Rendering Log Aesthetic Curves Using Runge-Kutta Methods
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Date
2014-07
Authors
Teh, Yee Meng
Journal Title
Journal ISSN
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Publisher
Universiti Sains Malaysia
Abstract
Log Aesthetic Curves (LAC) are visually pleasing curves which have been developed using monotonic curvature profile since 2005. Hence, it can be easily implemented in product design environment, such as Rhino 3D CAD systems. LAC is generally represented in an integral form of its turning angle. Traditionally, Gaussian-Kronrod method has been used to evaluate this curve as the formulation of LAC involves double integration. Recently, incomplete gamma function was proposed to represent LAC analytically, which decreases the computation time up to 13 times. In this research, numerical method known as Runge-Kutta methods (consisting of classical Runge-Kutta and adaptive Runge-Kutta) are proposed to evaluate LAC that reduces computation time for arbitrary, α. The difference between classical method and adaptive method is classical method renders curve using constant step size while the adaptive method controls step size to satisfy given tolerance. There are four types of adaptive methods used to render LAC namely Dormand-Prince, Fehlberg, Sarafyan and Kutta-Merson respectively. This research proves the existence of a numerical solution exists for LAC using Picard’s Existence and Uniqueness Theorem. The entire methods are examined in terms of computation time (O(n)) and truncation error (O(h)). The classical RK method reduces the computation time up to 9.3 times faster than incomplete gamma function with the curve segment defined for θ∈[0,1]. All the adaptive methods shorten the rendering time to about 26 ~ 27.3 times as compared to incomplete gamma function with the curve segment defined for θ∈[0,3]. The classical RK method has been divided into three different step sizes (h = 0.1, 0.01, 0.001). As expected, the result shows that the measured value moves towards the exact value when the step size reduces. Dormand-Prince method is the best among adaptive methods and closest to the exact value. The results obtained are promising where the evaluation time is decreased tremendously regardless in rendering curve segment or shape and at the same time preserving the LAC’s family
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Keywords
Rendering log aesthetic curves , using runge-kutta methods