Analytic parametric equations of log-aesthetic curves in terms of incomplete gamma functions

Ziatdinov, Rushan; Yoshida, Nakahiro; Kim, Tae‐Wan

doi:10.1016/j.cagd.2011.11.003

Cited by 51 publications

(29 citation statements)

References 15 publications

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“…We have compared the performance metric of classical RK method with Incomplete Gamma Function which was introduced in [17] ) to show the accuracy when the step size reduces. The classical RK method renders LAC segment up to 9.3 times faster than Incomplete Gamma Function and 99.4 times faster than Gaussian-Kronrod.…”

Section: Discussionmentioning

confidence: 99%

“…Incomplete Gamma Function is an analytic function, which is interpreted by simple and exact series representation [17]. Now that Incomplete Gamma Function is the exact solution for some , we have distinguished the truncation error for classical RK method and Gaussian-Kronrod by comparing with Incomplete Gamma Function.…”

Section: Truncation Errormentioning

confidence: 99%

“…They further extended LAC to Generalized Log-Aesthetic Curve (GLAC) which has extra degree of freedom [2]. In 2012, Ziatdinov et al [17] presented an analytic solution of LAC in terms of Incomplete Gamma Function which reduced the computation time up to 13 times. Recent progresses include shape analysis on GLAC [18], its drawable region which is wider than LAC [19] and the implementation of G2 LAC as plug-in for Rhino CAD system [20].…”

Section: Introductionmentioning

confidence: 99%

“…This paper proposes a numerical method which is called classical Runge-Kutta method to compute LAC. The final section reports measurement of computation time and its truncation error using classical Runge-Kutta method are compared with the analytic method, proposed by Ziatdinov et al [17] for a better way to improve the performance of LACs' computation.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Rendering log aesthetic curves via Runge-Kutta method

Gobithaasan¹,

Meng²,

Piah³

et al. 2014

AIP Conference Proceedings

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Abstract. Log Aesthetic Curves (LAC) are visually pleasing curves which has been developed using monotonic curvature profile. Hence, it can be easily implemented in product design environment, e.g, Rhino 3D CAD systems. LAC is generally represented in an integral form of its turning angle. Traditionally, Gaussian-Kronrod method has been used to render this curve which consumes less than one second for a given interval. Recently, Incomplete Gamma Function was proposed to represent LAC analytically which decreases the computation time up to 13 times. However, only certain value of shape parameters (denoted as α) which dictates the types of curves generated for LAC, can be used to compute LAC. In this paper, the classical Runge-Kutta (RK4) method is proposed to evaluate LAC numerically to reduce the LAC computation time for arbitrary, α. The preliminary result looks promising where the evaluation time is decreased tremendously. This paper also demonstrates the accuracy control of LAC by reducing the stepsize of RK4. The computation time and the accuracy for various α, are also illustrated in the last section of this paper.

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Section: Discussionmentioning

confidence: 99%

Section: Truncation Errormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Rendering log aesthetic curves via Runge-Kutta method

Gobithaasan¹,

Meng²,

Piah³

et al. 2014

AIP Conference Proceedings

View full text Add to dashboard Cite

show abstract

“…They also reported that the LCG gradient of Generalized Cornu spiral [5] can be written as a linear function [3]. In 2012, Ziatdnov et al [18] showed that some LA curves can be expressed by incomplete Gamma functions analytically which shortens the computation time up to 10 times. Recently Meek et al [10] proved that an unique solution exists for G 1 interpolation by using an LA curve segment when ߙ < 0.…”

Section: Introductionmentioning

confidence: 99%

Designing Log-aesthetic Splines withG²Continuity

Miura¹,

Shibuya²,

Gobithaasan³

et al. 2013

Computer-Aided Design and Applications

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This paper elucidates the possibilities to interactively generate and deform Logaesthetic (LA) curves regardless of their integral form. The methods proposed are twofold; in the first section, we propose new method to generate an S-shaped LA spline. In the next section, we propose a novel method to solve the G 2 Hermite interpolation problem with LA curves which is in the form of LA triplets. These methods have been implemented as a plug-in module for a commercial CAD system and are successfully used for practical design. This paper proofs that LA curve has matured and ready for industrial design.

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Aesthetic Design with Log-Aesthetic Curves and Surfaces

Miura

Gobithaasan

2016

Mathematics for Industry

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Bézier, B-Spline and NURBS are de facto flexible curves developed for various design intent. However, these curves possess complex curvature function complicating the process of aesthetic shape design. In order to shorten the process, we introduce the fundmental equations of aesthetic curves and surfaces. This paper elaborates on the technicalities of Log-Aesthetic (LA) curves and surfaces along with its practical application for industrial design. It is anticipated that the emerging LA curves and surfaces have good prospects to be the standards for designing aesthetic shapes.

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Analytic parametric equations of log-aesthetic curves in terms of incomplete gamma functions

Cited by 51 publications

References 15 publications

Rendering log aesthetic curves via Runge-Kutta method

Rendering log aesthetic curves via Runge-Kutta method

Designing Log-aesthetic Splines withG²Continuity

Aesthetic Design with Log-Aesthetic Curves and Surfaces

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Analytic parametric equations of log-aesthetic curves in terms of incomplete gamma functions

Cited by 51 publications

References 15 publications

Rendering log aesthetic curves via Runge-Kutta method

Rendering log aesthetic curves via Runge-Kutta method

Designing Log-aesthetic Splines withG2Continuity

Aesthetic Design with Log-Aesthetic Curves and Surfaces

Contact Info

Product

Resources

About

Designing Log-aesthetic Splines withG²Continuity