# Research Area

## Reseach Topics of My Interest

Computer-Aided Geometric Design (CAGD)

Computer-Aided Design (CAD) and its Applications

Computational Design

Computing

## Research Summary

Since 2010, I have been focusing on the study of geometric models of free-form and non-linear curves and surfaces and their applications in industrial design and styling. While traditional CAD systems rely on polynomial or rational parametric forms such as Bézier or NURBS curves and surfaces, these models are not always suffcient to meet the extreme aesthetic requirements of many design applications. To address this challenge, I have been exploring the use of log-aesthetic curves (LACs), a relatively new type of curve developed by Japanese researchers specifically for use in computer-aided aesthetic design (CAAD). LACs have the unique ability to capture the subtleties of visually pleasing shapes. They have been used to design objects like exterior surfaces of automobiles to other objects with significant aesthetic considerations.

In addition to my work on LACs, I have been researching fair curves, or monotonic curvature curves. This area of research has many potential applications in fields such as industrial design and CAD, where smooth and aesthetically pleasing curves are often required. Through my work in this area, I have aimed to develop new methods and tools for creating and manipulating fair curves, and to explore how these curves can be used in different contexts. Overall, my research aims to push the boundaries of what is possible with these and other geometric models, and to find new and innovative ways to apply them to real-world problems.

## Main Research Achievements

1. Parametric equations expressed in terms of incomplete gamma functions are presented. They allow us to find an exact analytic representation of a curve segment for any real value of α. The computational time for generating a LAC segment using the incomplete gamma functions is up to 13 times faster than using direct numerical integration. The results are generalizations of the well-known Cornu, Nielsen, and logarithmic spirals and involutes of a circle.

2. An algorithm for generating a C-shaped G2 multispiral transition curve between two non-parallel straight lines is presented. The G2 multispiral is a curve consisting of two or more log-aesthetic curve segments connected by curvature continuity, and it has inflection endpoints. The two log-aesthetic curve segments with shape parameter α < 0 are connected at the origin and form a multispiral. Depending on the parameter α, the multispiral transition curve G2 can have different shapes; moreover, the shape of the curve approaches a circular arc as α decreases. One of the main purposes of such a curve can be found in the design of high-speed railroad tracks. The curves obtained also find applications in gear design, where G2 continuity is essential to reduce noise and vibration, and in fillet modeling.

3. Superspirals, a new and very general family of fair curves, are introduced. The radius of curvature of the superspiral is given in terms of a completely monotone Gauss hypergeometric function. These curves are generalizations of log-aesthetic curves, as well as other curves whose radius of curvature is a special case of a completely monotonic Gauss hypergeometric function. The high-precision computation of a superspiral segment is performed using the Gauss-Kronrod integration method. The proposed curves, despite their complexity, are candidates for the generation of G2 and G3 nonlinear superspiral splines.

4. I have presented superspiraloids, which are surfaces generated by rotating a two-dimensional superspiral curve segment about an axis, and they, probably, can be applied to computer-aided design of, for example, canals or pipe surfaces. I have also studied families of curves with a monotonic curvature function and their applications in geometric modeling and aesthetic design. Aesthetic analysis and evaluation of the structure and plastic qualities of pseudospirals, i.e. curves with a monotonic curvature function, were carried out for the first time in the field of geometric modeling from the point of view of technical aesthetic laws.

5. I have also studied different families of curves for implementation as transition curves in road design. The combined model of transition curve-circular arc-transition curve was used for implementation. We discovered that log-aesthetic curves can be implemented as transition curves, but they do not have more specifications than the clothoid in terms of road vehicle kinematics. If they had many more specifications than the clothoid, I would create a new type of transition curve with better kinematic properties.

6. For the first time in computational mathematics, I proposed G1 Hermite interpolation curves in biangular coordinates and provided sufficient conditions for their convexity. In the biangular coordinate system, the problem reduces to choosing suitable functions for interpolating the biangular coordinates of the curve at its endpoints. The simplest linear equations in biangular coordinates correspond to the sectrix of Maclaurin, which was also extended by introducing two shape parameters that pull the curve towards the sides of its triangular envelope. I also considered a class of curves whose biangular coordinates have a constant sum, and analyzed their shape and curvature.