**Journal for Geometry and Graphics**

**Volume 6 (2002)
Abstracts**

L. Acs, E. Molnar: Algorithm for D-V Cells and Fundamental Domains, E^{4}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 E
^{4}-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 A
_{2}is called an isotropic plane I_{2}, if in A_{2}a metric is induced by an absolute figure (f, F), consisting of the line f at infinity of A_{2}and a point F in f. This paper gives a complete classification of the second order curves in the isotropic plane I_{2}. Although conics in A_{2}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 I_{2}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 L
_{2}. These motions are surfaces on the Lie group SO(2, 1). In the first part the basic properties of motions in L_{2}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 R_{4}. 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.