Miklós Hoffmann

• PhD, DSc
• Head of Department at Eszterhazy Karoly Catholic University

In his essay Science as a vocation, Max Weber has seen the essence of scientific activities in specialisation and enthusiasm. His arguments, together with works of Michael Polányi and others, are studied and compared with recent results and expectations of automatised, artificial intelligence driven scientific discovery in this paper. Our aim is to...

Providing additional information for parts or items usually means to enclose it next to the object or affix it on the component when that is possible. However, another solution is available by gaining the benefits of an additive manufacturing technology, 3D printing. This technology makes it possible to embed the additional information onto the sur...

Surface reconstruction from scattered data using Kohonen neural network is presented in this paper. The network produces a topologically predefined grid from the unordered data which can be applied as a rough approximation of the input set or as a base surface for further process. The quality and computing time of the approximation can be controlle...

In this paper, I will argue that with the emergence of digital virtual worlds (in video games, animation movies, etc.) by the animation industry, we need to rethink the role and authority of mathematics, also from an ontological point of view. First I will demonstrate that the application of mathematics to the creation and description of the digita...

A cube is one of the most fundamental shapes we can draw and can observe from a drawing. The two visualization methods most commonly applied in mathematics textbooks and education are the axonometric and the perspective representations. However, what we see in the drawing is really a cube or only a general cuboid (i.e., a polyhedron with different...

Spatial abilities, which are sources of our capacity to understand visual and spatial relations among objects, as well as the abilities to generate, retain, retrieve, and transform well-structured visual information are important in several scientific fields and workplaces. Various tests have already been prepared to measure these abilities, includ...

Several different methods are used to test the spatial abilities or visual skills of people. One of them is the Mental Cutting Test (MCT), the exercises of which offer a 2D projection of a 3D shape and a 2D plane, and testees should determine the shape of their intersection. MCT exercises need various 2D and 3D assets that should be developed befor...

In the last decade, various mobile applications have been developed to improve and measure spatial abilities using different spatial tests and tasks through augmented reality (AR), Virtual Reality (VR), or embedded 3D viewers. The Mental Cutting Test (MCT) is one of the most well-known and popular tests for this purpose, but it needs a vast number...

Mental Cutting Test is a widely used format to develop or measure the spatial skills of people in various situations, having different purposes. An exercise consists of a 2D projection of a 3D shape and an intersection plane, denoted by a frame. The task is to choose the shape of their intersection from the set of five shapes. In this paper, we inv...

Developing or measuring the spatial skills of people is still an interesting research topic nowadays. Well-known assignments such as the exercises of Mental Cutting Test are still popular tools of performing measurements or validating the knowledge of our students. However, we found that researchers, instructors and students are not entirely suppor...

While considering a mirror and light rays coming either from a point source or from infinity, the reflected light rays may have an envelope, called a caustic curve. In this paper, we study developable surfaces as mirrors. These caustic surfaces, described in a closed form, are also developable surfaces of the same type as the original mirror surfac...

QR code is a widely used format to encode information through images that can be easily decoded using a smartphone. These devices play a significant role in most people's everyday lives, making the encoded information widely accessible. However, decoding the QR code becomes challenging when significant deformations occur in the label. An easy and q...

The new National Core Curriculum came into force in September 2020 in Hungarian schools. The COVID-19 pandemic has had a deep impact on the final stages of its development. In this paper we have selected two areas for analysis: the fundamental principles of mathematics curriculum and the matriculation exam in mathematics. We propose improvements in...

The traditional way of presenting mathematical knowledge is logical deduction, which implies a monolithic structure with topics in a strict hierarchical relationship. Despite many recent developments and methodical inventions in mathematics education, many curricula are still close in spirit to this hierarchical structure. However, this organisatio...

A matematika mint performatív észjáték Éppen húsz éve annak, hogy megjelent Paul Ernest Social Constructivism as a Philosophy of Mathematics című könyve (Ernest 1998), ami újra felszította a matematika alapvető természetéről, létrejöttéről és alakulásáról szóló, matematikusokat és filozófusokat egyaránt megmozgató vitát, új lendületet adva az Émi...

In the field of aesthetic design, log-aesthetic curves have a significant role to meet the high industrial requirements. In this paper, we propose a new interactive $G^1$ Hermite interpolation method based on the algorithm of Yoshida et al. with a minor boundary condition. In this novel approach, we compute an extended log-aesthetic curve segment t...

Minkowski Pythagorean hodograph curves are widely studied in computer-aided geometric design, and several methods exist which construct Minkowski Pythagorean hodograph (MPH) curves by interpolating Hermite data in the R^{2,1} Minkowski space. Extending the class of MPH curves, a new class of Rational Envelope (RE) curve has been introduced. These a...

We started to develop an Android application which implements tasks in Augmented Reality, analogous to the classic Mental Cutting Test. Our goal is to offer practicing exercises to our users with the support of gamified elements and AR technology. Thus, users can switch their 2D scenarios into the AR, rotate and scale them to determine the current...

Information visualization is the science and art of visualizing information that flows around us in many disciplines, intending to make the data more digestible and understandable for non-expert users. The objective of this paper is to introduce a new and somewhat surprising application of the Poly-Universe system, as an educational tool of underst...

Recently there has been a growing interest in the topic of skinning of circles and spheres, since modeling based on these objects has been found useful in areas such as medical applications and character animation. Among others, an efficient method was presented by Kunkli and Hoffmann [1], whilst Bastl et al. also provided an effective skinning alg...

The isoptic surface of a three-dimensional shape is recently defined by Csima and Szirmai (2016) as the generalization of the well-known notion of isoptics of curves. In that paper, an algorithm has also been presented to determine isoptic surfaces of convex polyhedra. However, the computation of isoptic surfaces by that algorithm requires extendin...

If we consider a curve as a mirror, then parallel light rays reflected by this mirror curve form a family of lines, which generally has an envelope. This envelope is called caustic curve of the given mirror curve. The aim of this paper is to describe the caustic curve of a free-form curve and study its alteration by the modification of a control po...

Color vision deficiency represents an inability to perceive differences between certain colors that can be distinguished in the case of regular color vision. This article proposes a new daltonization method for re-coloring image segments perceived as confusingly colored by color deficient observers and, thus, to improve their perceived image qualit...

Computation of the blending surface of two given spheres is discussed in this paper. The blending surface (or skin), although not uniquely defined in the literature, is normally required to touch the given spheres in plane curves (i.e., in circles). The main advantage of the presented method over the existing ones is the minimization of unwanted di...

This paper aims at creating the right pedagogical attitudes in term of teaching a new technology, Radio Frequency Identification (RFID) by evaluating the social acceptance of this new method. Survey of future teachers, students of teacher master studies and students from informatics oriented secondary schools were surveyed comparing their attitudes...

A generalization of a recently developed trigonometric Bézier curve is presented in this paper. The set of original basis functions are generalized also for non-trigonometric functions, and essential properties, such as linear independence, nonnegativity and partition of unity are proved. The new curve—contrary to the original one—can be defined by...

Beside classical point based surface design, sphere based creation of characters and other surfaces has been introduced by some of the recently developed modeling tools in Computer Graphics. ZSpheres® by Pixologic, or Spore™ by Electronic Arts are just two prominent examples of these softwares. In this paper we introduce a new sphere based modeling...

Having examined the obstacles to the spread of radio frequency identification technology, we have found that the most significant was the low level of social acceptance, due to consumers' fear of privacy invasion. People worry—in some cases, with good reason—that this technology might detect their whereabouts and movements, as well as observe their...

Given a planar curve s(t), the locus of those points from which the curve can be seen under a fixed angle is called isoptic curve of s(t). Isoptics are well-known and widely studied, especially for some classical curves such as e.g. conics (Loria, 1911). They can theoretically be computed for a large class of parametric curves by the help of their...

For the optimal design of composites, with the aid of advancement of interdisciplinary and data analysis tools, a series of criteria including mechanical, electrical, chemical, cost, life cycle assessment and environmental aspects are now able to be simultaneously considered. As one of the most efficient approach, the MCDM applications can provide...

For the optimal design of composites, with the aid of advancement of interdisciplinary and data analysis tools, a series of criteria including mechanical, electrical, chemical, cost, life cycle assessment and environmental aspects are now able to be simultaneously considered. As one of the most efficient approach, the MCDM applications can provide...

In most of the real-world optimal design problems of engineering and business processes, in order to improve the functionality, the operating parameters need to be accurately tuned with the aid of the multiobjective optimization algorithms for which many conflicting objectives have to be traded off in selecting the preferred solution(s). For solvin...

In this article the criteria of mechanical behavior of the woven textile composites during the draping and the further involved simulations and analysis are included in the process of the optimal design and decision making. For this purpose, the advanced software architecture of Grapheur for interactive optimization and MCDM is utilized. In this so...

The notion of Gergonne point was generalized in several ways during the last decades. Given a triangle V 1 V 2 V 3 , a point I, and three arbitrary directions q i , we find a distance x=IQ 1 =IQ 2 =IQ 3 along these directions for which the three cevians V i Q i are concurrent. If I is the incenter, q i are the direction of the altitudes, and x is t...

Skinning of an ordered set of discrete circles is discussed in this paper. By skinning we mean the geometric construction of two G1 continuous curves touching each of the circles at a point, separately. After precisely defining the admissible configuration of initial circles and the desired geometric properties of the skin, we construct the touchin...

New types of quadratic and cubic trigonometrial polynomial curves have been introduced in [2] and [3]. These trigonometric curves have a global shape parameter λ. In this paper the geometric effect of this shape parameter on the curves is discussed. We prove that this effect is linear. Moreover we show that the quadratic curve can interpolate the c...

We prove that for a given axis the centers of all central collineations which transform a given proper conic c into a circle, lie on one conic cc confocal to the original one. The conics c and cc intersect into real points and their common diametral chord is conjugate to the direction of the given axis. Furthermore, for a given center S the axes of...

We define a cyclic basis for the vectorspace 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...

In this paper a model based approach is presented for generating 3D face models from two stereo images. The proposed method works with a small number of feature points of the face. The automatically registrated point pairs are used to reconstruct their 3D correspondence, then a predefined general face model is adjusted to the reconstructed 3D point...

The purpose of this paper is to introduce an interactive surface interpo- lation method by spline surfaces, which is a generalization of the method presented in (2). The technique is based on linear blending and works for a large class of surfaces including bicubic Bézier, B-spline, NURBS surfaces and the recently developed trigonometric surfaces a...

The quartic curve of Han [X. Han, Piecewise quartic polynomial curves with shape parameter, Journal of Computational and Applied Mathematics 195 (2006) 34–45] can be considered as the generalization of the cubic B-spline curve incorporating shape parameters into the polynomial basis functions. We show that this curve can be considered as the linear...

In parametric curve interpolation there is given a sequence of data points and corresponding parameter values (nodes), and we want to find a parametric curve that passes through data points at the associated parameter values. We consider those interpolating curves that are described by the combination of control points and blending functions. We st...

FB-spline curves are the unification of recently developed trigonometric CB-spline and hyperbolic HB-spline curves, including the classical B-spline curves. These generalized curves overcome some restrictions of B-spline curves and allow to design some important curves like helix, cycloids or catenary. Their properties, however, have been studied o...

Interpolation of a sequence of points by spline curves generally requires the solution of a large system of equations. In this paper we provide a method which requires only local computation instead of a global system of equations and works for a large class of curves. This is a generalization of a method which previously developed for B-spline, NU...

In this paper we further extend the generalization of the concept of Gergonne point for circles concentric to the inscribed circ le. Given a triangle V1V2V3, a point I and three arbitrary directions q1, q2, q3 from I, we find a dis- tance x = IQ1 = IQ2 = IQ3 along these directions, for which the three cevians ViQi are concurrent. Types and number o...

Spatial ability of first year university students is measured and evaluated in this paper. We used standard Mental Cutting Test (MCT), where a body is given by perspective view and correct cross section has to be chosen. While gender differences in MCT are reported by several papers including our earlier results, much less known are the reasons of...

Extensions of the classical B-spline curves by shape parameters are discussed in this paper. Using a method of linear blending, first we extend the abilities of GB-splines, originally defined by uniform shape parameter, to multiple shape parameters. Then two methods, having different types of continuity, are examined in terms of their curvature beh...

GB-spline curves can be considered as the generalization of B-spline curve incorporating a shape parameter into the polynomial basis functions. The geometric effect of the alteration of the shape parameter is discussed in this paper, including constrained shape control of the curve.

C-Bezier and C-B-spline curves, as the trigonometric extensions of cubic uniform spline curves are well-known in geometric modeling. These curves depend on a shape parameter fi in a way that fi ! 0 yields the cubic polynomial curves. The geometric efiect of the alteration of this parameter is discussed in this paper by the help of relative parametr...

Abstract Spatial visualization,of engineering,students,is of greatest importance,in terms of their professional achievement, thus evaluation of this skill is essen- tial. Mental Cutting Test (MCT) is one of the most,widely,used evaluation method,for spatial abilities. In this study,we present,an analysis of MCT results of first-year engineering stu...

B-spline surfaces are piecewisely defined surfaces where the section points of the domain of definition are called knots. In [2] the authors proved some theorems in terms of knot modification of B-spline curves. Here we generalize these results for one- and two-parameter family of surfaces. An additional result concerning a higher order contact of...

IntroductionTheoretical ResultsShape ControlSummaryReferences

B-spline curves are defined in a piecewise way over a closed interval, where the interval section points are called knots. In some recent publications geometrical properties of the modification of one knot value are discussed. The aim of this paper is to describe an effect of the symmetric modification of two knots.

In this paper we give a new, synthetic condition under which a central axonometric mapping is a central projection. This condition is applied for controlling the change of unit points of a central axonometric reference system. Correction of a central axonometric system to be of central projection type is also discussed by the help of our condition.

When a knot of a B-spline curve is modified its points move along curves called paths. We extend these paths and prove that points of these extended paths tend to control points. We also show that the family of paths possess an envelope which is a B-spline curve.

The effect of the modification of knot values on the shape of B-spline curves is examined in this paper. The modification of a knot of a B-spline curve of order k generates a one-parameter family of curves.This family has an envelope which is also a B-spline curve with the same control polygon and of order k−1. Applying this theoretical result, thr...

Given a polygon and one of its inner points P, the orthogonal projections of P onto the sides of the polygon are called pedal points of P. Here we prove different results concerning configurations by attaching different types of polygons to the segments of the sides defined by the pedals. These theorems can be considered as the generalizations of B...

We introduce planar grouping, a technique where planar relationship information is gathered for a set of rigid points. This information is used to accelerate affine transformations and clipping. The planar grouping technique is an optimization problem, implemented in a best-first greedy search. We present two error metrics, one simple and fast, one...

The modification of a knot of a B-spline curve of order k generates a family of B-spline curves. We show that an envelope of this family is a B-spline curve defined by the same control polygon, and its order is k − m ,w herem is the multiplicity of the modified knot. Moreover, their arbitrary order derivatives differ only in a multiplier.  2003 El...

This paper is devoted to the problem of creating and updating a B-spline surface based on a set of scattered data, i.e. totally unorganized points locally distributed over a 3D area. The crucial point of the problem is the parametrization of the given points, where earlier approaches introduced a base surface and the spatial points were mapped onto...

In a recent publication we described the effect of knot modifications of B-spline curves. The aim of this paper is the generalization of these results for surfaces. Altering one or two knot values of a B-spline surface, the paths of the points of the surface are discussed first, among which special ruled surfaces can be found. Then we prove that th...

The effect of knot modifications on the shape of B-spline and NURBS curves is discussed in this paper. Theoretical results include the description of the path of curve points, obtained by the modification of a knot value, and the examination of the one-parameter family of curves. It is shown that this family has an envelope which is a lower order B...

A multistep interpolation method is presented in this paper by which one can compute a planar functional B-spline curve from its gradient function with special endconditions. The method is applied in turbine blade section curve design where the curve consists of two interpolating B-spline arcs connecting two predefined circular arcs.

Presents shape control methods for cubic B-spline and NURBS curves by the modification of their knot values and by the simultaneous modification of weights and knots. Theoretical aspects of knot modification are also discussed, concerning the paths of points on a curve and the existence of an envelope for the family of curves resulting from a knot...

This paper is devoted to the geometrical examination of a family of B-spline curves resulted by the modifiaction of one of their knot values. These curves form a surface, the other parameter lines of which are the paths of the points of the original curve at a fixed parameter value. The first and second derivatives of these curves are examined yiel...

This paper is devoted to the shape control of B-spline curves achieved by the modification of one of its knot values. At first those curves are described along which the points of a B-spline curve move under the modification of a knot value. Then we show that the one-parameter family of k th order B-spline curves obtained by modifying a knot value...

This paper is devoted to the shape control of B-spline curves achieved by the modification of one of its knot values. At first those curves are described along which the points of a B-spline curve move under the modification of a knot value. Then we show that the one-parameter family of kth order B-spline curves obtained by modifying a knot value h...

General and special ruled surfaces play a central role in several CAD applications. The purpose of this paper is to develop a method with the help of which one can construct free-form ruled surface for given lines as rulings. The Kohonen neural network and the Plücker coordinates of projective geometry help us to construct a standard free-form surf...

The aim of this paper is to improve the method of modelling scattered data by free-from surfaces presented in [5]. In that method a neural network was used for ordering the data and forming a quadrilateral control grid from the scattered points, hence the standard free-form methods like Bézier-surface or NURBS could be applied to approximate or int...

Kohonen neural networks, also known as self-organizing maps are applied frequently for the handling of the problems of unordered data structures. In this paper a modified algorithm is presented which is more suitable for our suface reconstruction problem, because with the help of the new, continuously diminishing neighborhood the surface will be sm...

The handling of scattered spatial points is an important question in computer graphics and there are several methods to construct surfaces from these type of data. The aim of our paper is to present a new method which produces standard free-form surfaces from the scattered data. Earlier methods normally construct a triangular control grid from the...

One of the important questions of central axonometry is to give a condition under which a central axonometric mapping is a central projection. The aim of this paper is to prove that the well-known Stiefel's condition can be considered as a limiting case of a recent theorem proved by Sza o, Stachel and Vogel.

Constrained modification of B-spline surfaces is essential in geometric design. Modification tools by purely knot alteration are presented in this paper: moving an isoparametric line to a specified location and dragging a point of the surface to a predefined point. Both methods yield smoother change than the well-known control point repositioning,...

The well-known triangle center, the Gergonne point is generalized for circles concentric to the inscribed circle in this paper. Given a triangle V 1 V 2 V 3 , a point I and three arbitrary directions q 1 , q 2 , q 2 , we consider a circle with radius r and center I, for which finding r = IQ 1 = IQ 2 = IQ 3 along the given directions, the three cevi...

B-spline görbék/felületek pontjai által, az alakzat két csomóértékének szimmetrikus változtatásakor leírt pályagörbéket vizsgáltuk, és olyan alakmódosítási eljárást adtunk, amivel a felület adott pontját/paramétervonalát előre megadott helyre mozgathatjuk a csomóértékek változtatásával. A C-Bézier, C-B-spline és F-B-spline görbék pályagörbéinek geo...