Aesthetic logarithmic curve

English page

Mar 25, 2009

Introduction of the "aesthetic logarithmic curve"

Norimasa Yoshida (norimasa@cit.nihon-u.ac.jp) (Nihon University)
Takafumi Saito (txsaito@cc.tuat.ac.jp) (Tokyo University of Agriculture and Technology)
* E-mail address, please substitute the half space filled with an @.

* "Aesthetic logarithmic curve" is the past, or were referred to as "aesthetic curve" "beautiful curve".
* related research has been added.

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

The family of Aesthetic Curves

To introduce the aesthetic curve video ( DivX format)

Research paper

1. N. Yoshida [09c 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. CAD Joint Symposium Norimasa Yoshida [09b Yoshida, Ryo Fukuda, T. Saito, interactive generation method of aesthetic curve segments logarithmic space, and Visual Computing / Graphics, to appear.
3. N. Yoshida [09a 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. Aesthetic curve segment (Section 2.3), Norimasa Yoshida [08b Yoshida, Takafumi Saito, and the overall picture of the aesthetic curve design digital style , edited by digital research subcommittee meeting precision engineering design style, Kaibundo, 2008.
5. Norimasa Yoshida [08a Yoshida, Takafumi Saito, complex rhythm generation of the curve in the logarithmic curve aesthetic, graphics and CAD Technical Committee on Information Processing Society, Vol.2008, No.109, pp.79-84, 2008.
   In this report, by the search by the bisection method similar to the curve generation aesthetic logarithm of the conventional, the Curve aesthetic logarithmic curve rhythm composite curve (logarithmic graph curvature is represented by two straight lines shaped to or shaped V) indicating that it is possible to generate efficient and stable. More specifically, the position of a point at both ends tangential direction (ie, point 3), α1 slope of the logarithmic graph curvature, α2, change of direction angle ratio of the change direction angle curve of both and the entire curve (curve rhythm complex indicate the method by specifying the draw) rate of change of direction angle of the curve of the α1 for.
6. Norimasa Yoshida elucidation [07d Yoshida, Takafumi Saito, of the overall picture of the aesthetic space curve, Vol.2007 of Information Processing Society Technical Committee on CAD and graphics, NO.111, pp-55-60, 2007.
   In this report, to clarify the overall picture of the aesthetic space curve curvature change rate changes and torsion is a monotonic curve. Aesthetic space curve is defined by equation also be used for the radius and arc length of the torsion rate, an equation of the arc length and radius of curvature in the aesthetic curve. Aesthetic classification of space curves, such as whether to rotate them in terms of infinite, or determined the direction of the binormal vector tangent vector direction and also, the length of the arc length of 0 or ∞ to the point of curvature and torsion rate performed according to the. It is also shown that just as the yen has been associated with all the aesthetic curve, all aesthetic space curve is associated with a normal spiral.
7. Norimasa Yoshida and Takafumi Saito [07c Yoshida], Classification of Aesthetic Space Curves, SIAM Conference on Geometric Design and Computing, 2007.
8. N. Yoshida and T. Saito [07b Yoshida], Quasi-Aesthetic Curves in Rational Cubic Bezier 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. Norimasa Yoshida [07a Yoshida, Takafumi Saito, Tomoyuki Hiraiwa, interactive control of curvature monotonic curve segment, and CAD Joint Symposium on Visual Computing / Graphics, pp.19-24, 2007. [ PDF ]

   Abstract: In this paper, we propose a method for interactive control of Class A Bezier curve and typical pseudo-aesthetic curve. Pseudo-aesthetic curve is a representation of a rational cubic Bezier curve approximated by the aesthetic curve, a curve is kept almost linearity of the logarithmic distribution diagram of curvature. To confirm the pseudo-monotonicity of curvature of the aesthetic curve, leading to conditional expression to verify the monotonicity of curvature of the Bezier curve next three rational. Class A Bezier curve is typical, but is a polynomial Bezier curve that is guaranteed to be monotonic curvature, lack of interactive control, such as do not know where to come to the end and I do not draw. Therefore, we constructed a method to interactively control by three points as well as the aesthetic curve. In addition, the logarithmic spiral is (α = 1) a special case of the aesthetic curve can be drawn to clarify the area of ​​typical Class A Bezier curve, when we increase the order n of the Class A Bezier curve typical indicating that the approach. Summarized about the interactive control of curvature monotonic curve segment at the end, also discussed the direction of future research

10. Norimasa Yoshida [06c Yoshida, Takafumi Saito:... Approximate rational Bezier cubic curve aesthetic, graphics and CAD Technical Committee on Information Processing Society, Vol 2 006, No 199, pp.25-30, two thousand and six [PDF]
    
    Abstract: aesthetic curve, but is a beautiful curve of monotone curvature change, not compatible with the CAD system to be represented in the form of traditional integration, must be represented in a special format. In this paper, we propose a method to approximate rational Bezier curve of one of the following three which guarantees monotonicity of curvature, a curve segment of one aesthetic. In order to ensure monotonicity of curvature, are also discussed methodology to determine the monotonicity of rational cubic Bezier curve curvature. In addition, we implement the proposed method was applied to the curve segment aesthetic of various, in the case of less than 90 degrees, the next three rational one while maintaining the monotonicity of the curvature curve segments aesthetic one change of direction angle it was confirmed that the curve segment can be approximated by Bezier.
    
11. N. Yoshida and T. Saito [06b Yoshida], 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. Norimasa Yoshida [06a Yoshida, Takafumi Saito. Control and interactive elucidation overall picture of the beautiful curve, Joint Symposium on VC / GCAD, pp.77-82, two thousand and six [PDF] [Video ( DivX format)]
    Abstract: In this paper, we propose a new "aesthetic curve segment" of the new category that can respond to a request for a higher level of beauty in industrial design and design style. Many of the "aesthetic curve" in the artificial and the natural world, it is logarithmic distribution diagram of curvature can be approximated by a straight line has been pointed out by Harada et al. In this study, called "aesthetic curve" to figure logarithmic distribution curve can be represented by a straight line curvature. Miura et al have shown the general formula of aesthetic curves, have not revealed the nature of the curve also, not self-evident that the curve itself. In this study, to elucidate the overall picture of the aesthetic curve, further, indicate that it is also aesthetic curve evolute of the curve aesthetic. In addition, in order to interactively control the aesthetic curve, we propose a method to draw the aesthetic curve segment by specifying the tangential direction at those points and the two endpoints. In other words, the aesthetic curve, you can specify the α slope of the straight line in logarithmic distribution diagram of curvature, can be controlled in the same way as the next two Bezier curve by three control points is possible. To draw the aesthetic curve segment is the numerical integration is necessary, also indicates that it is possible to draw at a rate sufficiently interactive.
    In this paper, unlike [05a Yoshida], rather than in the direction angle, by the newly introduced parameter Λ, we explore the curve segment. There is feature in the case of Λ = 0, that the arc can be represented exactly. We are also evaluated by reflection lines and aesthetic nature of the evolute of the curve.
13. Norimasa Yoshida [05a Yoshida, Takafumi Saito:.. Committee on beautiful curve segment CAD, graphics and the Information Processing Society, one hundred and twenty-one No, pp.97-102, 2005 [PDF ]
    Abstract: Many of the "aesthetic curve" in the artificial and the natural world, it is logarithmic distribution diagram of curvature can be approximated by a straight line has been pointed out by Harada et al. Miura has shown by the general formula aesthetic arc length of a curve, in order to draw a curve is a numerical integration must be done, it is difficult to control interactive curve shape. In this report, as well as to formulate in each of the arc length angle and the direction the curve "aesthetic", to clarify the nature of the curve aesthetic of when there is a change in the slope of the straight line in the figure logarithmic distribution curvature, change its shape for α showing a state of. In addition, in order to control interactively shape curve, we propose a method to specify the direction in which the tangential point and the ends of the segment, to draw the aesthetic curve segment.

Page of Yoshida laboratory

Page of the laboratory to Saito

1. Tomoyuki Hiraiwa, Norimasa Yoshida, Takafumi Saito, Class A Bezier curve generation interactive general, JSPE Autumn Conference Annual Conference, pp.353-354, 2007.
2. Takafumi Saito, Norimasa Yoshida, Takumi Kaseda, Hiroko (Nakamura) Miyamura, inflection reflection function: a unified analytical theory CAD Joint Symposium of shape distortion of reflection surfaces for design evaluation, and Visual Computing / Graphics, pp.291-296, , 2007.
3. Tomoyuki Hiraiwa, Norimasa Yoshida, Takafumi Saito: Class A Bezier curve interactive control plane, the 69th National Convention of Information Processing Society, 2007.