Abstract
Lines of curvatures (LoCs) are curves on a surface that are derived from the first and second fundamental forms, and have been used for shaping various types of surface. In this paper, we investigated the LoCs of two types of log aesthetic (LA) surfaces; i.e., LA surfaces of revolution and LA swept surfaces. These surfaces are generated with log aesthetic curves (LAC) which comprise various families of curves governed by α. First, since it is impossible to derive the LoCs analytically, we have implemented the LoC computation numerically using the Central Processing Unit (CPU) and General Processing Unit (GPU). The results showed a significant speed up with the latter. Next, we investigated the curvature distributions of the derived LoCs using a Logarithmic Curvature Graph (LCG). In conclusion, the LoCs of LA surface of revolutions are indeed the duplicates of their original profile curves. However, the LoCs of LA swept surfaces are LACs of different shapes. The exception to this is when this type of surface possesses LoCs in the form of circle involutes.
Highlights
We presented two types of log aesthetic (LA) surface along with their respective Lines of curvatures (LoCs) in this paper
In order to speed up the computation time, we implemented LoC computation on the General Processing Unit (GPU), which resulted in the GPU performing approximately 20 times faster as compared
GPU, which resulted in the GPU performing approximately 20 times faster as compared to the Central Processing Unit (CPU) for LA surfaces of revolution
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. B-Spline (NURBS) became the de facto standard for CAD/CAM since it can represent conics analytically [3] These curves cannot be used directly for manufacturing due to their complex forms of curvature; various fairing algorithms were devised to reduce the oscillation in their curvature profile [4,5,6]. The LAC family has a monotonic curvature profile which can be used to represent well known spirals, e.g., clothoid, logarithmic spiral, circle involute and Nielsen’s spiral. Variations of aesthetic surfaces include the work of Inoue et al [19] who applied LAC as profile curves to generate a log aesthetic (LA) surface. The subsequent two sections introduce the necessary LAC and LoC equations used for the study This is followed by the generation of LA surface of revolution and LA swept surface. The features of LoCs are elaborated before concluding this work
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