## About me

### Personal details

**Born:**April 1, 1981 in Odorheiu Secuiesc (Székelyudvarhely), Romania;**Nationality:**Hungarian;**Work:**Babeș-Bolyai University, Department of Mathematics and Computer Science of the Hungarian Line, RO-400084 Cluj-Napoca, Romania;**University position:**associate professor.

### Research interests

- Computer Aided Geometric Design;
- Computer Graphics;
- Computational Geometry;
- Evolutionary Optimization;
- Monte Carlo methods.

## This site...

- does not cover all aspects of Computer Graphics, Computer Aided Geometric Design, Computational Geometry or Probability and Mathematical Statistics;
- besides my scientific resume, it is created and maintained for the use of my students.

## List of selected publications

**Ágoston Róth**, **October 11, 2018**. *Algorithm xyz: An OpenGL and C++ based function library for curve and surface modeling in a large class of extended Chebyshev spaces*, **ACM Transactions on Mathematical Software**, **x**(y):z–w, **IF**** _{2017/2018}** ≅ 2.905,

**AIS****≅ 1.845. [@publisher | @arXiv | accepted, article in press]**

_{2017/2018}**Abstract**

We propose a platform-independent multi-threaded function library that provides data structures to generate, differentiate and render both the ordinary basis and the normalized B-basis of a user-specified extended Chebyshev (EC) space that comprises the constants and can be identified with the solution space of a constant-coefficient homogeneous linear differential equation defined on a sufficiently small interval. Using the obtained normalized B-bases, our library can also generate, (partially) differentiate, modify and visualize a large family of so-called B-curves and tensor product B-surfaces. Moreover, the library also implements methods that can be used to perform dimension elevation, to subdivide B-curves and B-surfaces by means of de Casteljau-like B-algorithms, and to generate basis transformations for the B-representation of arbitrary integral curves and surfaces that are described in traditional parametric form by means of the ordinary bases of the underlying EC spaces. Independently of the algebraic, exponential, trigonometric or mixed type of the applied EC space, the proposed library is numerically stable and efficient up to a reasonable dimension number and may be useful for academics and engineers in the fields of Approximation Theory, Computer Aided Geometric Design, Computer Graphics, Isogeometric and Numerical Analysis.

Imre Juhász, **Ágoston Róth, 2019**. *Adjusting the energies of curves defined by control points*, **Computer-Aided Design**, **107**:77–88, **IF**** _{2017/2018}** ≅ 2.947,

**AIS****≅ 0.796. [@publisher]**

_{2017/2018}**Abstract**

We consider curves which are determined by the combination of control points and blending functions, and adjust their shape by energy functionals. We move selected control points of the curve while the rest of them are fixed, and find those positions of the movable control points which minimize the given energy functional. If just a single control point is moved, we show that the locus of the moving control point for which the energy of the curve has a prescribed value is a sphere the center of which minimizes the energy of the curve. On the basis of this, we provide procedures:

**i****)**to increase/maintain the order of continuity of linked curves at their joint, while decreasing their energy as low as possible,

**ii****)**for gap filling subject to energy minimization and continuity constraints,

**iii****)**for energy minimizing Hermite-type interpolating spline curves. We shortly outline how to extend this curve fairing method to tensor product surfaces as well.

**Abstract**

Extended Chebyshev spaces that also comprise the constants represent large families of functions that can be used in real-life modeling or engineering applications that also involve important (e.g. transcendental) integral or rational curves and surfaces. Concerning CAGD, the unique normalized B-bases of such vector spaces ensure optimal shape preserving properties, important evaluation or subdivision algorithms and useful shape parameters. Therefore, we propose global explicit formulas for the entries of those transformation matrices that map these normalized B-bases to the traditional (or ordinary) bases of the underlying vector spaces. Then, we also describe general and ready to use control point configurations for the exact representation of those traditional integral parametric curves and (hybrid) surfaces that are specified by coordinate functions given as (products of separable) linear combinations of ordinary basis functions. The obtained results are also extended to the control point and weight based exact description of the rational counterpart of these integral parametric curves and surfaces. The universal applicability of our methods is presented through polynomial, trigonometric, hyperbolic or mixed extended Chebyshev vector spaces.

**Abstract**

Using the normalized B-bases of vector spaces of trigonometric and hyperbolic polynomials of finite order, we specify control point configurations for the exact description of the zeroth and higher order (mixed partial) derivatives of integral curves and (hybrid) multivariate surfaces determined by coordinate functions that are exclusively given either by traditional trigonometric or hyperbolic polynomials in each of their variables. Based on homogeneous coordinates and central projection, we also propose algorithms for the control point and weight based exact description of the zeroth order (partial) derivative of the rational counterpart of these integral curves and surfaces. The core of the proposed modeling methods relies on basis transformation matrices with entries that can be efficiently obtained by order elevation.

Imre Juhász, **Ágoston Róth**, **2014**. *A scheme for interpolation with trigonometric spline curves*, **Journal of Computational and Applied Mathematics**, **263(C)**:246–261, **IF**** _{2014}** ≅ 1.266,

**AIS****≅ 0.630. [@publisher]**

_{2014}**Abstract**

We present a method for the interpolation of a given sequence of data points with

*Cⁿ*continuous trigonometric spline curves of order

*n*+ 1 (

*n*≥ 1) that are produced by blending elliptical arcs. Ready to use explicit formulae for the control points of the interpolating arcs are also provided. Each interpolating arc depends on a global parameter

*α*∈ (0,

*π*) that can be used for global shape modification. Associating non-negative weights with data points, rational trigonometric interpolating spline curves can be obtained, where weights can be used for local shape modification. The proposed interpolation scheme is a generalization of the Overhauser spline, and it includes a

*Cⁿ*Bézier spline interpolation method as the limiting case

*α*→ 0.

Imre Juhász, **Ágoston Róth**, **2013**. *A class of generalized B-spline curves*, **Computer Aided Geometric Design**, **30**(1):85–115, **IF**** _{2013}** ≅ 0.894,

**AIS****≅ 0.848. [@publisher]**

_{2013}**Abstract**

The classical B-spline functions of order

*k*≥ 2 are recursively defined as a special combination of two consecutive B-spline functions of order

*k*- 1. At each step, this recursive definition is based, in general, on different reparametrizations of the strictly increasing identity (linear core) function

*φ*(

*u*) =

*u*.

This paper generalizes the concept of the classical normalized B-spline functions by considering monotone increasing continuously differentiable nonlinear core functions instead of the classical linear one. These nonlinear core functions are not only nteresting from a theoretical perspective, but they also provide a large variety of shapes.

We show that many advantageous properties (like the non-negativity, local support, the partition of unity, the effect of multiple knot values, the special case of Bernstein polynomials and endpoint interpolation conditions) of the classical normalized B-spline functions remain also valid for this generalized case, moreover we also provide characterization theorems for not so obvious (geometrical) properties like the first and higher order continuity of the generalized normalized B-spline functions,

*C*¹

*continuous envelope contact property of the family of curves obtained by altering a selected knot value between its neighboring knots. Characterization theorems are illustrated by test examples.*

We also outline new research directions by ending our paper with a list of open problems and conjectures underpinned by numerous successful numerical tests.

**Ágoston Róth**, Imre Juhász, **2011**. *Constrained surface interpolation by means of a genetic algorithm*, **Computer-Aided Design**, **43**(9):1194–1210, **IF**** _{2011}** ≅ 1.234,

**AIS****≅ 0.603. [@publisher]**

_{2011}**Abstract**

We propose an evolutionary technique (a genetic algorithm) to solve heavily constrained optimization problems defined on interpolating tensor product surfaces by adjusting the parameter values associated with the data points to be interpolated. Throughout our study we assume that the functional, which operates on these types of interpolating surfaces, is described by a surface integral and fulfills the following conditions: it is not necessarily a smooth functional (i.e., it may have vanishing gradient vectors), it is bounded (i.e., the optimization algorithm can converge in a finite number of steps), it is invariant under parametrization, rigid body transformation and uniform scaling (i.e., different surface parametrization at different scales should generate the same optimized shape). We have successfully tested the proposed algorithm for functionals that involve: minimal surface area, minimal Willmore, umbilic deviation and total curvature energies, minimal third-order scale invariant weighted Mehlum–Tarrou energies, and isoperimetric like problems. In general, our algorithm can be used in the case of any kind of not necessarily smooth surface fairing functionals. The run-time and memory complexities of the suggested algorithm are reasonable. Moreover, the algorithm is independent of the type of tensor product surface.

Imre Juhász, **Ágoston Róth**, **2010**. *Closed rational trigonometric curves and surfaces*, **Journal of Computational and Applied Mathematics**, **234**(8):2390–2404, **IF**** _{2010}** ≅ 1.030,

**AIS****≅ 0.571. [@publisher]**

_{2010}**Abstract**

In CAGD curves are described mostly by means of the combination of control points and basis functions. If we associate weights with basis functions and normalize them by their weighted sum, we obtain another set of basis functions that we call quotient bases. We show some common characteristics of curves defined by such quotient basis functions. Following this approach we specify the rational counterpart of the recently introduced cyclic basis, and provide a ready to use tool for control point based exact description of a class of closed rational trigonometric curves and surfaces. We also present the exact control point based description of some famous curves (Lemniscate of Bernoulli, Zhukovsky airfoil profile) and surfaces (Dupin cyclide and the smooth transition between the Boy surface and the Roman surface of Steiner) to illustrate the usefulness of the proposed tool.

**Ágoston Róth**, Imre Juhász, **2010**. *Control point based exact description of a class of closed curves and surfaces*, **Computer Aided Geometric Design**, **27**(2):179–201, **IF**** _{2010}** ≅ 0.859,

**AIS****≅ 0.651. [@publisher]**

_{2010}**Abstract**

Based on cyclic curves/surfaces introduced in Róth et al. (2009), we specify control point configurations that result an exact description of those closed curves and surfaces the coordinate functions of which are (separable) trigonometric polynomials of finite degree. This class of curves/surfaces comprises several famous closed curves like ellipses, epi- and hypocycloids, Lissajous curves, torus knots, foliums; and surfaces such as sphere, torus and other surfaces of revolution, and even special surfaces like the non-orientable Roman surface of Steiner. Moreover, we show that higher order (mixed partial) derivatives of cyclic curves/surfaces are also cyclic curves/surfaces, and we describe the connection between the cyclic and Fourier bases of the vector space of trigonometric polynomials of finite degree.

**Ágoston Róth**, Imre Juhász, Josef Schicho, Miklós Hoffmann, **2009**. *A cyclic basis for closed curve and surface modeling*, **Computer Aided Geometric Design**, **26**(5):528–546, **IF**** _{2009}** ≅ 1.330,

**AIS****≅ 0.789. [@publisher | technical report]**

_{2009}**Abstract**

We define a cyclic basis for the vector space of truncated Fourier series. The basis has several nice properties, such as positivity, summing to 1, that are often required in computer aided design, and that are used by designers in order to control curves by manipulating control points. Our curves have cyclic symmetry, i.e. the control points can be cyclically arranged and the curve does not change when the control points are cyclically permuted. We provide an explicit formula for the elevation of the degree from

*n*to

*n*+

*r*(

*r*≥ 1), and prove that the control polygon of the degree elevated curve converges to the curve itself if

*r*tends to infinity. Variation diminishing property of the curve is also verified. The proposed basis functions are suitable for the description of closed curves and surfaces with

*C*

^{∞}continuity at all of their points.

Imre Juhász, **Ágoston Róth**, **2008**. *Bézier surfaces with linear isoparametric lines*, **Computer Aided Geometric Design**, **25**(6):385–396, **IF**** _{2008 }**≅ 1.512,

**AIS****≅ 0.895. [@publisher]**

_{2008}**Abstract**

We provide control point based necessary and sufficient conditions for (

*n*,

*m*) Bézier surfaces to have linear isoparametric lines.

Alexandru Kristály, Gheorghe Moroşanu, **Ágoston Róth**, **2008**. *Optimal placement of a deposit between markets: a Riemann-Finsler geometrical approach*, **Journal of Optimization Theory and Applications**, **139**(2):263–276, **IF**** _{2008 }**≅ 0.860,

**AIS****≅ 0.723. [@publisher]**

_{2008}**Abstract**

By using the Riemann-Finsler geometry, we study the existence and location of the optimal points for a general cost function involving Finsler distances. Our minimization problem provides a model for the placement of a deposit within a domain with several markets such that the total transportation cost is minimal. Several concrete examples are studied either by precise mathematical tools or by evolutionary (computer assisted) techniques.

## Doctoral thesis

**Title:** Recent problems in Evolutionary Optimization and Computer Aided Geometric Design

**Keywords:**

- evolutionary techniques;
- general cost functions;
- Riemann-Finsler manifolds;
- optimal deposit placements;
- elliptic equations;
- ruled and developable Bézier surfaces;
- trigonometric basis function;
- closed cyclic curves and surfaces;
- cyclic variation diminishing;
- order/degree elevation;
- basis transformation;
- control point based exact description of integral closed trigonometric curves and surfaces.

**Scientific supervisor: **Prof. dr. D. Dumitrescu (Babeș-Bolyai University)

**Date of public defending: **June 5, 2009