Log-aesthetic Curves Research Articles

Log-aesthetic curves as similarity geometric analogue of Euler's elasticae

In this paper we consider the log-aesthetic curves and their generalization which are used in CAGD. We consider those curves under similarity geometry and characterize them as stationary integrable flow on plane curves which is governed by the Burgers equation. We propose a variational formulation of those curves whose Euler–Lagrange equation yields the stationary Burgers equation. Our result suggests that the log-aesthetic curves and their generalization can be regarded as the similarity geometric analogue of Euler's elasticae.

Fairness metric of plane curves defined with similarity geometry invariants

ABSTRACTA curve is considered fair if it consists of continuous and few monotonic curvature segments. Polynomial curves such as Bezier and B-spline curves have complex curvature function, hence the curvature profile may oscillate easily with a little tweak of control points. Thus, bending energy and shear deformation energy are common fairness metrics used to produce curves with monotonic curvature profiles. The fairness metrics are used not just to evaluate the quality of curves, but it also aids in reaching to the final design. In this paper, we propose two types of fairness metric functionals to fair plane curves defined by the similarity geometry invariants, i.e. similarity curvature and its reciprocal to extend a variety of aesthetic fairing metrics. We illustrate numerical examples to show how log-aesthetic curves change depending on and G1 constraints. We extend LAC by modifying the integrand of the functionals and obtain quasi aesthetic curves. We also propose -curve to introduce symmetry concept ...

On Path Planning using Log-Aesthetic Curves

On Path Planning using Log-Aesthetic Curves

The approximation of generalized Log-aesthetic curves using Quintic Bezier curves

Generalized Log Aesthetic Curve segments (GLAC) are aesthetic curves that have monotonic curvature profile and hence they are considered fair. In the field of Computer-Aided Design (CAD), there exists a demand to construct fair curves for various design intent. However, we cannot implement GLAC in CAD system partly due to its transcendental form. A viable solution is to approximate GLACs using a quintic polynomial curve in the form of Bezier using curvature error measure. The problem of this approach is that it requires a formidable size of computations due to arc length reparametrization. In this paper, we introduce a new method of calculating curvature error measure using natural spline interpolation function to minimize computation effort while preserving the accuracy. The final section shows numerical examples depicting the proposed approximation of two types of the GLAC, which clearly indicate the efficiency of proposed method.

Minimum variation log-aesthetic surfaces and their applications for smoothing free-form shapes

AbstractThe log-aesthetic curve, which includes the logarithmic (equiangular) spiral, clothoid, and involute of a circle, achieves a control over curvature distribution by defining its shape as an integral form of its curvature and they are expected to be utilized for the field of design. However, it is very difficult to extend it to surfaces and the existing formulations have some problems that they cannot use arbitrary boundary curves. In this paper, we propose “minimum variation log-aesthetic surface” as a new formulation for the log-aesthetic surface. Based on variational principle our method can generate surfaces by minimizing the objective function newly proposed in this paper for given arbitrary boundary curves.Highlights We propose a “minimum variation log-aesthetic surface” as a new formulation of log-aesthetic surface. We derived a new objective function based on variational principle and our method can generate surfaces by minimizing the objective function. We generate bi-cubic B-spline surfaces by minimizing the new objective function. Numerical examples indicate that it can be used for practical aesthetic design.

Log-aesthetic flow governed by heat conduction equations

ABSTRACTIn this research, based on the concept of the smoothing by curve shortening flow and curvature flow, we will propose log-aesthetic flow to make free-form curves “log-aesthetic.” We will discuss smoothing methods that deal with continuous curves as well as discrete ones. We have one degree of freedom to control smoothing for log-aesthetic flow and can expect the completely smoothed shape of a given curve, which means the shape obtained by fairing, to be log-aesthetic curve. This paper shows that log-aesthetic flow is basically governed by the heat conduction equation, which has been well studied in both physics and mechanical engineering, its effects are easily inferred by the designers and it will be useful for practical aesthetic design.

Quadratic log-aesthetic curves

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.

Generalization of log-aesthetic curves by Hamiltonian formalism

Generalization of log-aesthetic curves by Hamiltonian formalism

Quintic polynomial approximation of log-aesthetic curves by curvature deviation

Log-aesthetic curves (LACs), possessing monotone curvature and including many classical curves, have been widely used to describe fair shapes in geometric modeling. However, they are generally represented in non-polynomial form and are thus not compatible with current CAD systems. In this paper we present quintic polynomial approximation of LAC segments. For a given LAC segment, a quintic G2 interpolating Bézier curve is obtained by minimizing a curvature-based error metric, with the advantage of being more likely to preserve the monotone curvature property. Numerical experiments demonstrate that our method can usually generate better results than the previous methods in terms of the deviation in positions and curvatures.

Reformulation of Generalized Log-aesthetic Curves with Bernoulli equations

AbstractA recent research has found out the relationship between the Log-aesthetic Curve and Riccati differential equations. The obtained differential equations are Riccati, but also Bernoulli type. Based on Sato and Shimizu's derivation of the formula of ρ-shift generalized log-aesthetic curve (GLAC), we derive the formula of the κ-shift GLAC as a solution of a Bernoulli differential equation.

The Polar-aesthetic Curve and Its Applications to Scissors Design

ABSTRACTThe origin of scissors is very old and a pair of scissors which was probably made in ancient Greece around B.C. 1000 is discovered. In Japan a new type of scissors was produced with newly designed blades and has become one of hit products in 2012. The shape of their blades is a Bernoulli curve. The Bernoulli curve is the logarithmic spiral and can be classified as a type of the log-aesthetic curves. One of its properties is that the angle between its tangent vector and the radial axis from the origin is kept constant. This is the reason why the logarithmic spiral is also called the equiangular spiral. Consequently if the blades of a pair of scissors are given by a logarithmic spiral, the cutting angle is always constant.In this paper, we extend the logarithmic spiral to make the cutting angle of a pair of scissors a linear function of the rotation angle of the cutting edges and newly define the polar-aesthetic curve and discuss about conditions for the monotonicity of its curvature. Furthermore we...

Log-aesthetic curves and Riccati equations from the viewpoint of similarity geometry

Log-aesthetic curves and Riccati equations from the viewpoint of similarity geometry

G2Continuous Generalized Log-Aesthetic Curves

ABSTRACTLog-Aesthetic (LA) curve has been claimed as one of a most promising curve for design purpose. However, LA curve has no shape variable which can be used to control its end curvatures directly. Generalized Log-Aesthetic Curve (GLAC) which is the family of LA curve has ability to control its curvature with an extra shape parameter. This paper highlights on designing two segments of GLACs in the form of C- and S-shapes with curvature continuity using interior point method. Designers will be able to design S-shapes and C-shapes by inputting control points, its direction of travel and estimated end curvatures.

Generation of Log-aesthetic curves using adaptive Runge–Kutta methods

Log aesthetic curve (LAC) has been explored extensively by many researchers since 2005. At first, Gaussian–Kronrod has been proposed to evaluate LAC as the formulation of LAC involves double integration. Recently, Incomplete Gamma Function (IGF) has been proposed to represent LAC analytically which decreases the computation time up to 13 times. This paper embarks on the representation of LAC using adaptive Runge–Kutta methods to decrease the LAC computation time. The famous adaptive methods such as Runge–Kutta Fehlberg, Dormand–Prince, Sarafyan and Kutta–Merson are employed to evaluate LAC so that a desired accuracy can be achieved. This paper ends with a detailed investigation on performance metric of IGF and adaptive RK methods to compute LAC. These methods will be compared in terms of computation time and truncation error. Numerical results indicate that the computation time of LAC can be greatly improved and at the same time preserving the LAC’s family.

Surface design based on direct curvature editing

This paper presents a novel method for modifying the shapes of existing uniform bi-cubic B-spline surfaces by interactively editing the curvatures along isoparametric curves. The method allows us to edit the curvatures of the two intersecting isoparametric curves at each knot with specified positions, unit tangents, and unit normals. The user adjusts the radii of circles, representing the radii of curvature in the u and v isoparametric directions directly via a GUI without having to work with control points and knots. Such shape specifications are converted into iterative repositionings of the control points on the basis of geometrical rules. Using these point-based curvature-editing techniques, we successfully embedded log-aesthetic curves into existing surfaces along their isoparametric curves. Moreover, we were able to distribute the cross curvature with log-aesthetic variation along the isoparametric curves. We applied our technique to the design of automobile hood surfaces to demonstrate the effectiveness of our algorithms.

Aesthetic Curves and Surfaces in Computer Aided Geometric Design

Aesthetic shapes are usually actualized as 3D objects represented by free-form surfaces. The main components used to achieve aesthetic surfaces are 2D and 3D curves, which are the elements most basic for determining the shapes and silhouettes of industrial products. Bézier, B-Spline and NURBS are types of flexible curves developed for various design intents. These curves, however produce complex curvature functions that may undermine the formulation of shape aesthetics. A viable solution to this problem is to formulate aesthetic curves and surfaces from well-defined curvatures to improve aesthetic design quality. This paper advocates formalizing aesthetic curve and surface theories to fill the gapmentioned above, which has existed since the 1970s. This paper begins by reviewing on fair curves and surfaces. It then extensively discusses on the technicalities of Log-Aesthetic (LA) curves and surfaces and touches on industrial design applications. These emerging LA curves have a high potential for being used as standards to generate, evaluate and reshape aesthetic curves and surfaces, thus revolutionizing efficiency in developing curve and shape aesthetics.

Transition Curve Modeling with Kinematical Properties: Research on Log-Aesthetic Curves

Log-aesthetic curves (LACs), which are generally used in industrial design for aesthetic shape modeling, are examined for implementation as transition curves despite their non-trivial representation in terms of incomplete gamma functions, or Fresnel integrals which appear in the parametric equations of a certain spirals. The family of log-aesthetic curves includes well-known spirals as Euler, Nielsen, logarithmic spiral, and involutes of a circle. The horizontal geometry of the route can contain classical transition curves, which are formed by transition curve-circular arc-transition curves. In order to compare the examined family of log-aesthetic transition curves with the classical transition curve in terms of vehicle-road kinematics, the curvature and superelevation functions are derived, and functions of lateral change of acceleration (LCA) curves are obtained and illustrated graphically using a constant motion model. The discontinuities in the form of jumps in the graphs of the lateral change of acce...

Aesthetic Curve Design with Linear Gradients of Logarithmic Curvature/Torsion Graphs

The quality of a curve for industrial design and computer graphics can be interrogated using Logarithmic Curvature Graph (LCG) and Logarithmic Torsion Graph (LTG). A curve is said to be aesthetic if it depicts linear LCG and LTG function. The Log-aesthetic curve (LAC) was developed bearing this notion and it was later extended to a Generalized Log-aesthetic curve (GLAC) using the -shift and -shift approach. This paper reformulates GLAC by representing the Logarithmic Curvature and Torsion graph’s gradient function as a nonlinear ordinary differential equation (ODE) with boundary conditions. The outputs of solving the ODEs result in a well defined Cesaro equation in the form of curvature function that is able to produce both planar as well as spatial curves with promising entities for industrial product design, computer graphics and more.

Log-Aesthetic Magnetic Curves and Their Application for CAD Systems

Curves are the building blocks of shapes and designs in computer aided geometric design (CAGD). It is important to ensure these curves are both visually and geometrically aesthetic to meet the high aesthetic need for successful product marketing. Recently, magnetic curves that have been proposed for computer graphics purposes are a particle tracing technique that generates a wide variety of curves and spirals under the influence of a magnetic field. The contributions of this paper are threefold, where the first part reformulates magnetic curves in the form of log-aesthetic curve (LAC) denoting it as log-aesthetic magnetic curves (LMC) and log-aesthetic magnetic space curves (LMSC), the second part elucidates vital properties of LMCs, and the final part proposes G2LMC design for CAD applications. The final section shows two examples of LMC surface generation along with its zebra maps. LMC holds great potential in overcoming the weaknesses found in current interactive LAC mechanism where matching a single segment with G2Hermite data is still a cumbersome task.

Log-Aesthetic Curves for Shape Completion Problem

An object with complete boundary or silhouette is essential in various design and computer graphics feats. Due to various reasons, some parts of the object can be missing hence increasing the complexity in designing process. It is therefore important to reconstruct the missing parts of an object while retaining its aesthetic appearance. In this paper, we propose Log-Aestheic Curves (LAC) for shape completion problem. We propose an algorithm to construct LAC segment and subsequently fit into the gap of the missing parts with C-shape or S-shape. For C-shape completion, we define LAC segment by specifying two endpoints and their respective tangent directions between the gaps while, for S-shape, the user defines an inflection point in between the endpoints. The final section illustrates three examples to showcase the efficiency of the proposed algorithm. The results are further compared with Kimia’s method to prove that the algorithm produces equally good result. Additionally, the proposed algorithm provides an extra degree of freedom in which the user would be able to choose the type of spiral that they desire to solve the shape completion problem.