Znanstveno-stručni časopis
Hrvatskog društva za geometriju i grafiku

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


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

Stitching B-spline Curves Symbolically

We present an algorithm for stitching B-spline curves, which is different from the generally used least square method. Our aim is to find a symbolic solution for unifying the control polygons of arcs separately described as 4th degree B-spline curves. We show the effect of interpolation conditions and fairing functions as well.

Key words: B-spline curve, B-spline surface, merging,
interpolation, fairing

Article in PDF.

Hiroshi Okumura (hiroshiokmr@gmail.com)

Lamoenian Circles of the Collinear Arbelos

We give an infinite sets of circles which generate Archimedean circles of a collinear arbelos.

Key words: arbelos, collinear arbelos, radical circle, Lamoenian circle
Article in PDF.



Norman John Wildberger, Ali Alkhaldi (n.wildberger@unsw.edu.au, aalkaldy@hotmail.com)

The Parabola in Universal Hyperbolic Geometry I

We introduce a novel definition of a parabola into the framework of universal hyperbolic geometry, show many analogs with the Euclidean theory, and also some remarkable new features. The main technique is to establish parabolic standard coordinates in which the parabola has the form xz = y2. Highlights include the discovery of the twin parabola and the connection with sydpoints, many unexpected concurrences and collinearities, a construction for the evolute, and the determination of (up to) four
points on the parabola whose normals meet.

Key words: universal hyperbolic geometry, parabola

Article in PDF.

Boris Odehnal (boris.odehnal@uni-ak.ac.at)

Conchoids on the Sphere

The construction of planar conchoids can be carried over to the Euclidean unit sphere. We study the case of conchoids of (spherical) lines and circles. Some elementary constructions of tangents and osculating circles are stil valid on the sphere. Further, we aim at the illustration and a precise description of the algebraic properties of the principal views of spherical conchoids, i.e., the conchoid's images under orthogonal projections onto their symmetry planes.

Key words: spherical curves, conchoids, algebraic curves, tangent, osculating circle, singularities, orthogonal projection

Article in PDF.




Géza Csima, Jenö Szirmai (csgeza@math.bme.hu, szirmai@math.bme.hu)

On the Isoptic Hypersurfaces in the
n-Dimensional Euclidean Space

The theory of the isoptic curves is widely studied in the Euclidean plane E2 (see [1] and [13] and the references given there). The analogous question was investigated by the authors in the hyperbolic H 2 and elliptic E2 planes (see [3], [4]), but in the higher dimensional spaces there is no result according to this topic.
In this paper we give a natural extension of the notion of the isoptic curves to the n-dimensional Euclidean space En (n≥ 3) which are called isoptic hypersurfaces. We develope an algorithm to determine the isoptic hypersurface HD of an arbitrary (n−1) dimensional compact parametric domain D lying in a hyperplane in the Euclidean n-space. We will determine the equation of the isoptic hypersurfaces of rectangles D  E2 and visualize them with Wolfram Mathematica. Moreover, we will show some possible applications of the isoptic hypersurfaces.

Key words: isoptic curves, hypersurfaces, differential ge-
ometry, elliptic geometry

Article in PDF.




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

Introduction to the Planimetry of the Quasi-
Hyperbolic Plane

The quasi-hyperbolic plane is one of nine projective-metric
planes where the absolute figure is the ordered triple {j1, j2,F}, consisting of a pair of real lines j1 and j2 through the real point F. In this paper some basic geometric notions of the quasi-hyperbolic plane are introduced. Also the classification of qh-conics in the quasi-hyperbolic plane with respect to their position to the absolute figure is given. The notions concerning the qh-conic are introduced and some selected constructions for qh-conics are presented.

Key words: quasi-hyperbolic plane, perpendicular points, central line, qh-conics classification, osculating qh-circle

Article in PDF.

Maria Čuljak (culjakmaria1@gmail.com)

Isometries in Escher's Work

For better understanding of M. C. Escher's tesselation graphics we provide an overview of planar isometries and classification of plane symmetry groups. Some of the plane symmetry groups are explained on prominent Escher's graphics.

Key words: Escher, isometries, tessellation, plane sym-
metry groups

Article in PDF.