Znanstveno-stručni časopis
Hrvatskog društva za geometriju i grafiku

Scientific and Professional Journal
of the Croatian Society for Geometry and Graphics

János Pallagi, Benedek Schultz, Jenö Szirmai (jpallagi@math.bme.hu, schultz.benedek@gmail.com, szirmai@math.bme.hu)

Equidistant Surfaces in H2×R Space

After having investigated the equidistant surfaces ("perpendicular bisectors" of two points) in S2×R space (see[6]) we consider the analogous problem in H2×R space from among the eight Thurston geometries. In [10] the third author has determined the geodesic curves, geodesic balls of H2×R space and has computed their volume, has defined the notion of the geodesic ball packing and its density. Moreover, he has developed a procedure to determine the dendity of the geodesic ball packing for deneralized Coxeter space groups of H2×R and he has applied this algorithm to them.
In this paper we introduce the notion of the equidistant surface to two points in H2×R geometry, determine its equation and we shall visualize it in some cases. The pictures have been made by the Wolfram Mathematica software

Key words: non-Euclidean geometries, geodesic curve, geodesic sphere, equidistant surface in H2×R geometry

Article in PDF.

Miljenko Lapaine (mlapaine@geof.hr)

Mollweide map Projection

Karl Brandan Mollweide (1774-1825) was German mathematician and astronomer. The formulas known after him as Mollweide's formulas are shown in the paper, as well as the proof "without words". Then, the Mollweide map projection is defined and formulas derived in different ways to show several possibilities that lead to the same result. A generalization of Mollweide projection is derived enabling to obtain a pseudocylindrical equal-area projection having the overall shape of an ellipse with any prescribed ratio of its semiaxes. The inverse equations of Mollweide projection has been derived, as well.
The most important part in research of any map projection is distortion distribution. That means that the paper continues with the formulas and images enabling us to get some filling about the linear and angular distortion of the Mollweide projection.
Finally, several applications of Mollweide projections are represented, with the International Cartographic Association logo as an example of one of its successful applications.

Key words: Mollweide, Mollweide's formula, Mollweide map projection

Article in PDF.

Márta Szilvási-Nagy, Szilvia Béla (szilvasi@math.bme.hu, belus@math.bme.hu)

B-spline Patches Fitting on Surfaces and Triangular Meshes

In this paper a technique for the construction of quartic polynomial B-spline patches fitting on analytical surfaces and triangle meshes is presented. The input data are curvature values and principal directions at a given surface point which can be computed directly, if the surface is represented by a vector function.
In the case of discrete surface representation, i.e. on a triangle mesh the required input data are computed from a circular neighborhood of a specified triangle facet. Such a surface patch may replace a well defined region of the mesh, and can be used e.g. in re-triangulation, mesh-simplification and rendering algorithms.

Key words: B-spline surface, local surface approximation, principal curvatures, triangle mesh

Article in PDF.


Norman John Wildberger (n.wildberger@unsw.edu.au)

Universal Hyperbolic Geometry III: First Steps in Projective Triangle Geometry

We initiate a triangle geometry in the projective metrical setting, based on the purely algebraic approach of universal geometry, and yielding in particular a new form of hyperbolic triangle geometry. There are three main strands: the Orthocenter, Incenter and Circumcenter hierarchies, with the last two dual. Formulas using ortholinear coordinates are a main objective. Prominent are five particular points, the b, z, x, h and s points, all lying on the Orthoaxis A. A rich kaleidoscopic aspect colours the subject.

Key words: universal hyperbolic geometry, triangle geometry, projective geometry, bilinear form, ortholinear coordinates, incenter, circumcenter, orthoaxis

Article in PDF.


Günter Wallner, Franz Gruber (guenter.wallner@uni-ak.ac.at, franz.gruber@uni-ak.ac.at)

Interactive Modeling and Subdivision of Flexible Equilateral Triangular Mechanisms

Based on the requests from architects, we developed a system which allows to interactively design and subdivide flexible triangular surfaces. Due to economical reasons the number of different types of building elements should be small. For that reason we only use equilateral base triangles of unique size with the possibility of subdivision. To allow to interactively move vertices and to ensure constant edge length we use force-directed methods instead of inverse kinematics. This paper describes the data structure, the algorithm and the influence of subdivision on the kinematic flexibility of the mesh.

Key words: subdivision, uniform 1-to-4 split, flexibile mechanism, force directed algorithm

Article in PDF.

Kristian Sabo, Sanja Scitovski (ksabo@mathos.hr, sanja@mathos.hr)

Location of Objects in a Plane

In the paper we consider a direct and the inverse location problem in the plane. Thereby we use different distance-like functions with appropriate illustrations. Several examples from various areas of applications are given.

Key words: data clustering, location problem, k-means, k-median, optimization

Article in PDF.

Ana Sliepčević, Ivana Božić (anas@grad.hr, ivana.bozic@tvz.hr)

Perspective Collineation and Osculating Circle of Conic in PE-plane and I-plane

All perspective collineations in a real affine plane are classified according to a constant cross-ratio and the position of the center and axis. A special attention will be given to the conditions which basic elements of perspective collineation have to fulfill in order to obtain the touch or osculation or hyperosculation of two conics. On the affine models of an isotropic and pseudo - Euclidean plane the osculating circle of a conic is constructed by using perspective collineations.

Key words: perspective collineation, homology, elation, constant cross - ratio, conic, osculating circle

Article in PDF.

Martina Triplat Horvat, Miljenko Lapaine, Dražen Tutić (mthorvat@geof.hr, mlapaine@geof.hr, dtutic@geof.hr)

Application of Bošković's Geometric Adjustment
Method on Five Meridian Degrees

In this paper, the first method of adjustment, proposed by Josip Ruder Bošković, is described in detail, on the example on five meridian degrees. Bošković sets three conditions on the data of the lengths of the meridian degrees to calculate corrections that would fix all degrees in order to get a better estimate of true values. The conditions that have to be satisfied are explained by geometric method which Bošković described in all his studies. For the purpose of this paper, in the process of computing these five meridian degrees, data from Bošković original book have been used.
Geometric solution, described by Bošković himself, is not easy to understand at fist, as this is noted by other authors who have studied Boškovć's method as well. Hence, geometric description of the Bošković's method is shown in analytical form as well.

Key words: Josip Ruđer Bošković, geometric adjustment method

Article in PDF.

Mirela Katić Žlepalo, Boris Uremović (mkatić@tvz.hr, boris.uremovic@tvz.hr)

Application of Elevational Projection in Defining
Scope of Construction Pit Excavation

One of the frequent problems that the civil engineering experts have to deal with during planning the process of building is to define the scope of construction pit excavation. The most common method to solve this problem is to use the so-called elevational projection. In this article we show the phases of defining the scope of construction pit excavation together with one example from the civil engineering practice and how to use some tasks from basic theory of elevational projection to solve such problems.

Key words: elevational projection, scope of construction pit excavation
Article in PDF.