The Development of Log Aesthetic Patch and Its Projection onto the Plane
The Development of Log Aesthetic Patch and Its Projection onto the Plane
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DOI：
10.3390/math10010160
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发表时间：
2022-01
期刊：
影响因子：
2.4
通讯作者：
Yee-Meng Teh;R. Gobithaasan;K. Miura;Diya’ J. Albayari;W. E. Ong
中科院分区：
文献类型：
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作者：
Yee-Meng Teh;R. Gobithaasan;K. Miura;Diya’ J. Albayari;W. E. Ong
在这项工作中，我们引入了一种称为原木美学补丁（LAP）的新型表面。该曲面是 Coons 曲面片的扩展，其中四个边界曲线是平面或空间对数美学曲线 (LAC)。为了确定其多功能性，我们使用曲率线 (LoC) 信息将双曲抛物面近似为 LAP。 LoC 的外部起到双曲抛物面边界的作用，在构建 LAP 之前用 LAC 替换。由于 LoC 在造船的热弯和冷弯工艺中至关重要，因此我们根据 LoC 曲率、曲率导数、扭转和对数曲率图 (LCG) 来研究 LAP。数值结果表明两个表面的 LoC 都具有单调曲率。 LAP 近似相对于其原始双曲抛物面的一个优点是，LAP 的 LoC 可以近似为 LAC，因此 LoC 的曲率一阶导数是单调的，而对于双曲抛物面来说，它们是非单调的。这证实了生产的 LAP 确实具有高质量。最后，我们使用测地曲率将 LAP 投影到平面上，以创建可以粘贴在一起的条带，模仿造船行业的热弯曲和冷弯曲过程。
In this work, we introduce a new type of surface called the Log Aesthetic Patch (LAP). This surface is an extension of the Coons surface patch, in which the four boundary curves are either planar or spatial Log Aesthetic Curves (LACs). To identify its versatility, we approximated the hyperbolic paraboloid to LAP using the information of lines of curvature (LoC). The outer part of the LoCs, which play a role as the boundary of the hyperbolic paraboloid, is replaced with LACs before constructing the LAP. Since LoCs are essential in shipbuilding for hot and cold bending processes, we investigated the LAP in terms of the LoC’s curvature, derivative of curvature, torsion, and Logarithmic Curvature Graph (LCG). The numerical results indicate that the LoCs for both surfaces possess monotonic curvatures. An advantage of LAP approximation over its original hyperbolic paraboloid is that the LoCs of LAP can be approximated to LACs, and hence the first derivative of curvatures for LoCs are monotonic, whereas they are non-monotonic for the hyperbolic paraboloid. This confirms that the LAP produced is indeed of high quality. Lastly, we project the LAP onto a plane using geodesic curvature to create strips that can be pasted together, mimicking hot and cold bending processes in the shipbuilding industry.