3rd Class Circular Curves in Quasi-Hyperbolic Plane Obtained by Projective Mapping

The metric in the quasi-hyperbolic plane is induced by an absolute figure F_QH = {F, f₁, f₂}, consisting of two real lines f₁ and f₂ incident with the real point F. A curve of class n is circular in the quasi-hyperbolic 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.

Key words: projectivity, circular curve of the 3rd class, quasi-hyperbolic plane

Introduction to Planimetry of Quasi-Elliptic Plane

The quasi-elliptic plane is one of nine projective-metric planes where the metric is induced by the absolute gure F_QE = { j₁, j₂,F} consisting of a pair of conjugate imagi- nary lines j₁ and j₂, intersecting at the real point F. Some basic geometric notions, de nitions, selected constructions and a theorem in the quasi-elliptic plane will be presented.

Key words: quasi-elliptic plane, perpendicular points, cen- tral line, qe-conic classication, hyperosculating qe-circle, envelope of the cental lines
circles

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Incenter Symmetry, Euler Lines, and Schiffler Points

We look at the four-fold symmetry given by the Incenter quadrangle of a triangle, and the relation with the cirum- circle, which in this case is the nine-point 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.

Key words:triangle geometry, Euclidean geometry, ra- tional trigonometry, bilinear form, Schiffler points, Euler lines, Incenter hierarchy, circumcircles

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Special Conics in a Hyperbolic Plane

In 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 (h-plane) { and similarly in any Cayley-Klein 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 h-plane 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.

Key words: conic section, hyperbolic plane, Thales conic, equilateral hyperbola

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Quadrangle Centroids in Universal Hyperbolic Geometry

We 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 three-fold perspective symme- tries. The diagonal triangles of these three quadrangles are also triply perspective.

Key words: Universal hyperbolic geometry, projective geometry, centroids, quadrangles, diagonal triangles, perspectivities

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On Algebraic Minimal Surfaces

We give an overiew on various constructions of algebraic minimal surfaces in Euclidean three-space. Especially low degree examples shall be studied. For that purpose, we use the different representations given by WEIERSTRASS including the so-called 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 one-parameter 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.

Key words: minimal surface, algebraic surface, rational parametrization, polynomial parametrization, meromorphic function, isotropic curve, Weierstrass-representation, Björling formula, evolute of a spacecurve, curve of constant slope

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Adjusting Curvatures of B-spline Surfaces by Operations on Knot Vectors

The knot vectors of a B-spline 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 B-spline 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 B-spline surface at a given point. We present an algorithm for the deformation of a B-spline 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 ball-end tool is pushed into a surface during processing.
In our algorithm we use only linear interpolation equations besides the repositioning of knot values, in order to get numerically stable and effective solutions.

Key words: surface representations, geometric algorithms

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Curves of Foci of Conic Pencils in pseudo- Euclidean Plane

In this article, it will be shown that the curve of foci of an order conic pencil in the pseudo-Euclidean 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.

Key words: pseudo-Euclidean plane, conic sections, foci, conic pencil

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k-means Algorithm

In this paper, k-means algorithm is presented. It is a heuristic algorithm for solving NP-hard 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 hyper-planes. The method of the first variation as a tool for escaping from a local minimum is also presented in the paper.

Key words: k-means algorithm, clustering, rst variation method

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Graph Colouring and its Application within Cartography

The 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.

Key words: graph, graph colouring, map, map colouring, the four colour theorem

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