Aesthetic logarithmic curve

English page

March 25, 2009

"Log-aesthetic curve," Introduction to

Norimasa Yoshida (norimasa@cit.nihon-u.ac.jp) (Nihon University)
Takafumi Saito (txsaito@cc.tuat.ac.jp) (TUAT)
* E-mail address @ Please replace the space with half the filling.

* "Log-aesthetic curve", formerly "aesthetic curve" or "beautiful curve" was called.
* Related studies have been added.

Overview of aesthetic curve is logarithmic, click here please visit.

The family of Aesthetic Curves

Curves introduces the aesthetic Video ( DivX format)

Study

1. [09c Yoshida] N. Yoshida, R. Fukuda, T. Saito, Logarithmic Curvature and Torsion Graphs, to appear.
   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.
2. [09b Yoshida], Norimasa Yoshida, Ryo Fukuda, T. Saito, An interactive method for generating aesthetic curve segment logarithmic space, and CAD Joint Symposium on Visual Computing / Graphics, to appear.
3. [09a Yoshida] 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. Compound - rhythm curves are curves whose logarithmic curvature graphs are repres e nted 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 comp o und-rhythm curves by specifying two endpoints, their tangential directions, $ \ alpha_0 $ and $ \ alpha_1 $ (which are the slopes of logarithmic curv a ture 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.

4. [08b Yoshida], Norimasa Yoshida, T. Saito, and overall aesthetic curve segment of the aesthetic curve (Section 2.3), Digital Style Design , Digital Style Design Study Guide Subcommittee JSPE,海文堂, 2008.
5. [08a Yoshida], Norimasa Yoshida, T. Saito, complex rhythm generation curves in the logarithmic curve aesthetic, IPSJ Graphics and CAD Study Group, Vol.2008, No.109, pp.79-84, 2008.
   In this paper, logarithmic curves aesthetic curves in complex rhythms (represented by two straight lines curve-shaped or V-shaped to the curvature logarithmic graph), and the binary search method similar to the traditional logarithmic curve generation aesthetic indicating that it is possible to produce efficient and stable. Specifically, the curves rhythm complex tangential position and both endpoints (ie, point 3), α1 slope of the logarithmic graph curvature, α2, the ratio of the change in direction angle curves of the two and the (change in angular orientation of the entire curve rate of change of angle of α1 on the direction of the curve) shows the method by specifying a draw.
6. [07d Yoshida], Norimasa Yoshida, T. Saito, elucidation of the overall aesthetic of a space curve, Vol.2007 IPSJ Graphics and CAD Study Group, NO.111, pp-55-60, 2007.
   In this report, to elucidate the overall aesthetic of a space curve is a monotonous curve changes curvature change rate and character. Aesthetic space curve, the equation of the curve arc length and radius of curvature of the aesthetic, it is also used to define the relations of the radius and arc length characters. Classification of aesthetic space curve, the length of the arc length is 0 or ∞ to the point of curvature and torsion rates, they also in terms of, and whether to rotate or infinitely determined direction and the direction of the tangent vector binormal vector performed as needed. In addition, similar to that associated with all the aesthetic circle curve, indicating that the helix is ​​always associated with all the aesthetic space curve.
7. [07c Yoshida] Norimasa Yoshida and Takafumi Saito, Classification of Aesthetic Space Curves, SIAM Conference on Geometric Design and Computing, 2007.
8. [07b Yoshida] No. Yoshida and T. Saito, in Quasi-Rational Cubic Bezier Curves Aesthetic Forms, Computer-Aided Design & Applications, Vol. 4, Nos.. 1-4, pp.477-486, 2 007. [ 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.
9. [07a Yoshida], Norimasa Yoshida, T. Saito, T. Hiraiwa, interactive control of curvature monotonic curve segment, and CAD Joint Symposium on Visual Computing / Graphics, pp.19-24, 2,007. [ PDF ]

   Abstract: In this paper, we propose a method for interactive control of a typical Class A Bezier curve and the pseudo-aesthetic curves. Pseudo-aesthetic curve is approximated by the representation of a rational cubic Bezier curves aesthetic curves is kept nearly linear curve of logarithmic curvature distribution. To verify the monotonicity of curvature of the pseudo-aesthetic curve leads to a condition to ensure monotonicity of curvature of the rational cubic Bezier curves. Class A Bezier curve is typically a polynomial Bezier curve that is guaranteed to be monotonic curvature, lack of interactive control, such as know where they come to and end drawing seen. Therefore, we constructed a method to interactively control the three-point curve as well as aesthetic. It also reveals the drawable area of ​​a typical Class A Bezier curve, and the special case of aesthetic curves continue to increase the degree n of a typical Class A Bezier curve (α = 1) in the logarithmic spiral indicating that the closer. Summarizing about the interactive control of curvature monotonic curve segment to end, also discusses future research directions

10. [06c Yoshida], Norimasa Yoshida and Takafumi Saito: rational cubic Bezier curve approximation aesthetic, Information Processing Society Technical Committee on Graphics and CAD, Vol. 2,006, No. One hundred and ninety-nine, pp.25-30, 2006. [PDF]
    
    Abstract: aesthetic curve, but the beautiful curves of the change in curvature monotony, lack of compatibility with existing CAD systems to be represented in integral form, must be represented in a special format. In this report, one aesthetic curve segment, we propose a method to approximate a rational cubic Bezier curve is guaranteed monotonic curvature. In order to guarantee the monotonicity of curvature, also discussed methods to ensure monotonicity of the curvature of the rational cubic Bezier curves. In addition, implementing the proposed method was applied to a variety of aesthetic curve segments, in the case of a change of direction angle within 90 degrees, while maintaining a rational cubic curvature monotonic curve segment in one aesthetic confirmed that the curve segment can be approximated by Bezier.
    
11. [06b Yoshida] No. Yoshida and T. Saito, Interactive Aesthetic Curve Segments, The Visual Computer (Pacific Graphics), Vol. 22, No.9-11, Pp.896-905, 2 006. [PDF ]
    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.
12. [06a Yoshida], Norimasa Yoshida and Takafumi Saito: Analysis and overview of the interactive control beautiful curves, symposium VC / Gcad, pp.77-82, 2 006. [PDF] [Video ( DivX format)
    Abstract: In this paper, a new category that can respond to requests for a high level of beauty in design and style industrial design "aesthetic curve segment" is proposed. Of artifacts and natural curve "aesthetic" A lot of has been pointed out by Harada et al can be approximated by a log-linear curvature distribution. In this study, the curve can be represented by a straight line logarithmic curvature distribution curve "aesthetic" call. Miura et al have shown a general formula of aesthetic curves, curve itself is rather obvious, it clearly is not the nature of the curve. In this study, to elucidate the overall aesthetic of the curve, and further indicates that the aesthetic curve evolute of the curve is also aesthetic. Moreover, in order to interactively control the aesthetic curve, we propose a method to draw aesthetic curve segment by specifying the tangential direction at the two end points and those points. That is, the aesthetic curve, and α specifies the slope of the logarithmic distribution diagram of curvature, which can be controlled like a quadratic Bezier curves by three control points. The aesthetic curve segment drawn is a numerical integration is required, the drawing also shows that the rate can be fully interactive.
    In this paper, [05a Yoshida] Unlike the orientation angle, rather than by the newly introduced parameter Λ, and explore the curve segment. In the case of Λ = 0, that is characterized by an arc can be represented exactly. In addition, we also evaluated using reflection lines and aesthetic nature of the evolute of the curve.
13. [05a Yoshida], Norimasa Yoshida and Takafumi Saito: Beautiful curved segments, Information Processing Society Technical Committee on Graphics and CAD, No. One hundred and twenty-one, pp.97-102, two thousand and five. [PDF ]
    Abstract: in an artificial and natural curve "aesthetic" A lot of has been pointed out by Harada et al can be approximated by a log-linear curvature distribution. Miura is a formula that shows the aesthetic curve by arc length, the curve must be made for numerical integration, it is difficult to control interactive curve shape. In this report, "Curve aesthetic," as well as the formulation in each of the arc length and angular direction, and to clarify the nature of the curve aesthetic when a change in the slope of the distribution logarithmic curvature changes its shape for α showing a state. Moreover, in order to control the curve shape interactively, specify the tangential direction at both ends of the segment and there, we propose a method for drawing an aesthetic curve segment.

Yoshida Lab Page

Saito Lab Page

1. T. Hiraiwa, Norimasa Yoshida, T. Saito, Class A Bezier curve generation interactive general, Autumn Meeting of JSPE Annual Conference, pp.353-354, 2007.
2. Takafumi Saito, Norimasa Yoshida, Takumi KASEDA, Miyamura (Nakamura), Hiroko, reflecting an inflection function: a unified theory of strain analysis for evaluation of the reflection surface shape design, and CAD Joint Symposium on Visual Computing / Graphics, pp.291-296, 2007.
3. T. Hiraiwa, Norimasa Yoshida and Takafumi Saito: Class A Bezier curve interactively control the plane, the 69th National Convention of IPSJ, 2007.