## Vol. 24 (2016), No. 1, February 2017

### Geometric loci – generalization of Monge’s circle and possibilities of current software [CZ]

 Author J. Blažek, P. Pech Abstract Searching for loci belongs to classical topics of mathematics school curricula all over the world. In the text investigation of loci by GeoGebra 5.0 using the command LocusEquation, the tool which was recently implemented into GeoGebra and which brings a new approach in searching for loci of points, is given. The use of the command LocusEquation is demonstrated on one concrete example of the Monge's circle, where this command fails. The probable reason of the failure of the software is shown. With the use of computer algebra system CoCoA the locus equation of the searched locus is derived. Further a synthetical approach of searching for the locus is performed. At the end the Monge's circle is generalized to the case when the tangents to a conic make an arbitrary angle. Article Download

### Intersection of curves – facts, computations, applications in blowup

 Author P. Chalmovianský Abstract We deal with application of intersection of curves in resolution of a singularity. The basic tool is a product of curves in homology group used on a certain set of generators producing as a result multiplicity of the infinitely near points of a singularity in the inverse of their proximity matrix. We explain the interconnection of several concepts related to a isolated singularity such as blowup, proximity relations, intersections and homology group of the blowup illustrated on an example. Article Download

### Usage of responsive systems for solving combinatorics problems at elementary school [CZ]

 Author J. Kaňková, J. Kopecký Abstract This paper deals with the graphical calculator Desmos and with the use of its module Activities. This tool combines advantages of computer algebra system and online student response system into an environment where it is possible to create interactive exercises. We perform an analysis of pupils' solutions to selected problems of probability theory which were completed through this application in math lessons. In the end we refer some teachers’ difficulties in using Desmos. Article Download

### Circle as rational curve [SK]

 Author A. Kolcun Abstract Author in the paper analyzes exact rational polynomial representation of circle. To do that, he uses Bezier representation. Analysis is limited for the curves to rank four – reason for this limitation will result in the end of this paper. Article Download

### Mathematical problems inspired by bridge over the Vltava [CZ]

 Author H. Mahnelová, M. Šuková Abstract It seems that the mathematics in primary education and non-specialized secondary education is lacking connection with the engineering practice and maybe this one of the reasons why there is the decline of interest among students in related fields. Paper provides sample of several exercises based on real life documentation for bridge construction in the Czech Republic. The exercises introduce professional sketches with various projections of building and simple professional terminology. When solving, students foster their spatial perception, apply knowledge from different areas such as right-angled triangle, quadratic equations, analytic geometry of parabola, construction of angle between two planes, similar triangles, arithmetic sequences. Authors got needed documentation and some useful information from online building magazine, other information was provided by project company. Article Download

### Constructions of quadrilateral meshes: a comparative study

 Author P. Surynková Abstract Polygonal meshes represent important geometric structures with a large number of applications. The study of polygonal meshes is motivated by many processing tasks in automotive/aerospace industry, engineering, architecture, engineering, construction industry, and industrial design. Much of literature on polygonal representations focuses on quadrilateral meshes which are composed of quadrilaterals as they possess several advantages compared to triangle meshes. In this short contribution we present a comparative study of known methods for constructions of quadrangulations of various classes and for different purposes. We suggest a new method for computing all unique quadrilateral meshes of a certain class based on sequential construction. Article Download

### Theory and applications of curves [CZ]

 Author Z. Šír Abstract Curves as mathematical objects are remarkably diverse and inspiring and yet time quite simple. This is the reason why they are suitable topic for education at various levels. At the same time they found great use in applications. In this paper the author focuses on several specific aspects of curves and different perspectives and ways to approach them. The author considers options of connection and mutual elucidation between theoretical and practical topics the most interesting. Because of that the paper deals for example with reparametrization, curvature, different frames along curves, polynomial and rational curves. Article Download

### Notes on properties and applications of Minkowski point set operations

 Author D. Velichová Abstract This paper provides information about some basic properties of Minkowski point set operations - sum, difference and product of two point sets, namely basic geometric objects, i.e. points, lines and curves. Resulting geometric figures are determined analytically by vector maps and their intrinsic geometric properties are derived by means of derivatives of vector functions representing operands of respective Minkowski operations. Interesting examples of generated complex geometric forms and structures are included, with illustrations of their applications. Article Download

### Classical problems from graph theory at schools and in the applications [CZ]

 Author Š. Voráčová Abstract Classical exercises of the planar graphs provide a large number of problems in simple language, many of which can be solved on the first stage of primary education. Due to this potential they have their unquestionable place at recreational mathematics and mathematical olympiads. Even though many of pedagogical experiments have shown successfulness of including topics from the graph theory in primary and secondary education, those topics are represented in textbooks only marginally. In doing so mathematical essence of problems of the natural world can be found in graph theory casually and naturally. Heuristically focused education inspired by the graph theory can be in authors’ opinion the most accessible way to introduce principals of the mathematical modelling. Article Download