Approximating Generalised Cornu Spiral With Low Energy Curve
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Date
2012-04
Authors
Chan, Chui Ling
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Journal ISSN
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Publisher
Universiti Sains Malaysia
Abstract
The computations of visual pleasing and mathematically fair curve are an ongoing process. In the earlier literature, researchers have been studying the smooth curve, or in a more technically term, the fair curve through the physical approach known as elastica, Elastica means a thin strip of elastic material. Daniel Bernoulli stated the elastica as variational problem in terms of strain energy and Malcom (1977) mentioned that the simplest way to characterize a spline mathematically is with the fact that a spline assumes a shape which minimizes its elastic strain energy. In the recent literature, curvature profile was the main focus of fair curve computation. Refer to Farin (1989), a curve is fair if it contains relatively few pieces of monotone curvature profile. The most used approaches in computing fair curve included: i. controlling the curve directly to enforce it to satisfy the monotone curvature variation and ii. using curve with define monotone curvature profile such as the spiral. The first approach will cause loss of flexibility to the curve. Whilst for the second approach, the spiral is non-polynomial, making it unsuitable to integrate with the existing computer aided design (CAD) system. Thus, we need to approximate the spiral with polynomial. There are various approximations approaches found in the literature. Among them, there is a lack of discussion about the strain energy of the approximating polynomial. Since strain energy is the basis of curve shape study, it makes sense for us to determine the approximating curve through this concept.
The objective of this study is to approximate the generalized Cornu spiral (GCS) using the concept of strain energy. Numerical method is use to adjust the strain energy of the approximating polynomial. Here, we adopted the quintic Hermite curve which is the simplest solution involving quintic polynomial. The Quintic Hermite curve is defined by end points position, end points first and second derivatives. We adjusted the curve by varying the magnitude of derivatives. The relative curvature error is use to demonstrate the accuracy of approximation. Result had shown that low energy polynomial curve can form good approximation of spiral segment.
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Keywords
Approximating generalised cornu spiral , with low energy curve