Journal for Geometry and Graphics

Volume 6 (2002)

Abstracts

L. Acs, E. Molnar: Algorithm for D-V Cells and Fundamental Domains, E4 Space Groups with Broken Translations in the Icosahedral Family, 6 (2002) 001--016
As a continuation of our earlier work we extend our algorithm for E4-space groups in the icosahedral family, where non-lattice translations (broken translations) occur as well. So we obtain new 4-polytopes as fundamental domains from D-V cells of crystallographic orbits. We illustrate our situations by some characteristic examples. A computer program will produce further results.

J. Beban-Brkic: Isometric Invariants of Conics in the Isotropic Plane --- Classification of Conics, 6 (2002) 017--026
A real affine plane A2 is called an isotropic plane I2, if in A2 a metric is induced by an absolute figure (f, F), consisting of the line f at infinity of A2 and a point F in f. This paper gives a complete classification of the second order curves in the isotropic plane I2. Although conics in A2 have been investigated earlier, this paper offers a new method based on Linear Algebra.
The definition of invariants of a conic with respect to the group of motions in I2 makes it possible to determine the type of a conic without reducing its equation to canonical form. The obtained results are summarized in an overview table. Such an approach can also be understood as an example of classifying quadratic forms in n-dimensional spaces with non-regular metric.

M. Hlavova: Two-Parametric Motions in the Lobatchevski Plane, 6 (2002) 027--036
We classify two-parametric motions in the Lobatchevski plane L2. These motions are surfaces on the Lie group SO(2, 1). In the first part the basic properties of motions in L2 are derived and it turns out that the kinematical space belonging to these motions is locally the space SO(2, 2) / SO(2, 1) realized as the unit quadric with signature (2, 2) in the vector space R4. The remaining part contains explicit expressions and graphic representations of surfaces induced by motions with constant invariants. We also present some special cases - developable surfaces.

P. Schreiber: Generalized Descriptive Geometry, 6 (2002) 037--060
Generalized Descriptive Geometry (GDG) denotes all techniques of imaging abstract objects and their relations, which are constructive in the same sense as the methods of classical descriptive geometry, i.e., there have to be algorithms concerning the abstract objects (may be decision processes) so that there are translations into algorithms working with the pictures instead of the abstract objects. Hence GDG is more than mere visualization though the borderlines sometimes are fuzzy. The paper presents an outline of standard methods. Some of them are very old and were originally developed outside mathematics.

A. Sliepcevic: A New Generalization of the Butterfly Theorem, 6 (2002) 061--068
The butterfly theorem and some of its generalizations deal with a specific point related to a quadrangle inscribed into a circle. By use of the Sturm-Desargues involution theorem it is proved that with any such quadrangle an infinite number of butterfly points is associated which are located on an equilateral hyperbola. Finally an infinite number of quadrangles sharing the same butterfly curve is presented.

K. Sugihara: Laguerre Voronoi Diagram on the Sphere, 6 (2002) 069--082
The Laguerre Voronoi diagram, also called the power diagram, is one of the important generalizations of the Voronoi diagram in the plane, in which the generating points are generalized to circles and the distance is generalized to the Laguerre distance. In this paper, an analogue of the Laguerre Voronoi diagram is introduced on the sphere. The Laguerre distance from a point to a circle on the sphere is defined as the geodesic length of the tangent line segment from the point to the circle. This distance defines a new variant of the Voronoi diagram on the sphere, and it inherits many characteristics from the Laguerre Voronoi diagram in the plane. In particular, a Voronoi edge in the new diagram is part of a great circle (i.e., the counterpart of a straight line), and the Voronoi edge is perpendicular to the great circle passing through the centers of the two generating circles. Furthermore, the construction of this diagram is reduced to the construction of a three-dimensional convex hull, and thus a worst-case optimal O(n\log n) algorithm is obtained. Applications of this diagram include the computation of the union of spherical circles and related problems.

M. Szilvasi-Nagy: Filling Holes with B-spline Surfaces, 6 (2002) 083--098
This paper presents an algorithm for filling holes with polynomial tensor product B-spline surfaces of degree (3, 2). The B-spline surface is constructed as a tube shaped surface which is attached to the boundary of the hole at one end and is tied up in a closing point at the other end. The patches around the closing point are degenerate three-sided patches. The unknown control points and shape influencing tangent magnitudes of the B-spline surface are computed from boundary conditions and fairness criteria by minimizing appropriate energy functions.

G. Prieto, A. Velasco: Predicting Academic Success of Engineering Students in Technical Drawing from Visualization Test Scores, 6 (2002) 099--109
While observing the difficulties of first-year engineering students toward learning technical drawing, taking into account the progressively reduced work time with them, and recognizing the importance of spatial aptitude in the engineering profession, we feel the necessity to improve the teaching methodologies in this subject. In our opinion, in order to effectively plan the didactic process, it is necessary to detect as early as possible those students who require more attention and support. This study proposes an investigation of a visualization psychometric test that could facilitate an early diagnosis concerning the academic performance of technical drawing students. To this end, a computerized version of the Mental Cutting Test (MCT) was carried out on a sample of Brazilian engineering students from the Paulista State University at Guaratingueta Campus (UNESP) and from the Polytechnic School of Sao Paulo University (EPUSP). The test was analyzed by the Item Response Theory, with the Rasch model, a measurement model with optimal properties in order to estimate the level in spatial aptitude of the examinees. The results suggest that MCT can be useful in detecting those students with different performance levels in technical drawing.

S. Gorjanc: Special Quartics with Triple Points, 6 (2002) 111--120
This paper deals with a special class of 4th-order surfaces in the 3-dimensional Euclidean space. The surfaces of this class contain the absolute conic, a double straight line and triple points. It is shown that such surfaces may contain on the double line at least two real triple points which are classified according to the type of their tangent cones. The selected examples of the surfaces are displayed using the program Mathematica 4.1.
Keywords: Algebraic surfaces of 4th order, triple point.
Classification: 51N35.

A. Hirsch: Extension of the 'Villarceau-Section' to Surfaces of Revolution with a Generating Conic, 6 (2002) 121--132
When a surface of revolution with a conic as meridian is intersected with a double tangential plane, then the curve of intersection splits into two congruent conics. This decomposition is valid whether the surface of revolution intersects the axis of rotation or not. It holds even for imaginary surfaces of revolution. We present these curves of intersection in different cases and we also visualize imaginary curves. The arguments are based on geometrical reasoning. But we also give in special cases an analytical treatment.
Keywords: Villarceau-section, ring torus, surface or revolution with a generating conic, double tangential plane.
Classification1: 51N05.

H. Stachel: Remarks on A. Hirsch's Paper concerning Villarceau Sections, 6 (2002) 133--140
When a surface of revolution with a conic as meridian is intersected with a bitangential plane, then the curve of intersection splits into two congruent conics. Conversely a necessary and sufficient condition is presented such that the rotation of a conic about a non-coplanar axis gives a surface with conics as meridians. Both results are proved by direct computation.
Keywords: Villarceau section.
Classification: 51N05; 51N35.

M. Hoffmann, I. Juhasz: Geometric Aspects of Knot Modification of B-spline Surfaces, 6 (2002) 141--150
In a recent publication we described the effect of knot modifications of B-spline curves. The aim of this paper is the generalization of these results for surfaces. Altering one or two knot values of a B-spline surface, the paths of the points of the surface are discussed first, among which special ruled surfaces can be found. Then we prove that the family of B-spline surfaces obtained by knot alteration, possesses an envelope which is a lower order B-spline surface.
Keywords: B-spline surfaces, knot modification.
Classification1: 53A05, 68U05.

E. Korczak: A Quadratic Transformation Based on a Straight Line and a Conic, 6 (2002) 151--166
The L-transformation is quadratic in the projective 3-space and originally based on an irreducible spatial cubic. Here the case is addressed where the cubic splits into a conic and a straight line. Two cases are distinguished depending on whether the conic and the line are disjoint or not. An analytic representation of the L-transformation is given and the images of planes and lines are studied in detail.
Keywords: Lambda-transformation, quadratic transformation.
Classification: 51N15; 51N35.

G. Weiss: Golden Hexagons, 6 (2002) 167--182
A "golden hexagon" is a set of six points, which is projectively equivalent to the vertices of a regular pentagon together with its center. Such a geometric figure generalizes in some sense the classical one-dimensional golden section to two dimensions. This paper deals with some remarkable properties of golden hexagons and with special Euclidean representatives as well as with further generalizations.
Keywords: golden section, golden ratio, golden cross ratio, geometrically defined iterative processes, Desargues' Theorem, polarity with respect to a conic, Moebius circle geometry, bio-geometry, regular polyhedra.
Classification: 51M04; 51M05, 51M20.

K. Mende: Concerning the Japanese Kabuki Stage, 6 (2002) 183--190
The depiction of the performance space in Japanese Kabuki theatre illustrations has been influenced by the Western perspective. The use of Western perspective in Kabuki drawings differs from the way in which it is employed in depictions of Western theatres. A characteristic of the Kabuki theatre is the wide stage front, which opens up on the sides. This research paper considers the special characteristics of the traditional Japanese Kabuki stage. It analyses dougu-cho (set drawings) which are backdrop paintings for Kabuki set designs.
Keywords: perspective drawing, backdrop painting, Kabuki.
Classification: 51N05.

G. S. Ivanov: The History and Perspectives of the Development of Applied Geometry in Russia, 6 (2002) 191--194
Here the development of Descriptive Geometry as a scientific discipline in Russia and its transition to Applied Geometry is presented. It includes the main authors and the directions of research as well as the tasks of the organization plan and the methodical plan for the nearest future.
Keywords: Descriptive Geometry, Applied Geometry, dissertations.
Classification: 51N05.

V. Y. Mikhailenko: Achievements of the Ukrainian School of Applied Geometry, 6 (2002) 195--200
This is a brief summary of recent scientific activities in Ukraine in the field of Applied Geometry.
Keywords: Descriptive Geometry, Applied Geometry.
Classification: 51N05.

V. Plosky: From the System Analysis of Applied Geometry Methods toward the Structure of the Ukrainian Geometrical School, 6 (2002) 201--212
The historical development, the internal structure and the information contents of the Ukrainian School of Applied Geometry are analysed using methods from the theory of organizations, from social psychology, and modern management. This is the base for estimating tendecies of its future development and determining operative and strategic targets for this scientific area under new market conditions.
Keywords: Applied Geometry, system analysis.
Classification: 51N99.

K. Furukawa, T. Yonemura, S. Nagae: Presenting Educational Contents by Using a Non-contact Viewer, 6 (2002) 213--220
Personal computers are now rapidly diffusing into public facilities as well as educational organizations and common families. To let variety of user groups handle software with ease, it is an urgent business and an essential factor to develop a digital society as well as to construct a human-friendly environment for operation. This article describes a method to effectively fabricate various formative models by means of paper craft and suggests an example of educational tools with which everybody can explore the environment to be acquainted with computer and joy of creation during enjoyment.
Keywords: computer graphics, educational contents, non-contact viewer.
Classification1: 51N05.

K. Suzuki: Activities of the Japan Society for Graphic Science -- Research and Education, 6 (2002) 221--229
We briefly review the research and educational activities of the Japan Society for Graphic Science (JSGS). The membership in the JSGS stands now at around 330 made of mostly individual members who are university and college instructors of graphics-related subjects. The JSGS holds nation-wide meetings twice a year, once in spring and the other in autumn. In each JSGS meeting a forum on graphics education is also held. The JSGS publishes the "Journal of Graphic Science of Japan" four times in a year. The JSGS also occasionally publishes books. In addition to these activities in Japan, the JSGS has been making efforts in promoting international cooperation.
Keywords: graphic science, descriptive geometry, computer graphics, spatial ability, spatial visualization.
Classification: 51N05.