# Log-Aesthetic Curves

12/10/2010

Log-Aesthetic Curves

The Familiy of log-aesthetic Curves
***Log-aesthetic curves were originally called aesthetic curves. At CAD conference 2007, I have talked about our research with Prof. Carlo H. Sequin. He proposed to use the name log-aesthetic curves instead of aesthetic curves because the curves use logarithmic graphs. Talking with Prof. Harada and Prof. Miura, we have decided to use the name log-aesthetic curves.
• N. Yoshida, R. Fukuda, T. Saito, Logarithmic Curvature and Torsion Graphs, in Mathematical Methods for Curves and Surfaces 2008 edited by Daehlen et al.,  LNCS 5862, Springer, pp.434-443, 2010.[PDF]This paper introduces logarithmic curvature and torsion graphs　for analyzing planar and space curves. We present a method for drawing　these graphs from any differentiable parametric curves and clarify　the characteristics of these graphs. We show several examples of theses　graphs drawn from planar and 3D Bezier curves. From the graphs, wecan see some interesting properties of curves that cannot be derived fromthe curvature or torsion plots.
• N. Yoshida, R. Fukuda, T. Saito, Log-Aesthetic Space Curve Segments, SIAM/ACM Joint Conference on Geometric and Physical Modeling, 2009, pp.35-46.[PDF]For designing aesthetic surfaces, such as the car bodies, it is very important to use aesthetic curves as characteristic lines.  In such curves, the curvature should be monotonically varying, since it dominates the distortion of reflected images on curved surfaces.  In this paper, we present an interactive control method of log-aesthetic space curves.  We define log-aesthetic space curves to be curves whose logarithmic curvature and torsion graphs are both linear.  The linearity of these graphs constrains that the curvature and torsion are monotonically varying.  We clarify the characteristics of log-aesthetic space curves and identify their family.  Moreover, we present a novel method for drawing a log-aesthetic space curve segment by specifying two endpoints, their tangents, the slopes, $\alpha$ and $\beta$, of straight lines of the logarithmic curvature and torsion graphs, and the torsion parameter $\Omega$.  Our implementation shows that log-aesthetic curve segments can be controlled fully interactively.
• N. Yoshida, T. Saito, Compound-Rhythm Log-Aesthetic Curves, Computer-Aided Design and Applications,Vol. 6, No.2, pp.243-252, 2009. [PDF]

This paper presents an efficient and stable method for drawing compound-rhythm log-aesthetic
curves. Compoundrhythm curves are curves whose logarithmic curvature graphs are represented by V-type or upside down V-type segments. We show that, once the continuity condition is derived, compound rhythm curves can be efficiently generated in a similar manner to generating monotonic rhythm curves. We also present a method for drawing compound-rhythm curves by specifying two endpoints, their tangential directions, $\alpha_0$ and $\alpha_1$ (which are the slopes of logarithmic curvature graphs) and the ratio $r_{\theta}$ of the change of the tangential angle of the curve $\alpha_0$ to the change of the tangential angle of the whole curve.
• N. Yoshida and T. Saito, Quasi-Aesthetic Curves in Rational Cubic Bezier Forms, Computer-Aided Design & Applications, Vol. 4, Nos. 1-4, pp.477-486, 2007. [PDF]Abstract: Designing aesthetically appealing models is vital for the marketing success of industrial products. In this paper, we propose quasi-Aesthetic Curves that can be used in CAD systems for aesthetic shape design. Quasi-Aesthetic Curves represented in rational cubic Bezier Forms are curves whose logarithmic curvature histograms (LCHs) become nearly straight lines. The monotonicity of curvature of quasi-Aesthetic Curves is checked by the proposed method. We generate quasi-Aesthetic Curves by approximating the Aesthetic Curves whose LCHs are strictly represented by straight lines. We show that one Aesthetic Curve segment whose change of tangential angle is less than 90 deg. can be replaced by one quasi-Aesthetic Curve segment guaranteeing the monotonicity of the curvature in most of practical situations.
• Norimasa Yoshida and Takafumi Saito, Interactive Aesthetic Curve Segments, The Visual Computer (Pacific Graphics), Vol. 22, No.9-11, pp.896-905, 2006.[Video(WindowsMedia)] [PDF]
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Abstract: To meet highly aesthetic requirements in industrial design and styling, we propose a new category of aesthetic curve segments. To achieve these aesthetic requirements, we use curves whose logarithmic curvature histograms(LCH) are represented by straight lines. We call such curves aesthetic curves. We identify the overall shapes of aesthetic curves depending on the slope of LCH $\alpha$, by imposing specific constraints to the general formula of aesthetic curves. For interactive control, we propose a novel method for drawing an aesthetic curve segment by specifying two endpoints and their tangent vectors. We clarify several characteristics of aesthetic curve segments.

Errata

• In Eq. (9),  “\int_0^{\theta} e^(\Lambda + i) \psi d \psi    if \alpha = 1”
• In Eq.(10),   ” (1 – e^(-Lambda * s)) / Lambda  if \alpha=0   “.   Eq. (11) should also be corrected for \alpha=0.