fumihiko Kimura ◽  
Hideki Aoyama
Keyword(s):  
Design System ◽  
Design Tools ◽  
Control Points ◽  
Modeling Framework ◽  
High Quality ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Class A ◽  
Monotone Curvature ◽  
Curvature And Torsion

In this paper, a style design system in which the conditions for Class A Bézier curves are applied is presented to embody designer’s intention by aesthetically high-quality shapes. Here, the term “Class A” means a high-quality shape that has monotone curvature and torsion, and the recent industrial design requires not only aesthetically pleasing aspect but also such high-quality shapes. Conventional design tools such as normal Bézier curves can represent any shapes in a modeling system; however, the system only provides a modeling framework, it does not necessarily guarantee high-quality shapes. Actually, designers do a cumbersome manipulation of many control points during the styling process to represent outline curves and feature curves; this hardship prevents designers from doing efficient and creative styling activities. Therefore, we developed a style design system to support a designer’s task by utilizing the Class A conditions of Bézier curves with monotone curvature and torsion.

A Bezier Curve Based Key Management Scheme for Hierarchical Wireless Sensor Networks

2012 ◽  
Vol 263-266 ◽  
pp. 2979-2985
Author(s):  
Yong Luo Shen ◽  
Jun Zhang ◽  
Di Wei Yang ◽  
Lin Bo Luo
Keyword(s):  
Key Management ◽  
Wireless Sensor ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Control Points ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Management Scheme ◽  
Secret Keys ◽  
Key Management Scheme

In this paper, we propose a novel key management scheme based on Bezier curves for hierarchical wireless sensor networks (WSNs). The design of our scheme is motivated by the idea that a Bezier curve can be subdivided into arbitrarily continuous pieces of sub Bezier curves. The subdivided sub Bezier curves are easily organized to a hierarchical architecture that is similar to hierarchical WSNs. The subdivided Bezier curves are unique and independent from each other so that it is suitable to assign each node in the WSN with a sub Bezier curve. Since a piece of Bezier curve can be presented by its control points, in the proposed key management scheme, the secret keys for each node are selected from the corresponding Bezier curve’s control points. Comparing with existing key management schemes, the proposed scheme is more suitable for distributing secret keys for hierarchical WSNs and more efficient in terms of computational and storage cost.

scholarly journals A MATRIX PRESENTATION OF HIGHER ORDER DERIVATIVES OF BÉZIER CURVE AND SURFACE

2021 ◽  
Vol 21 (1) ◽  
pp. 77-90
Author(s):  
TUBA AĞIRMAN AYDIN
Keyword(s):  
Higher Order ◽  
Control Points ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Algebraic Form ◽  
Important Place ◽  
Curves And Surfaces ◽  
The Matrix ◽  
Special Matrix ◽  
Derivatives Of

In this study, the Bézier curves and surfaces, which have an important place in interactive design applications, are expressed in matrix form using a special matrix that gives the coefficients of the Bernstein base polynomial. The matrix forms of higher order derivatives of the Bézier curves and surfaces are obtained. It is demonstrated by numerical examples that the bidirectional transition between the control points and parametric equations of the Bézier curves and surfaces can be easily achieved using these matrix forms. In addition, it is demonstrated that this type of curve and surface, whose control points are known, its higher order derivatives can be calculated without it's parametric equations. In this study, the Bézier curves and surfaces are presented in a more easily understandable and easy to use format in algebraic form for designers.

scholarly journals Parametric Hull Design with Rational Bézier Curves

2021 ◽  
pp. 221-227
Author(s):  
Antonio Mancuso ◽  
Antonio Saporito ◽  
Davide Tumino
Keyword(s):  
Graphical User Interface ◽  
Early Stage ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Rational Bézier Curves ◽  
Hull Design ◽  
Geometrical Constraints ◽  
Sailing Yacht ◽  
Definition Of ◽  
Different Characteristics

AbstractIn this paper, a tool able to support the sailing yacht designer during the early stage of the design process has been developed. Quadratic and cubic Rational Bézier curves have been selected to describe the main curves defining the hull of a sailing yacht. The adopted approach is based upon the definition of a set of parameters, say the length of water line, the beam of the waterline, canoe body draft and some dimensionless coefficients according to the traditional way of the yacht designer. Some geometrical constraints imposed on the curves (e.g. continuity, endpoint angles) have been conceived aimed to avoid unreasonable shapes. These curves can be imported in any commercial CAD software and used as a frame to fit with a surface. The algorithm and the related Graphical User Interface (GUI) have been written in Visual Basic for Excel. To test the usability and the precision of the tool, two sailboats with different characteristics have been replicated. The rebuilt version of the hulls is very close to the original ones both in terms of shape and dimensionless coefficients.

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