Generalized Fractional Bézier Curve with Shape Parameters

The construction of new basis functions for the Bézier or B-spline curve has been one of the most popular themes in recent studies in Computer Aided Geometric Design (CAGD). Implementing the new basis functions with shape parameters provides a different viewpoint on how new types of basis functions can develop complex curves and surfaces beyond restricted formulation. The wide selection of shape parameters allows more control over the shape of the curves and surfaces without altering their control points. However, interpolated parametric curves with higher degrees tend to overshoot in the process of curve fitting, making it difficult to control the optimal length of the curved trajectory. Thus, a new parameter needs to be created to overcome this constraint to produce free-form shapes of curves and surfaces while still preserving the basic properties of the Bézier curve. In this work, a general fractional Bézier curve with shape parameters and a fractional parameter is presented. Furthermore, parametric and geometric continuity between two generalized fractional Bézier curves is discussed in this paper, as well as demonstrating the effect of the fractional parameter of curves and surfaces. However, the conventional parametric and geometric continuity can only be applied to connect curves at the endpoints. Hence, a new type of continuity called fractional continuity is proposed to overcome this limitation. Thus, with the curve flexibility and adjustability provided by the generalized fractional Bézier curve, the construction of complex engineering curves and surfaces will be more efficient.

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Explicit Continuity Conditions for G1 Connection of S-λ Curves and Surfaces

Mathematics ◽

10.3390/math8081359 ◽

2020 ◽

Vol 8 (8) ◽

pp. 1359 ◽

Cited By ~ 1

Author(s):

Gang Hu ◽

Huinan Li ◽

Muhammad Abbas ◽

Kenjiro T. Miura ◽

Guoling Wei

Keyword(s):

Computer Aided Design ◽

Linear Independence ◽

Basis Functions ◽

Geometric Continuity ◽

Complex Curves ◽

Curves And Surfaces ◽

Computer Aided ◽

Cad Cam ◽

Geometric Representations ◽

Aided Design

The S-λ model is one of the most useful tools for shape designs and geometric representations in computer-aided geometric design (CAGD), which is due to its good geometric properties such as symmetry, shape adjustable property. With the aim to solve the problem that complex S-λ curves and surfaces cannot be constructed by a single curve and surface, the explicit continuity conditions for G1 connection of S-λ curves and surfaces are investigated in this paper. On the basis of linear independence and terminal properties of S-λ basis functions, the conditions of G1 geometric continuity between two adjacent S-λ curves and surfaces are proposed, respectively. Modeling examples imply that the continuity conditions proposed in this paper are easy and effective, which indicate that the S-λ curves and surfaces can be used as a powerful supplement of complex curves and surfaces design in computer aided design/computer aided manufacturing (CAD/CAM) system.

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

Mathematics ◽

10.3390/math8060967 ◽

2020 ◽

Vol 8 (6) ◽

pp. 967 ◽

Cited By ~ 3

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.

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A Novel Generalization of Trigonometric Bézier Curve and Surface with Shape Parameters and Its Applications

Mathematical Problems in Engineering ◽

10.1155/2020/4036434 ◽

2020 ◽

Vol 2020 ◽

pp. 1-25 ◽

Cited By ~ 4

Author(s):

Sidra Maqsood ◽

Muhammad Abbas ◽

Gang Hu ◽

Ahmad Lutfi Amri Ramli ◽

Kenjiro T. Miura

Keyword(s):

Basis Functions ◽

Geometric Continuity ◽

Shape Parameters ◽

Bernstein Basis ◽

Bezier Curves ◽

Bézier Curves ◽

Surface Design ◽

Curves And Surfaces ◽

Bernstein Basis Functions ◽

Parametric Continuity

Adopting a recurrence technique, generalized trigonometric basis (or GT-basis, for short) functions along with two shape parameters are formulated in this paper. These basis functions carry a lot of geometric features of classical Bernstein basis functions and maintain the shape of the curve and surface as well. The generalized trigonometric Bézier (or GT-Bézier, for short) curves and surfaces are defined on these basis functions and also analyze their geometric properties which are analogous to classical Bézier curves and surfaces. This analysis shows that the existence of shape parameters brings a convenience to adjust the shape of the curve and surface by simply modifying their values. These GT-Bézier curves meet the conditions required for parametric continuity (C0, C1, C2, and C3) as well as for geometric continuity (G0, G1, and G2). Furthermore, some curve and surface design applications have been discussed. The demonstrating examples clarify that the new curves and surfaces provide a flexible approach and mathematical sketch of Bézier curves and surfaces which make them a treasured way for the project of curve and surface modeling.

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A novel extension to the polynomial basis functions describing Bezier curves and surfaces of degree n with multiple shape parameters

Applied Mathematics and Computation ◽

10.1016/j.amc.2013.07.073 ◽

2013 ◽

Vol 223 ◽

pp. 1-16 ◽

Cited By ~ 18

Author(s):

Xinqiang Qin ◽

Gang Hu ◽

Nianjuan Zhang ◽

Xiaoli Shen ◽

Yang Yang

Keyword(s):

Basis Functions ◽

Polynomial Basis ◽

Shape Parameters ◽

Bezier Curves ◽

Bézier Curves ◽

Curves And Surfaces

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Optimization of Tooth Root Profile Using Bezier Curve with G2 Continuity to Reduce Bending Stress of Asymmetric Spur Gear Tooth

MATEC Web of Conferences ◽

10.1051/matecconf/201823703010 ◽

2018 ◽

Vol 237 ◽

pp. 03010 ◽

Cited By ~ 2

Author(s):

Priyakant Vaghela ◽

Jagdish Prajapati

Keyword(s):

Stress Concentration ◽

Gear Tooth ◽

Spur Gear ◽

Bézier Curve ◽

Bezier Curve ◽

Tooth Root ◽

Geometric Continuity ◽

Bending Stress ◽

Gear Design ◽

G2 Continuity

This research describes simple and innovative approach to reduce bending stress at tooth root of asymmetric spur gear tooth which is desire for improve high load carrying capacity. In gear design at root of tooth circular-filleted is widely used. Blending of the involute profile of tooth and circular fillet creates discontinuity at root of tooth causes stress concentration occurs. In order to minimize stress concentration, geometric continuity of order 2 at the blending of gear tooth plays very important role. Bezier curve is used with geometric continuity of order 2 at tooth root of asymmetric spur gear to reduce bending stress.

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Designing Developable C-Bézier Surface with Shape Parameters

Mathematics ◽

10.3390/math8030402 ◽

2020 ◽

Vol 8 (3) ◽

pp. 402 ◽

Cited By ~ 1

Author(s):

Caiyun Li ◽

Chungang Zhu

Keyword(s):

Architectural Design ◽

Basis Functions ◽

Bézier Curve ◽

Bezier Curve ◽

Developable Surface ◽

Rational Bézier Curves ◽

Polynomial Representations ◽

Rational Bézier Curve ◽

Shape Controlling ◽

Two Parameters

Developable surface plays an important role in geometric design, architectural design, and manufacturing of material. Bézier curve and surface are the main tools in the modeling of curve and surface. Since polynomial representations can not express conics exactly and have few shape handles, one may want to use rational Bézier curves and surfaces whose weights control the shape. If we vary a weight of rational Bézier curve or surface, then all of the rational basis functions will be changed. The derivation and integration of the rational curve will yield a high degree curve, which means that the shape of rational Bézier curve and surface is not easy to control. To solve this problem of shape controlling for a developable surface, we construct C-Bézier developable surfaces with some parameters using a dual geometric method. This yields properties similar to Bézier surfaces so that it is easy to design. Since C-Bézier basis functions have only two parameters in every basis, we can control the shape of the surface locally. Moreover, we derive the conditions for C-Bézier developable surface interpolating a geodesic.

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