Znanstvenostručni
časopis Hrvatskog društva za geometriju i grafiku Scientific and Professional Journal 
 Helena Halas, Ema Jurkin (hhalas@grad.hr, ejurkin@rgn.hr)
3rd Class Circular Curves in QuasiHyperbolic Plane Obtained by Projective MappingThe metric in the quasihyperbolic plane is induced by
an absolute figure F_{QH} = {F, f_{1}, f_{2}}, consisting of two real
lines f_{1} and f_{2} incident with the real point F. A curve of
class n is circular in the quasihyperbolic plane if it contains at least one absolute line.
The curves of the 3rd class can be obtained by projective mapping, i.e. obtained by projectively linked pencil of
curves of the 2nd class and range of points. In this article
we show that the circular curves of the 3rd class of all
types, depending on their position to the absolute gure,
can be constructed with projective mapping. 
 Ana Sliepčević, Ivana Božić Dragun (ivana.bozic@tvz.hr, anasliepcevic@gmail.com)
Introduction to Planimetry of QuasiElliptic PlaneThe quasielliptic plane is one of nine projectivemetric
planes where the metric is induced by the absolute gure F_{QE} = { j_{1}, j_{2},F} consisting of a pair of conjugate imagi
nary lines j_{1} and j_{2}, intersecting at the real point F. Some
basic geometric notions, de nitions, selected constructions
and a theorem in the quasielliptic plane will be presented. 
 Nguyen Le, N J Wildberger (nguyenlecm2009@gmail.com, n.wildberger@unsw.edu.au)
Incenter Symmetry, Euler Lines, and Schiffler PointsWe look at the fourfold symmetry given by the Incenter quadrangle of a triangle, and the relation with the cirum circle, which in this case is the ninepoint conic of the quadrangle. By investigating Euler lines of Incenter tri angles, we show that the classical Schiffler point extends to a set of four Schiffler points, all of which lie on the Euler line. We discover also an additional quadrangle of Incenter Euler points on the circumcircle and investigate its interesting diagonal triangle. The results are framed in purely algebraic terms, so hold over a general bilinear form. We present also a mysterious case of apparent symmetry breaking in the Incenter quadrangle.

 Gunter Weiss (weissgunter@hotmail.com)
Special Conics in a Hyperbolic PlaneIn Euclidean geometry we find three types of special conics, which are distinguished with respect to the Euclidean
similarity group: circles, parabolas, and equilateral hyperbolas. They have on one hand special elementary geometric properties (c.f. [7]) and on the other they are strongly
connected to the \absolute elliptic involution" in the ideal
line of the projectively enclosed Euclidean plane. There
fore, in a hyperbolic plane (hplane) { and similarly in any
CayleyKlein plane { the analogue question has to consider projective geometric
properties as well as hyperbolic elementary geometric properties. It
turns out that the classical concepts "circle", "parabola", and
"(equilateral) hyperbola" do not suit very well to the many cases of conics in a hyperbolic plane (c.f. e.g. [10]). Nevertheless, one
can consider conics in a hplane systematicly having one
ore more properties of the three Euclidean special conics.
Place of action will be the \universal hyperbolic plane" p,
i.e., the full projective plane endowed with a hyperbolic
polarity ruling distance and angle measure. 
 Sebastian Blefari, N J Wildberger (sebastian.eduard.blefari@gmail.com, n.wildberger@unsw.edu.au)
Quadrangle Centroids in Universal Hyperbolic GeometryWe study relations between the eight projective quadrangle centroids of a quadrangle in universal hyperbolic geometry which are analogs of the barycentric centre of a Euclidean quadrangle. We investigate the number theoretical con ditions for such centres to exist, and show that the eight centroids naturally form two quadrangles which together with the original one have threefold perspective symme tries. The diagonal triangles of these three quadrangles are also triply perspective. 
 Boris Odenhal (boris.odehnal@uniak.ac.at) On Algebraic Minimal SurfacesWe give an overiew on various constructions of algebraic minimal surfaces in Euclidean threespace. Especially low degree examples shall be studied. For that purpose, we use the different representations given by WEIERSTRASS including the socalled Björling formula. An old result by LIE dealing with the evolutes of space curves can also be used to construct minimal surfaces with rational parametrizations. We describe a oneparameter family of rational minimal surfaces which touch orthogonal hyperbolic paraboloids along their curves of constant Gaussian curvature. Furthermore, we find a new class of algebraic and even rationally parametrizable minimal surfaces and call them cycloidal minimal surfaces.

 Szilvia B.S. Béla, Márta SzilvásiNagy (belus@math.bme.hu, silvasi@math.bme.hu)
Adjusting Curvatures of Bspline Surfaces by Operations on Knot VectorsThe knot vectors of a Bspline surface determine the ba sis functions
hereby, together with the control points, the shape of the surface. Knot
manipulations and their influence on the shape of curves have been investigated in
several papers (see e.g. [4] and [5]). The computations
can be made very efficiently, if the basis functions and
the vector function of the Bspline surface are represented in matrix form (see
[1] and [6]). In our latest work [2] we summarized the knot manipulation
techniques and the corresponding computations in matrix form. We also developed an algorithm for a direct knot sliding, how a knot
can be repositioned in one step instead of inserting a new
knot value, then removing an old one from the knot vector.
In this paper we analyse the effect of varying knot intervals on the Gaussian curvature of a Bspline surface at a
given point. We present an algorithm for the deformation
of a Bspline surface, so that it should go through a given
point with a given Gaussian curvature. The result of this
deformation is, that a sphere with a given radius will fit
tangential the reshaped surface at the given point with
equal Gaussian curvatures. In applications the same situation arises, when a ballend tool is pushed into a surface
during processing.

Mirela KatićŽlepalo (mkatic@tvz.hr)
Curves of Foci of Conic Pencils in pseudo Euclidean PlaneIn this article, it will be shown that the curve of foci of an order conic pencil in the pseudoEuclidean plane is generally a bicircular curve of 6th order. In some cases, de pending on a position of four base points of the pencil, this curve is of 5th, 4th or 3rd order and in some cases it is even a conic or only a line.

 Ivančica Mirošević (ivancica.mirosevic@fesb.hr)
kmeans AlgorithmIn this paper, kmeans algorithm is presented. It is a heuristic algorithm for solving NPhard optimisation problem of classifying a given data into clusters, with a number of clusters fixed apriori. The algorithm is simple and it's convergence is fast, what makes it widely used, despite its tendency of stopping in a local minimum and inability of recognizing clusters not separated by hyperplanes. The method of the first variation as a tool for escaping from a local minimum is also presented in the paper.

 Mate Glaurdić, Jelena BebanBrkić, Dražen Tutić (mate.glaurdic@gmail.com, jbeban@geof.hr, dtutic@geof.hr)
Graph Colouring and its Application within CartographyThe problem of colouring geographical political maps has historically been associated with the theory of graph colouring. In the middle of the 19th century the following question was posed: how many colours are needed to colour a map in a way that countries sharing a border are coloured differently. The solution has been reached by linking maps and graphs. It took more than a century to prove that 4 colours are sufficient to create a map in which neighbouring countries have different colours.
