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

Logarithmic Curvature Graph as a Shape Interrogation Tool

Abstract A compact formula for Logarithmic Curvature Graph(LCG) and its gradient for planar curves has been shown which can be used as shape interrogation tool. Using these entities, the mathematical definition for a curve to be aesthetic has been introduced to overcome the ambiguity that occurs in measuring the aesthetic value of a curve. Detailed examples are shown how LCG and its gradient can be used to identify curvature extrema and measure the aesthetic value of curves. Mathematics Subject Classification: 65D17, 68U0

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 G2 LMC 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 G2 Hermite data is still a cumbersome task

Preserving monotone or convex data using quintic trigonometric Bézier curves

Bézier curves are essential for data interpolation. However, traditional Bézier curves often fail to detect special features that may exist in a data set, such as monotonicity or convexity, leading to invalid interpolations. This study aims to improve the deficiency of Bézier curves by imposing monotonicity or convexity-preserving conditions on the shape parameter and control points. For this purpose, the quintic trigonometric Bézier curves with two shape parameters are used. These techniques constrain only one of the shape parameters, leaving the other free to provide users with more freedom and flexibility in modifying the final curve. To guarantee smooth interpolation, the curvature profiles of the curves are analyzed, which aids in selecting the optimal shape parameter values. The effectiveness of the developed schemes was evaluated by implementing real-life data and data obtained from the existing schemes. Compared with the existing schemes, the developed schemes produce low-curvature interpolation curves with unnoticeable wiggles and turns. The proposed methods also work effectively for both nonuniformly spaced data and negative-valued convex data in real-life applications. When the shape parameter is correctly chosen, the developed interpolants exhibit continuous curvature plots, assuring $C^2$ continuity

Aesthetic spiral for design

A planar spiral called Generalized Log Aesthetic Curve segment (GLAC) has been proposed using the curve synthesis process with two types of formulation; ρ-shift and κ-shift. Both methods were carried out by extending the formulation of Generalized Cornu Spiral (GCS) in a similar manner to the Log Aesthetic Curve (LAC). The family of GLAC comprises of planar curves of high quality such as GCS, LAC, clothoid, logarithmic spiral and circle involute. The GLAC segment has an additional parameter to determine its shape as compared to GCS and LAC segment, hence an extra constraint can be satisfied when shaping the GLAC segment. The last section of the paper shows a numerical example

On the Variety of Planar Spirals and Their Applications in Computer Aided Design

In this paper we discuss the variety of planar spiral segments and their applications in objects in both the real and artificial world. The discussed curves with monotonic curvature function are well-known in geometric modelling and computer aided geometric design as fair curves, and they are very significant in aesthetic shape modelling. Fair curve segments are used for two-point G1 and G2 Hermite interpolation, as well as for generating aesthetic splines