Log-aesthetic Curves Research Articles

Drawable Region of the Generalized Log Aesthetic Curves

The main characteristic of visually pleasing curves used for product design is a monotonic curvature profile. Recently, a planar curve called Generalized Log Aesthetic Curve (GLAC) has been extended from the Log Aesthetic Curve (LAC), and it has an additional shape parameter,ν. This curve preserves the monotonicity of curvature and is said to produce visually pleasing curves. This paper delves on the drawable region of the GLAC segment which indicates the probable solutions of shape parameters from given interpolating points and the direction of travel at those points. The first section reviews the formulation of GLAC and its related bounds. The section describes the algorithm for identifying the drawable region. It is followed by the section describing how small changes ofνwiden the drawable boundaries. The final section discusses the superiority of GLAC compared to LAC for use in industrial product design.

対数型美的曲線を用いた<i>G</i><sup>2</sup> Hermite 内挿法

Although the log-aesthetic curves are generally expressed only by an integral form of the tangent vector, it is possible to interactively generate and deform them and they are expected to be utilized for practical use of industrial and graphical design. In this paper we propose a novel method to solve the G2 Hermite interpolation problem with the log-aesthetic plane curve as well as a new method to generate an S-shaped curve for G1 Hermite interpolation. The methods proposed in the paper have been implemented as a plug-in module for a commercial CAD system and are successfully used for practical design.

Shape analysis of generalized log-aesthetic curves

This paper elucidates the properties of Generalized Log-Aesthetic Curves (GLAC) for CAD practicalities. It is an extension of the emerging Log-Aesthetic (LA) curve with an extra degree of freedom (ν) encompassing clothoid, circle involute, logarithmic spiral and etc. This paper analyzes the bounds of GLAC with respect to its arc length and turning angle. Inflection points occurrence details based on the shape parameters (ν, Λ and α) are highlighted in this paper. Mathematics Subject Classification: 65D17, 68U07

Family of superspirals with completely monotonic curvature given in terms of Gauss hypergeometric function

We present superspirals, a new and very general family of fair curves, whose radius of curvature is given in terms of a completely monotonic Gauss hypergeometric function. The superspirals are generalizations of log-aesthetic curves, as well as other curves whose radius of curvature is a particular case of a completely monotonic Gauss hypergeometric function. High-accuracy computation of a superspiral segment is performed by the Gauss–Kronrod integration method. The proposed curves, despite their complexity, are the candidates for generating G2, and G3 non-linear superspiral splines.

Fitting [formula omitted] multispiral transition curve joining two straight lines

In this paper, we describe an algorithm for generating a C-shaped G2 multispiral transition curve between two non-parallel straight lines. The G2 multispiral is a curve that consists of two or more log-aesthetic curve segments connected with curvature continuity, and it has inflection endpoints. Compound-rhythm log-aesthetic curves are not directly applicable to the generation of transition curves between two straight lines, which is important in highway and railroad track design, because both endpoints are required to be inflection points. Thus, a new approach for generating transition curves is necessary. The two log-aesthetic curve segments with shape parameter α<0 are connected at the origin, and they form a multispiral. The problem is to find a similar triangle, as in the given data. Depending on the parameter α, the G2 multispiral transition curve may have different shapes; moreover, the shape of the curve approximates a circular arc as α decreases. The obtained curves also find applications in gear design and fillet modeling.

The Evolutes of Log-Aesthetic Planar Curves and the Drawable Boundaries of the Curve Segments

AbstractThis paper investigates the two characteristics of log-aesthetic curves. We first show that the evolutes of log-aesthetic curves are also log-aesthetic curves. We provide a proof that the evolute of a log-aesthetic curve with the shape parameter α is a log-aesthetic curve with the shape parameter −1/(α−2). Then, we present a method for drawing the theoretical drawable boundaries of log-aesthetic curve segments with α 1. We compare the theoretical drawable boundaries with the experimental drawable regions and show that they agree well except when α is close to 0.

Variational Formulation of the Log-Aesthetic Surface and Development of Discrete Surface Filters

The log-aesthetic curves include the logarithmic (equiangular) spiral, clothoid, and involute curves. Although most of them are expressed only by an integral form of the tangent vector, it is possible to interactively generate and deform them and they are expected to be utilized for practical use of industrial and graphical design. The discrete log-aesthetic filter based on the formulation of the log-aesthetic curve has successfully been introduced not to impose strong constraints on the designer's activity, to let him/her design freely and to embed the properties of the log-aesthetic curves for complicated ones with both increasing and decreasing curvature. In this paper, in order to define the log-aesthetic surface and develop surface filters based on its formulation, at first we reformulate the log-aesthetic curve with variational principle. Then we propose several new functionals to be minimized for free-form surfaces and define the log-aesthetic surface. Furthermore we propose new discrete surface filters based on the log-aesthetic surface formulation

Analytic parametric equations of log-aesthetic curves in terms of incomplete gamma functions

Log-aesthetic curves (LACs) have recently been developed to meet the requirements of industrial design for visually pleasing shapes. LACs are defined in terms of definite integrals, and adaptive Gaussian quadrature can be used to obtain curve segments. To date, these integrals have only been evaluated analytically for restricted values ( 0 , 1 , 2 ) of the shape parameter α. We present parametric equations expressed in terms of incomplete gamma functions, which allow us to find an exact analytic representation of a curve segment for any real value of α. The computation time for generating an LAC segment using the incomplete gamma functions is up to 13 times faster than using direct numerical integration. Our equations are generalizations of the well-known Cornu, Nielsen, and logarithmic spirals, and involutes of a circle.

複合リズム対数型美的曲線の入力とその意匠デザインへの応用(&lt;小特集&gt;第18回設計工学・システム部門講演会)

This paper proposes a method of inputting a planar log-aesthetic curve with compound-rhythm by use of four control points. The log-aesthetic curve does not exhibit any undulations of curvature since its curvature monotonically increases or decreases and it is suitable to be used for practical product design. The input methods by use of control points presented so far can generate only one log-aesthetic curve and it is not capable enough to represent characteristic lines of industrial products. We show that the compound rhythm log-aesthetic curve made up of two log-aesthetic curve segments connected with C^3 continuity has an enough expression ability to represent character lines of cars by illustrating practical design examples.

Discrete Log-Aesthetic Filter

AbstractThe aesthetic curves include the logarithmic (equiangular) spiral, clothoid, and involute curves. Although most of them are expressed only by an integral form of the tangent vector, it is possible to interactively generate and deform them and they are expected to be utilized for practical use of industrial and graphical design. However, the curvature of a log-aesthetic curve segment must be monotonically increasing or decreasing and a strong constraint as a function of arc length is imposed on its curvature. For geometrical design, it is desirable not to impose strong constraints on the designer’s activity, to let him/her design freely and to embed the properties of the log-aesthetic curves for complicated curves with both increasing and decreasing curvature. Hence we develop discrete filters named discrete log-aesthetic filters to fair a sequence of points with noises and to fit it locally to log-aesthetic curves. Furthermore we extend them for surfaces.

Discrete Log-Aesthetic Filter

AbstractThe aesthetic curves include the logarithmic (equiangular) spiral, clothoid, and involute curves. Although most of them are expressed only by an integral form of the tangent vector, it is possible to interactively generate and deform them and they are expected to be utilized for practical use of industrial and graphical design. However, the curvature of a log-aesthetic curve segment must be monotonically increasing or decreasing and a strong constraint as a function of arc length is imposed on its curvature. For geometrical design, it is desirable not to impose strong constraints on the designer’s activity, to let him/her design freely and to embed the properties of the log-aesthetic curves for complicated curves with both increasing and decreasing curvature. Hence we develop discrete filters named discrete log-aesthetic filters to fair a sequence of points with noises and to fit it locally to log-aesthetic curves. Furthermore we extend them for surfaces.

Compound-Rhythm Log-Aesthetic Curves

AbstractThis paper presents an efficient and stable method for drawing compound-rhythm log-aesthetic curves. Compound-rhythm 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, α0 and α1 (which are the slopes of logarithmic curvature graphs) and the ratio rθ of the change of the tangential angle of the curve α0 to the change of the tangential angle of the whole curve.

3315 対数型美的曲線の意匠デザインへの応用

This paper proposes the method of inputting a planar log-aesthetic curve with compound-rhythm by use of four control points. The log-aesthetic curve does not exhibit any undulations of curvature since its curvature monotonically increases or decreases and it is suitable to be used for practical product design. The input methods by use of control points presented so far can generate only one log-aesthetic curve and it is not capable enough to represent characteristic lines of industrial products. We show that the compound-rhythm log-aesthetic curve made up of two log-aesthetic curve segments connected with G^3 continuity has an enough expression ability to represent character lines of cars by illustrating practical design examples.

Input of Log-aesthetic Curve Segments with Inflection End Points and Generation of Log-aesthetic Curves withG2continuity

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