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

Yee Teh,Wen Ong,R Gobithaasan,Diya’ Albayari,Kenjiro Miura

doi:10.3390/math10010160

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.

Highlights

The introduction of Bezier curves and surfaces representation was a significant breakthrough of Computer Aided Geometric Design (CAGD), which was later extended to B-spline representation [1,2]
The complex form of curvature of these curves, which are not suitable for direct manufacturing, led to the introduction of a variety of efficient fairing algorithms to reduce the oscillation in the curvature profile of these curves [4,5,6]
The research on Log Aesthetic Curves (LACs) has been active since Miura [10] introduced a linear Logarithmic Curvature Graph (LCG) as its fundamental equation

Summary

Introduction

The introduction of Bezier curves and surfaces representation was a significant breakthrough of Computer Aided Geometric Design (CAGD), which was later extended to B-spline representation [1,2]. In 2014, Joo et al [25] proposed an algorithm for computing LoCs on parametric surfaces as well as to derive its curvature and torsion They further showed that these LoCs may aid in designing ship hulls. In 2019, Takezawa et al [23] proposed an interactive method to control LoCs on a doubly curved surface They smoothed the experiment surfaces by implementing smoothed directions on the LoCs instead of using true principal directions. The details on how the LAC and LASC can be drawn and controlled interactively are fully discussed in [12,13]

Lines of Curvature

The Projection of a Surface onto a Plane

Numerical Example

LoCs on Surfaces

Conclusions and Future Work