Quintic generalized Hermite interpolation curves: construction and shape optimization using an improved GWO algorithm

The construction of new basis functions for the Bezier 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 Bezier curve. In this work, a general fractional Bezier curve with shape parameters and a fractional parameter is presented. Furthermore, parametric and geometric continuity between two generalized fractional Bezier 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 Bezier curve, the construction of complex engineering curves and surfaces will be more efficient.

Generalized Fractional Bézier Curve with Shape Parameters

This paper introduces a new approach for the fabrication of generalized developable cubic trigonometric Bezier (GDCT-Bezier) surfaces with shape parameters to address the fundamental issue of local surface shape adjustment The GDCT-Bezier surfaces are made by means of GDCT-Bezier-basis-function-based control planes and alter their shape by modifying the shape parameter value The GDCT-Bezier surfaces are designed by maintaining the classic Bezier surface characteristics when the shape parameters take on different values In addition, the terms are defined for creating a geodesic interpolating surface for the GDCT-Bezier surface The conditions appropriate and suitable for G 1 , Farin–Boehm G 2 , and G 2 Beta continuity in two adjacent GDCT-Bezier surfaces are also created Finally, a few important aspects of the newly formed surfaces and the influence of the shape parameters are discussed The modeling example shows that the proposed approach succeeds and can also significantly improve the capability of solving problems in design engineering

Generalized Developable Cubic Trigonometric Bézier Surfaces

A novel generalized trigonometric Bézier curve: Properties, continuity conditions and applications to the curve modeling

In this article, an algorithm to obtain a smooth path (free from obstacles) that can be optimized by shortest path distance, bending energy and curvature variation energy will be presented. Previously, scholars used various tools to generate smooth path such as Clothoid, Log-Aesthetic curves (LACs), and Bezier curves. The limited number of solutions from the aforementioned curves become one of the drawback to generate smooth path planning. Therefore, providing a number of solutions that can generate smooth path planning becomes the objective of this study. In this paper to generate a smooth path, five templates of spiral transition curves having three different shape parameters with monotone curvature (either increase or decrease) by cubic GHT-Bezier curves are proposed. Moreover, few examples of path planning technique via cubic GHT-Bezier spiral curve to show the flexibility of smooth path by minimization of the shortest path (minimum arc length) L , bending energy E and curvature variation energy V are presented. The superiority of cubic GHT-Bezier spiral path smoothing techniques as compared to Clothoid and LACs is also demonstrated.

Smooth path planning via cubic GHT-Bézier spiral curves based on shortest distance, bending energy and curvature variation energy

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

Muhammad Ammad

Papers

Construction of Local Shape Adjustable Surfaces Using Quintic Trigonometric Bézier Curve

Generalized Developable Cubic Trigonometric Bézier Surfaces