# Log-aesthetic curves

The family of log-aesthetic curves clarified in our first paper.

Norimasa Yoshida and Takafumi Saito, Interactive Aesthetic Curve Segments, The Visual Computer (Pacific Graphics), Vol. 22, No.9-11, pp.896-905, 2006.

### The first use of log-aesthetic curves

Log-aesthetic curves were originally called aesthetic curves. At CAD conference 2007, I made a presentation about approximating aesthetic curves. Prof. Carlo H. Sequin of U. C. Berkeley 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, and Prof. Saito, we have decided to use the name log-aesthetic curves. The left is the picture of the first use of log-aesthetic curves.

### Shape Information of Curves and its Visualization using Two-tone Pseudo Coloring

Norimasa Yoshida and Takafumi Saito, Shape Information of Curves and its Visualization using Two-tone Pseudo Coloring, Computer-Aided Design and Applications, Vol. 21, No.1, pp.11-28, Jan. 2024. doi: 10.14733/cadaps.2024.11-28

Abstract. This paper presents a method for computing and visualizing the shape information of dierentiable parametric curves. The shape information is the slope of the logarithmic curvature graph and the slope of the logarithmic torsion graph. We derive the equations for computing and in terms of curvature and torsion, respectively. The value of $\alpha$ is related to the specific curvature function, such as the linear function when $\alpha=-1$. For space curves, the value of is also related to the specific torsion function. Using the two-tone pseudo coloring for the visualization of the shape information, users can read out the approximate value of and for each point of the curve. For some planar curves, we clarify the similarities with log-aesthetic curves by taking the limit of as the parameter approaches the limit value.

Quadratic Log-Aesthetic Curves

Norimasa Yoshida, Takafumi Saito, Quadratic Log-Aesthetic Curves, Computer-Aided Design and Applications, Volume 14, Issue 2, pp. 219-226, March 2017. DOI: 10.1080/16864360.2016.1223434

Abstract: This paper proposes quadratic log-aesthetic curves that are curves whose logarithmic curvature graphs are quadratic. In previous work, generalized log-aesthetic curves are derived by shifting either the curvature or the radius of curvature of log-aesthetic curves. Quadratic log-aesthetic curves are another generalization of log-aesthetic curves by making logarithmic curvature graphs quadratic. We derive the equations of quadratic log-aesthetic curves and clarify their characteristics. For drawing quadratic log-aesthetic curves, we need to compute the inverses of the error and imaginary error functions. We present a method for computing these inverses and confirm that the curves can be generated in real time.

Logarithmic curvature and torsion graphs

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. DOI: 10.1007/978-3-642-11620-9_28

Abstract. 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 B´ezier curves. From the graphs, we can see some interesting properties of curves that cannot be derived from the curvature or torsion plots.

Quasi-Log-Aesthetic Curves

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.

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 Bézier 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.

*Now, aesthetic curves are called log-aesthetic curves

### Log-Aesthetic Curves

Norimasa Yoshida and Takafumi Saito, Interactive Aesthetic Curve Segments, The Visual Computer (Pacific Graphics), Vol. 22, No.9-11, pp.896-905, 2006.

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.

*Now, aesthetic curves are called log-aesthetic curves