The modeling of Bezier curves and surfaces with their shape parameters is the most popular area of research in computer aided geometric design/computer aided manufacturing (CAGD/CAM) due to their geometric characteristics. In this paper, we propose an important idea to tackle the problem in construction of some engineering symmetric revolutionary curves and symmetric rotation surfaces by using the generalized hybrid trigonometric Bezier (or GHT-Bezier, for short) curve. The shape of the curves and surfaces can be modified by the alteration of shape parameters. The free-form complex curves using GHT-Bezier curves with constraints of parametric continuity are constructed. Finally, by using the GHT-Bezier curves with their continuity conditions and symmetric formulas, we construct different types of symmetric figures, symmetric revolutionary curves and symmetric rotation surfaces in $\mathbb {R}^{2}$ and $\mathbb {R}^{3}$ to show the efficiency of modeling. These symmetric examples show that the proposed method is time saving, effective and efficient in construction of complex engineering symmetric curves and surfaces.

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A Novel Approach of Hybrid Trigonometric Bézier Curve to the Modeling of Symmetric Revolutionary Curves and Symmetric Rotation Surfaces

/pdf/quintic-trigonometric-bezier-curve-with-two-shape-parameters-3qx92ba4zc.pdf

Quintic Trigonometric Bezier Curve with Two Shape Parameters

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 Bezier (GHT-Bezier) curves. The GHT-Bernstein basis functions and Bezier curve with shape parameters are presented. The parametric and geometric continuity constraints for GHT-Bezier 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 Bezier curve and some other applications by using the proposed GHT-Bezier 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 Novel Generalized Hybrid Trigonometric Bézier-Like Curve with Shape Parameters and Its Applications

The construction of free-form complex curves is very hot topic in engineering and computer aided design/computer aided manufacturing (CAD/CAM). The trigonometric Bézier-like curves got more attention in the fields of mathematics and engineering in recent years because of their useful geometric properties as compared to classical Bézier curves as well as ordinary Bernstein basis functions. The new trigonometric cubic Bézier-like (or TC Bézier-like, for short) curves along with new trigonometric cubic Bernstein-like (or TC Bernstein-like, for short) basis functions with single shape parameter with continuity conditions are presented in this paper. The necessary and sufficient constraints for C2 and G2 between two contiguous TC Bézier-like curves are constructed in order to remove the difficulty to composite curves which they cannot often be constructed by means of single curve. The role of shape parameter on connected curves with detailed smooth continuity steps is also part of this study. The ellipses and parabolas can also be represented exactly by using the TC Bézier-like curves. Some important applications of TC Bézier-like curves as: approximate some conic curves, font designing and free form complex curves are discussed. Some modelling examples show that the proposed TC Bézier-like curve technique is time saving, very effective and efficient which they can easily be modelled and give a powerful tool to design engineering complex curves.

Some engineering applications of new trigonometric cubic Bézier-like curves to free-form complex curve modeling

Based on quintic trigonometric Bezier like basis functions, the biquintic Bezier surfaces are modeled with four shape parameters that not only possess the key properties of the traditional Bezier surface but also have exceptional shape adjustment. In order to construct Bezier like curves with shape parameters, we present a class of quintic trigonometric Bezier like basis functions, which is an extension of a traditional Bernstein basis. Then, according to these basis functions, we construct three different types of shape adjustable surfaces such as general surface, swept surface and swung surface. In addition to the application of the proposed method, we also discuss the shape adjustment of surfaces showing with curvature nephogram (with and without fixing the boundaries). However, the modeling examples shows that the suggested approach is efficient and easy to implement.

https://www.mdpi.com/2073-8994/12/8/1205/pdf

Construction of Local Shape Adjustable Surfaces Using Quintic Trigonometric Bézier Curve

This paper describes a new function called quintic trigonometric Bezier curve that has the potential to estimate the maximum driving speed allowed for safe driving on roads. The shape parameters present in this trigonometric Bezier function gives more flexibility for users in designing highways. Since the curves do interpolate the points, small changes in shape parameters will affect the curvature of the curve, therefore creating a bigger difference towards speed estimation. The objective of this articles is to conclude curvature information of quintic trigonometric Bezier curves with the goal of approximating roads’ maximum speed and estimating the maximum allowed speed.

Quintic trigonometric Bezier curve and its maximum speed estimation on highway designs

This paper describes the general method to develop S-shaped and C-shaped transition curves using Cubic Trigonometric Bezier Curve with two shape parameters which satisfy G2 Hermite condition. Some numerical examples of S-shaped and C-shaped transition curve segment are presented.

S-shaped and C-shaped transition curve using cubic trigonometric Bezier

Abstract—Bezier curves have useful properties for path generation problem, for instance, it can generate the reference trajectory for vehicles to satisfy the path constraints. Both algorithms join cubic Bezier curve segment smoothly to generate the path. Some of the useful properties of Bezier are curvature. In mathematics, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line. Another extrinsic example of curvature is a circle, where the curvature is equal to the reciprocal of its radius at any point on the circle. The smaller the radius, the higher the curvature thus the vehicle needs to bend sharply. In this study, we use Bezier curve to fit highway-like curve. We use different approach to find the best approximation for the curve so that it will resembles highway-like curve. We compute curvature value by analytical differentiation of the Bezier Curve. We will then compute the maximum speed for driving using the curvature information obtained. Our research works on some assumptions; first, the Bezier curve estimates the real shape of the curve which can be verified visually. Even though, fitting process of Bezier curve does not interpolate exactly on the curve of interest, we believe that the estimation of speed are acceptable. We verified our result with the manual calculation of the curvature from the map.

Approximating Maximum Speed on Road from Curvature Information of Bezier Curve

Introduction: Curve tting is crucial, especially when dealing with a function that can t data with a high degree of smoothness. Curve tting has been applied in a variety of areas, including computer-aided geometric design, manufacturing, medical imaging, and animation [2]. Curves such as log aesthetic curve [4] and the classical Bézier curve are widely utilised tools in the curve tting process due to their advantageous qualities. On the other hand, because the shape of the curve cannot be changed without changing its control points, the Bézier curve has limited exibility and adjustability. Furthermore, shifting or altering control points to get the correct shape is time-consuming. Hence, in the present work,the generalized fractional Bézier curve will be used. Since the generalized fractional Bézier curve comprises shape and fractional parameters, and the picture outline can be constructed using the same degree of curve, the curve tting technique will become easier. Various levels of continuity will be used, notably fractional continuity. Because of the fractional continuity, the position of common points or related joints may be changed along the rst curve, giving more options for common point adjustment. Shape parameters contribute to the adjustment of shape locally and globally, hence enhance further the exibility of curves.

M. Y. Misro

Papers

Quintic Trigonometric Bezier Curve with Two Shape Parameters

Quintic trigonometric Bezier curve and its maximum speed estimation on highway designs

S-shaped and C-shaped transition curve using cubic trigonometric Bezier

Approximating Maximum Speed on Road from Curvature Information of Bezier Curve

Curve Fitting Using Generalized Fractional Bézier Curve