The Development of Log Aesthetic Patch and Its Projection onto the Plane

Yee Meng Teh, R. U. Gobithaasan, Kenjiro T. Miura, Diya’ J. Albayari and Wen Eng Ong
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Yee Meng Teh: School of Mathematical Sciences, Universiti Sains Malaysia, Minden 11800, Malaysia
R. U. Gobithaasan: Special Interest Group on Modelling & Data Analytics, Faculty of Ocean Engineering Technology and Informatics, University Malaysia Terengganu, Kuala Nerus 21030, Malaysia
Kenjiro T. Miura: Graduate School of Science & Technology, Shizuoka University, Hamamatsu 432-8561, Shizuoka, Japan
Diya’ J. Albayari: Mathematics Department, Ibn Rushd National Academy-International Section IB, Amman 17126, Jordan
Wen Eng Ong: School of Mathematical Sciences, Universiti Sains Malaysia, Minden 11800, Malaysia

Mathematics, 2022, vol. 10, issue 1, 1-17

Abstract: 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.

Keywords: log aesthetic curves; log aesthetic space curves; surface design; Coons patch; lines of curvature (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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