Log-aesthetic Curves Research Articles

Formula omitted]-Curves: Extended log-aesthetic curves with variable shape parameter

This paper proposes a novel curve called the α-curve, which is specified by a variable shape parameter α and can be applied in curve reconstruction involving curve fairing, fitting, and segmentation. In contrast to classical log-aesthetic curves in which α is constant, this study contributes a novel formulation of α that varies monotonically within a specified segment. By introducing the variable α, the logarithmic curvature graph (LCG) becomes piecewise linear and parabolic, or any other function if necessary. Consequently, in comparison to conventional log-aesthetic curves that have limited field usage based on their rigidity, α-curves have various LCGs and flexible representations, thereby opening up many practical applications. Experimental results and comparative studies on curves created by CAD experts in the automotive industry demonstrate the theoretical and practical validity of α-curves.

The Approximation of Generalized Log-Aesthetic Curves with G 2 Cubic Trigonometric Bézier Function

One of the requirements of curves in computer-aided design (CAD) is a curve with monotonic curvature profiles. Generalized log-aesthetic curves (GLACs) comprise a family of aesthetic curves which possesses a monotonic curvature profile. However, we cannot directly implement GLAC in CAD systems since it is in the form of a transcendental form. In this paper, we used cubic trigonometric Bézier (T-Bézier) curves with two shape parameters to approximate GLAC with G2 continuity. The final approximation formula inherits the shape parameters of GLAC whereas T-Béziers’ shape parameters are utilized to satisfy G2 constraints. Numerical results indicate that the proposed algorithm is capable of approximating GLAC within the given tolerance in (at least) two iterations.

Log-aesthetic curves: Similarity geometry, integrable discretization and variational principles

In this paper, we consider a class of plane curves called log-aesthetic curves and their generalization which are used in computer aided geometric design. In the framework of similarity geometry, those curves are characterized as invariant curves under the integrable flow on plane curves governed by the Burgers equation. They also admit a variational formulation leading to the stationary Burgers equation as the Euler-Lagrange equation. As an application of the formulation, we propose a discretization of these curves and the associated variational principle which preserves the underlying integrable structure. We finally present algorithms for generating discrete log-aesthetic curves for given G1 data based on similarity geometry. Our method is able to generate S-shaped discrete curves with an inflection as well as C-shaped curves according to the boundary condition. The resulting discrete curves are regarded as self-adaptive discretization and thus high-quality even with the small number of points. Through the continuous representation, those discrete curves provide a flexible tool for the generation of high-quality shapes.

Fairing of planar curves to log-aesthetic curves

We present an algorithm to fair a given planar curve by a log-aesthetic curve (LAC). We show how a general LAC segment can be uniquely characterized by seven parameters and present a method of parametric approximation based on this fact. This work aims to provide tools to be used in reverse engineering for computer-aided geometric design. Finally, we show an example of usage by applying this algorithm to the data points obtained from 3D scanning a model-car roof.

Analysis of Drag Coefficients around Objects Created Using Log-Aesthetic Curves

A fair curve with exceptional properties, called the log-aesthetic curves (LAC) has been extensively studied for aesthetic design implementations. However, its implementation in terms of functional design, particularly hydrodynamic design, remains mostly unexplored. This study examines the effect of the shape parameter α of LAC on the drag generated in an incompressible fluid flow, simulated using a semi-implicit backward difference formula coupled with P2−P1 Taylor–Hood finite elements. An algorithm was developed to create LAC hydrofoils that were used in this study. We analyzed the drag coefficients of 47 LAC hydrofoils of three sizes with various shapes in fluid flows with Reynolds numbers of 30, 40, and 100, respectively. We found that streamlined LAC shapes with negative α values, of which curvature with respect to turning angle are almost linear, produce the lowest drag in the incompressible flow simulations. It also found that the thickness of LAC objects can be varied to obtain similar drag coefficients for different Reynolds numbers. Via cluster analysis, it is found that the distribution of drag coefficients does not rely solely on the Reynolds number, but also on the thickness of the hydrofoil.

High-quality approximation of log-aesthetic curves based on the fourth-order derivative

Abstract We propose a new method for approximating log-aesthetic curves ${\boldsymbol C}_{\mathrm{LA}}$ 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, ${\boldsymbol C}_{\mathrm{LA}}$ 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 ${\boldsymbol C}_{\mathrm{LA}}$ with Bézier curves from two aspects. One aspect entails the uncertainty of how accurately the low-degree Bézier curves can approximate ${\boldsymbol C}_{\mathrm{LA}}$. The other aspect is the existence of a subset of ${\boldsymbol C}_{\mathrm{LA}}$ that is inherently difficult to approximate with such conventional parametric curves. The experimental results and comparisons demonstrated the validity of the proposed method.

Interactive G1 and G2 Hermite Interpolation Using Coupled Log-aesthetic Curves

Computer-Aided Design and Applications is an international journal on the applications of CAD and CAM. It publishes papers in the general domain of CAD plus in emerging fields like bio-CAD, nano-CAD, soft-CAD, garment-CAD, PLM, PDM, CAD data mining, CAD and the internet, CAD education, genetic algorithms and CAD engines. The journal is aimed at all developers and users of CAD technology to ptovide CAD solutions for various stages of design and manufacturing. The journal publishes all about Computer-Aided Design and Computer-Aided technologies.

The Development of Log Aesthetic Patch and Its Projection onto the Plane

In this work, we introduce a new type of surface called the Log Aesthetic Patch (LAP). This surface is an extension of the Coons surface patch, in which the four boundary curves are either planar or spatial Log Aesthetic Curves (LACs). To identify its versatility, we approximated the hyperbolic paraboloid to LAP using the information of lines of curvature (LoC). The outer part of the LoCs, which play a role as the boundary of the hyperbolic paraboloid, is replaced with LACs before constructing the LAP. Since LoCs are essential in shipbuilding for hot and cold bending processes, we investigated the LAP in terms of the LoC’s curvature, derivative of curvature, torsion, and Logarithmic Curvature Graph (LCG). The numerical results indicate that the LoCs for both surfaces possess monotonic curvatures. An advantage of LAP approximation over its original hyperbolic paraboloid is that the LoCs of LAP can be approximated to LACs, and hence the first derivative of curvatures for LoCs are monotonic, whereas they are non-monotonic for the hyperbolic paraboloid. This confirms that the LAP produced is indeed of high quality. Lastly, we project the LAP onto a plane using geodesic curvature to create strips that can be pasted together, mimicking hot and cold bending processes in the shipbuilding industry.

Lines of Curvature for Log Aesthetic Surfaces Characteristics Investigation

Lines of curvatures (LoCs) are curves on a surface that are derived from the first and second fundamental forms, and have been used for shaping various types of surface. In this paper, we investigated the LoCs of two types of log aesthetic (LA) surfaces; i.e., LA surfaces of revolution and LA swept surfaces. These surfaces are generated with log aesthetic curves (LAC) which comprise various families of curves governed by α. First, since it is impossible to derive the LoCs analytically, we have implemented the LoC computation numerically using the Central Processing Unit (CPU) and General Processing Unit (GPU). The results showed a significant speed up with the latter. Next, we investigated the curvature distributions of the derived LoCs using a Logarithmic Curvature Graph (LCG). In conclusion, the LoCs of LA surface of revolutions are indeed the duplicates of their original profile curves. However, the LoCs of LA swept surfaces are LACs of different shapes. The exception to this is when this type of surface possesses LoCs in the form of circle involutes.

Smooth path planning via cubic GHT-Bézier spiral curves based on shortest distance, bending energy and curvature variation energy

<abstract> In this article, an algorithm to obtain a smooth path (free from obstacles) that can be optimized by shortest path distance, bending energy and curvature variation energy will be presented. Previously, scholars used various tools to generate smooth path such as Clothoid, Log-Aesthetic curves (LACs), and Bézier curves. The limited number of solutions from the aforementioned curves become one of the drawback to generate smooth path planning. Therefore, providing a number of solutions that can generate smooth path planning becomes the objective of this study. In this paper to generate a smooth path, five templates of spiral transition curves having three different shape parameters with monotone curvature (either increase or decrease) by cubic GHT-Bézier curves are proposed. Moreover, few examples of path planning technique via cubic GHT-Bézier spiral curve to show the flexibility of smooth path by minimization of the shortest path (minimum arc length) <bold>L</bold>, bending energy <bold>E</bold> and curvature variation energy <bold>V</bold> are presented. The superiority of cubic GHT-Bézier spiral path smoothing techniques as compared to Clothoid and LACs is also demonstrated. </abstract>

Log-aesthetic curves and their relation to fluid flow patterns in terms of streamlines☆

Abstract The log-aesthetic curve (LAC) is a family of aesthetic curves with linear logarithmic curvature graphs (LCGs). It encompasses well-known aesthetic curves such as clothoid, logarithmic spiral, and circle involute. LAC has been playing a pivotal role in aesthetic design. However, its application for functional design is an uncharted territory, e.g. the relationship between LAC and fluid flow patterns may aid in designing better ship hulls and breakwaters. We address this problem by elucidating the relationship between LAC and flow patterns in terms of streamlines at a steady state. We discussed how LAC pathlines form under the influence of pressure gradient via Euler's equation and how LAC streamlines are formed in a special case. LCG gradient ($\alpha $) for implicit and explicit functions is derived, and it is proven that the LCG gradient at the inflection points of explicit functions is always 0 when its third derivative is nonzero. Due to the complexity of the parametric representation of LAC, it is almost impossible to derive the general representation of LAC streamlines. We address this by analyzing the streamlines formed by incompressible flow around an airfoil-like obstacle generated with LAC having various shapes, ${\alpha _r} = \ \{ { - 20,{\rm{\ }} - 5,{\rm{\ }} - 1,{\rm{\ }} - 0.5,{\rm{\ }} - 0.15,{\rm{\ }}0,{\rm{\ }}1,{\rm{\ }}2,{\rm{\ }}3,{\rm{\ }}4,{\rm{\ }}20} \}$, and simulating the streamlines using FreeFem++ reaching a steady state. We found that the LCG gradient of the resultant streamlines is close to that of a clothoid. When the obstacle shape is almost the same as that of a circle ($\alpha \ = \ 20$), the streamlines adjacent to the obstacles have numerous curvature extrema despite nearing steady state. The flow speed variation is the lowest for $\alpha \ = \ - 1.43$ and gets higher as $\alpha$ is increased or decreased from $\alpha \ = \ - 1.43$.

Interpolation of Point Sequences with Extremum of Curvature by Log-aesthetic Curves with G2 continuity

Interpolation of Point Sequences with Extremum of Curvature by Log-aesthetic Curves with G2 continuity

Mathematical Generalization of Log-Aesthetic Curves and Aesthetic Curves Hierarchy Via Similarity Geometry

Mathematical Generalization of Log-Aesthetic Curves and Aesthetic Curves Hierarchy Via Similarity Geometry

A Note on Superspirals of Confluent Type

Superspirals include a very broad family of monotonic curvature curves, whose radius of curvature is defined by a completely monotonic Gauss hypergeometric function. They are generalizations of log-aesthetic curves, and other curves whose radius of curvature is a particular case of a completely monotonic Gauss hypergeometric function. In this work, we study superspirals of confluent type via similarity geometry. Through a detailed investigation of the similarity curvatures of superspirals of confluent type, we find a new class of planar curves with monotone curvature in terms of Tricomi confluent hypergeometric function. Moreover, the proposed ideas will be our guide to expanding superspirals.

τ-curve: introduction of cusps to aesthetic curves☆

Abstract Yan, Schiller, Wilensky, Carr, and Schaefer pointed out that one of the demerits of clothoid interpolation is a jumping behavior during the deformation of the curve. This phenomenon occurs because the clothoid curve cannot have a cusp, where the curve is kinked or the direction of the curve is abruptly changed. We discuss how to introduce cusps for the log-aesthetic curve including the clothoid curve and propose to use for the representation of a curve the direction angle instead of curvature and define a new curve named τ-curve, which is defined by the direction angle of the curve.

G1 Hermite Interpolating with Discrete Log-aesthetic Curves and Surfaces

G1 Hermite Interpolating with Discrete Log-aesthetic Curves and Surfaces

Optimal Path Smoothing with Log-aesthetic Curves Based on Shortest Distance, Minimum Bending Energy or Curvature Variation Energy

Optimal Path Smoothing with Log-aesthetic Curves Based on Shortest Distance, Minimum Bending Energy or Curvature Variation Energy

Generalization of log-aesthetic curves via similarity geometry

The class of log-aesthetic curves includes the logarithmic spiral, clothoid, and involute of a circle. Although most of these curves are expressed only by an integral form of the tangent vector, it is possible to interactively generate and deform them, thereby presenting many applications in industrial and graphic design. The use of the log-aesthetic curves in practical design, however, is still limited. Therefore, we should extend its formula to obtain curves that solve various practical design problems such as $$G^n$$ Hermite interpolation, deformation, smoothing, data-point fitting, and blending plural curves. In this paper, we present a systematic approach to representing log-aesthetic curves via similarity geometry. In turn, this research provides a unified framework for various studies on log-aesthetic curves, particularly of log-aesthetic curve formulation.

Universal software platform for visualizing class F curves, log-aesthetic curves and development of applied CAD systems

This article describes the capabilities of a universal software platform for visualizing class F curves and developing specialized applications for CAD systems based on Microsoft Excel VBA, the software complex FairCurveModeler, and computer algebra systems. Additionally, it demonstrates the use of a software platform for visualizing functional and log-aesthetic curves integrated with CAD Fusion360. The value of the curves is evident in visualizing the qualitative geometry of the product shape in industrial design. Moreover, the requirements for the characteristics of class F curves are emphasized to form a visual purity of shape in industrial design and to provide a positive emotional perception of the visual image of the product by a person.

A new formulation of the minimum variation log-aesthetic surface for scale-invariance and parameterization-independence

ABSTRACTThe log-aesthetic curve, which includes the logarithmic (equiangular) spiral, clothoid, and circular involute, achieves control over the curvature distribution by defining its shape as an integral of its curvature, and is expected to be utilized for the field of aesthetic design.Some formulations of the log-aesthetic surface as extensions of the log-aesthetic curve have been proposed. The minimum variation surface is one of them, and has a feature that it can be used for arbitrary four boundary curves. The minimum variation log-aesthetic surface is defined as a surface which minimizes an objective function. However, it is not scale-invariant and parameterization-independent.In this study, we propose a new formulation of the minimum variation log-aesthetic surface for scale-invariance and parameterization-independence.