1 Department of Mathematics, University of Sargodha, Sargodha, Pakistan.
2 Department of Mathematics, College of Science, Hawassa University, Hawassa, Ethiopia.
PLoS One. 2024 Jun 3;19(6):e0303340. doi: 10.1371/journal.pone.0303340. eCollection 2024.
The subject matter of surfaces in computer aided geometric design (CAGD) is the depiction and design of surfaces in the computer graphics arena. Due to their geometric features, modeling of Bézier curves and surfaces with their shape parameters is the most well-liked topic of research in CAGD/computer-aided manufacturing (CAM). The primary challenges in industries such as automotive, shipbuilding, and aerospace are the design of complex surfaces. In order to address this issue, the continuity constraints between surfaces are utilized to generate complex surfaces. The parametric and geometric continuities are the two metrics commonly used for establishing connections among surfaces. This paper proposes continuity constraints between two generalized Bézier-like surfaces (gBS) with different shape parameters to address the issue of modeling and designing surfaces. Initially, the generalized form of C3 and G3 of generalized Bézier-like curves (gBC) are developed. To check the validity of these constraints, some numerical examples are also analyzed with graphical representations. Furthermore, for a continuous connection among these gBS, the necessary and sufficient G1 and G2 continuity constraints are also developed. It is shown through the use of several geometric designs of gBS that the recommended basis can resolve the shape and position adjustment problems associated with Bézier surfaces more effectively than any other basis. As a result, the proposed scheme not only incorporates all of the geometric features of curve design schemes but also improves upon their faults, which are typically encountered in engineering. Mainly, by changing the values of shape parameters, we can alter the shape of the curve by our choice which is not present in the standard Bézier model. This is the main drawback of traditional Bézier model.
计算机辅助几何设计（CAGD）中的曲面主题是在计算机图形学领域中对曲面的描绘和设计。由于其几何特征，使用形状参数对 Bézier 曲线和曲面进行建模是 CAGD/计算机辅助制造（CAM）中最受欢迎的研究主题。汽车、造船和航空航天等行业的主要挑战是复杂曲面的设计。为了解决这个问题，利用曲面之间的连续性约束来生成复杂曲面。参数和几何连续性是用于建立曲面之间连接的两个常用度量标准。本文提出了两种不同形状参数的广义 Bézier 样条曲面（gBS）之间的连续性约束，以解决曲面建模和设计的问题。首先，开发了广义 Bézier 样条曲线（gBC）的 C3 和 G3 的广义形式。为了验证这些约束的有效性，还通过图形表示分析了一些数值示例。此外，为了实现这些 gBS 之间的连续连接，还开发了必要和充分的 G1 和 G2 连续性约束。通过使用几种 gBS 的几何设计，表明所建议的基可以比任何其他基更有效地解决与 Bézier 曲面相关的形状和位置调整问题。因此，所提出的方案不仅包含了曲线设计方案的所有几何特征，而且还改进了它们在工程中通常遇到的缺陷。主要是，通过改变形状参数的值，我们可以根据自己的选择改变曲线的形状，而这在标准 Bézier 模型中是不存在的。这是传统 Bézier 模型的主要缺点。
