Logarithmic Curvature and Torsion Graphs

Yoshida, Nakahiro; Fukuda, Ryo; Saito, Takafumi

doi:10.1007/978-3-642-11620-9_28

Cited by 11 publications

(7 citation statements)

References 9 publications

(17 reference statements)

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“…In line with the above, Harada studied the class of curves appearing in natural entities such as bird's eggs, butterflies' wings, and man-made products such as Japanese swords and automobile shapes, and coined the term of aesthetic curves to define the class of curves whose logarithmic distribution of curvature is approximated by a straight line [22]. Miura proposed the analytical equations of aesthetic curves by using generalizations of the logarithmic spiral and the clothoid spiral, and studied the properties of the logarithmic curvature graph [2,23]. Here, the horizontal axis measures log ρ and the vertical axis measures log |dL/d log ρ|, where L is arc length of the curve and ρ is its radius of curvature.…”

Section: Introductionmentioning

confidence: 99%

On a Class of Polar Log-Aesthetic Curves

Parque¹

2021

Preprint

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Curves are essential concepts that enable compounded aesthetic curves, e.g., to assemble complex silhouettes, match a specific curvature profile in industrial design, and construct smooth, comfortable, and safe trajectories in vehicle-robot navigation systems. New mechanisms able to encode, generate, evaluate, and deform aesthetic curves are expected to improve the throughput and the quality of industrial design. In recent years, the study of (log) aesthetic curves have attracted the community's attention due to its ubiquity in natural phenomena such as bird eggs, butterfly wings, falcon flights, and manufactured products such as Japanese swords and automobiles.A (log) aesthetic curve renders a logarithmic curvature graph approximated by a straight line, and polar aesthetic curves enable to mode user-defined dynamics of the polar tangential angle in the polar coordinate system. As such, the curvature profile often becomes a by-product of the tangential angle.In this paper, we extend the concept of polar aesthetic curves and establish the analytical formulations to construct aesthetic curves with user-defined criteria. In particular, we propose the closed-form analytic characterizations of polar log-aesthetic curves meeting user-defined criteria of curvature profiles and dynamics of polar tangential angles. We present numerical examples portraying the feasibility of rendering the logarithmic curvature graphs represented by a straight line. Our approach enables the seamless characterization of aesthetic curves in the polar coordinate system, which can model aesthetic shapes with desirable aesthetic curvature profiles.

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Section: Introductionmentioning

confidence: 99%

On a Class of Polar Log-Aesthetic Curves

Parque¹

2021

Preprint

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“…Kanaya et al [5] presented Kvector and later renamed it as Logarithmic Curvature Graph (LCG) [9,10]. If the gradient of LCG is constant, then the curve is categorized as aesthetic curves [5,7].…”

Section: Introductionmentioning

confidence: 99%

“…The research on combination of two linear LCG under certain condition had been done by Yoshida and Saito [13]. Furthermore, Yoshida et al proposed LCG and Logarithmic Torsion Graph (LTG) for analysing the properties of arbitrary parametric curves [10]. In 2009, Levien and Sequin [14] proofed that LAC is the most promising curve for aesthetic design.…”

Section: Introductionmentioning

confidence: 99%

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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“…Planar log-aesthetic curves [3,4,5,8] are curves with linear logarithmic curvature graphs (LCGs) [12]. The curves were originally proposed for the shape design of highly aesthetic objects.…”

Section: Introductionmentioning

confidence: 99%

Quasi-Log-Aesthetic Curves in Polynomial Bézier Form

Yoshida

Fukuda

Saitô

et al. 2013

Computer-Aided Design and Applications

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This paper presents planar quasi-log-aesthetic curves in polynomial Bézier form. Logaesthetic curves are curves that can be considered as the generalization of the Clothoid, Nielsen's spiral, logarithmic spirals and circle involute. By deriving the Taylor polynomials of log-aesthetic curves and converting the basis to Bernstein basis, we obtain quasi-log-aesthetic curves in polynomial Bézier form. We show the implementation results with logarithmic curvature graphs and a 1 G Hermite interpolation method.

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Logarithmic Curvature and Torsion Graphs

Cited by 11 publications

References 9 publications

On a Class of Polar Log-Aesthetic Curves

On a Class of Polar Log-Aesthetic Curves

Rendering log aesthetic curves via Runge-Kutta method

Quasi-Log-Aesthetic Curves in Polynomial Bézier Form

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