# Research

### Visualization of Curvature Monotonicity Regions

Takafumi Saito and Norimasa Yoshida, Curvature monotonicity evaluation functions on rational Bézier curves. Computers & Graphics, Vol. 114, pp.219-229, Aug. 2023. doi: https://doi.org/10.1016/j.cag.2023.05.019.

Abstract: We have derived functions of the lowest possible degree that enable us to evaluate curvature monotonicity for any 2D and 3D rational Bézier curves. We proved that the degree of the function is at most 8n − 12 for planar rational Bézier curves of degree n, and is at most 11n − 18 for space rational Bézier curves of degree n. These functions are derived in the Bernstein basis, allowing for efficient checking of curvature monotonicity using subdivision or Bézier clipping. As an application, we present real-time visualization of the region of a particular control point that guarantees monotonic variation of curvature over the entire segment of the rational Bézier curve. This allows users to identify where to move the control point to ensure that the curvature changes monotonically.

Norimasa Yoshida, Seiya Sakurai, Hikaru Yasuda, Taisei Inoue and Takafumi Saito, Visualization of the Curvature Monotonicity Regions of Polynomial Curves and its Application to Curve Design, Computer-Aided Design and Applications, Vol. 21, No.1, pp.75-87, Jan. 2024. doi: 10.14733/cadaps.2024.75-87 (The PDF is available on the site since it is open access. )

Abstract: Freeform curves, such as Bézier curves or B-spline curves are widely used in many applications. Although freeform curves have many nice properties, controlling the curvature variation by manually moving a control point is not easy. For polynomial Bézier and Bspline curves, this paper proposes a real-time method to visualize the region of a control point where the curvature becomes monotonically varying. By representing the numerator of the curvature derivative in Bernstein form, the proposed method checks the curvature monotonicity of a specific control point for every pixel on the screen using a GPU. With this approach, a user can determine where to move the control point to achieve monotonically varying curvature. Additionally, two applications of the proposed method are presented.

### 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 (The PDF is available on the site since it is open access. )

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.

### High-quality approximation of log-aesthetic curves

Shoichi Tsuchie, Norimasa Yoshida, High-quality approximation of log-aesthetic curves based on the fourth order derivative, Journal of Computational Design and Engineering, Volume 9, Issue 6, pp. 2439–2451, Dec. 2022. https://doi.org/10.1093/jcde/qwac117 (The PDF is available on the site since it is open access. )

Abstract: We propose a new method for approximating log-aesthetic curves $C_LA$ using high-degree Bézier curves. By leveraging the property that higher order derivatives are more sensitive to the quality of approximation, the method minimizes an objective function based on the fourth-order derivative; consequently, $C_LA$ is approximated with high accuracy. In addition, the proposed method is composed of two steps to ensure stable optimization so as not to be negatively affected because of a local minimum and to evaluate the fourth-order derivative. Furthermore, we reveal the difficulty in sufficiently approximating $C_LA$ with Bézier curves from two aspects. One aspect entails the uncertainty of how accurately the low-degree Bézier curves can approximate $C_LA$ . The other aspect is the existence of a subset of $C_LA$ that is inherently difficult to approximate with such conventional parametric curves. The experimental results and comparisons demonstrated the validity of the proposed method.

### Intrinsically defined Planar Curves

Norimasa Yoshida, Takafumi Saito, Intrinsically defined Planar Curves based on Explicit B-spline Curvature Functions, Computer-Aided Design and Applications, Vol. 19, No. 1, pp.152-163, Jan. 2022. DOI:https://doi.org/10.14733/cadaps.2022.164-17 (The PDF is available on the site since it is open access. )

Abstract: In the design of aesthetic objects, controlling the curvature variation of a curve segment is an important task. Freeform curves, such as Bézier curves or B-spline curves, are widely used in many CAD systems. Controlling the curvature variation of freeform curves, however, is not easy because how the curvature behaves by moving its control points is not predictable. This paper introduces an intrinsically defined planar curve based on an explicit polynomial B-spline curve and its G1 and G2 Hermite interpolation method. The advantage of using explicit B-spline curvature functions instead of explicit Bézier curvature functions is that a wider variety of curvature variation can be represented by increasing the number of segments and modifying the knots. The proposed method can theoretically match any G2 Hermite conditions. In the proposed approach, the curvature plot is specified in terms of an explicit B-spline curve and the explicit B-spline curve is integrated to generate a curve segment such that given G1 or G2 Hermite interpolation conditions are satisfied. As an application of the proposed curve, we show a method of controlling the curvature variation interactively by modifying control curvatures shown on the curvature comb satisfying G1 or G2 Hermite conditions.

Norimasa Yoshida, Takafumi Saito, Planar Curves based on Explicit Bézier Curvature Functions, Computer Aided Design and Applications, Vol. 17, No. 1, pp.77-87, Jan. 2020. DOI: https://doi.org/10.14733/cadaps.2020.77-87 (The PDF is available on the site since it is open access. )

For controlling the curvature variation with given 𝐺1 or 𝐺2 Hermite interpolation conditions, we propose intrinsically defined planar curves based on explicit Bézier curvature functions. In the proposed curve, the curvature variation is specified by an explicit Bézier curve. To perform 𝐺1 or 𝐺2 Hermite interpolation, some of control curvatures of the explicit Bézier curve are modified to fit the given conditions. We have implemented the method in C++ and confirmed that curves can be generated in fully interactively. We clarify how the viable regions for 𝐺2 Hermite interpolation changes depending on the degree of explicit polynomial Bézier curves. Applications of the proposed curves include the design of aesthetic curves for aesthetic shape desing as well as 2D illustrations.

### 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 (The PDF is available on the site since it is open access. ) (The PDF is available on the site since it is open access. )

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.

### Log-aesthetic space curves

N. Yoshida, R. Fukuda, T. Saito, Log-Aesthetic Space Curve Segments, SIAM/ACM Joint Conference on Geometric and Physical Modeling (GDSPM), pp.35-46 2009.

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, α and β, of straight lines of the logarithmic curvature and torsion graphs, and the torsion parameter Ω. Our implementation shows that log-aesthetic curve segments can be controlled fully interactively.

### 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.

### Interactive Control of 3D Class A Bézier curves

N. Yoshida, R. Fukuda, T. Saito, Interactive Generation of 3D Class A Bezier Curve segments, Computer-Aided Design and Application, Vol. 7, No.2, pp.163-172, 2010. DOI: 10.3722/cadaps.2010.163-172 (The PDF is available on the site since it is open access. )

Abstract This paper presents a method for interactively generating a 3D class A Bézier curve segment by specifying two endpoints and their tangents. We clarify geometric properties of 3D class A Bézier curves and use them for efficiently generating 3D class A Bézier curve segments satisfying the specified positional and tangential constraints. The characteristics of typical 3D class A Bézier curves are also clarified.

### Interactive Control of 2D Class A Bézier curves

N. Yoshida , T. Hiraiwa, and T. Saito, Interactive Control of Planar Class A Bezier Curves using Logarithmic Curvature Graphs, Computer-Aided Design & Applications, Vol.5, Nos.1-4, pp.121-130, 2008.

Abstract: We present a method for interactively drawing a planar class A Bézier curve segment. First, we present a method for interactively drawing a typical class A Bézier curve segment by specifying three points like a quadratic Bézier curve segment. We show that as the degree of a typical class A Bézier curve segment is elevated, the curve converges to a logarithmic spiral segment. At the limit of infinite degree, the curve segment becomes a logarithmic spiral segment. We also present a method for drawing a general class A Bézier curve segment by perturbing the elements of the typical class A matrix so that the endpoint constraints are satisfied. To see the characteristics of the generated curves, we propose to use logarithmic curvature graphs.

### 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.

*Currently, 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.

*Currently, aesthetic curves are called log-aesthetic curves

Instead of http://www.yoshida-lab.net/aesthetic/iacs.html go https://www.yoshida-lab.net/english/research-e/iacs