**Journal for Geometry and Graphics**

**Volume 5 (2001)
Abstracts**

A. G. Horvath, I. Prok: Packing Congruent Bricks into a Cube, 5 (2001) 001--012- L. Lovasz raised the problem whether 27 congruent bricks of edge lengths
a, b, c (0 < a < b < c, a+b+c = s) can be packed into a cube of edge
length s without overlaps so that the arrangement is universal, in other
words, it should be independent from the choice of a, b and c. If
that were possible, we could obtain a geometric proof of the inequality
a+b+c ≥ 3 Squareroot (abc) between the arithmetic and geometric
means of three positive numbers. (This would be an analogous method to the
well-known proof of the inequality a+b ≥ 2 Squareroot (ab), (a, b > 0),
concerning the packing or four rectangles of edge lengths a, b into a
square of edge length a+b.)

Hence, fundamentally, this is a special packing problem: some bricks having fixed volume must be put into a container of given volume. From the combinatorial point of view, similar container problems were investigated by D. Jennings. The first author has found a possible universal arrangement, and someone else has found an additional one which has proved to be different under the symmetries of the cube. In the paper we introduce an algorithm for finding all the different universal arrangements. As a result we obtain 21 possibilities (listed in Section 4) by the corresponding computer program. Our method seems to be suitable for solving the analogous problem in higher dimensions.

P. Pech: The Harmonic Analysis of Polygons and Napoleon's Theorem, 5 (2001) 013--022- Plane closed polygons are harmonically analysed, i.e., they are expressed in the form of the sum of fundamental k-regular polygons. From this point of view Napoleon's theorem and its generalization, the so-called theorem of Petr, are studied. By means of Petr's theorem the fundamental polygons of an arbitrary polygon have been found geometrically.

N. Ando, N. Yamahata, S. Masumi, M. Chatani: Shape Grammar and Form Properties of Architectural Figures, 5 (2001) 023--034- Since the beginning of history, various forms of architecture have been designed in the world. The characteristics of those forms were considered in the figure of elevation. In this study, architectural figures as the simple expression of architectural forms are analyzed to understand the shape grammar, i.e., the geometrical composition of figures, and the form properties of architecture. Firstly, the shape grammar of architectural figures is verified by presenting 74 figures of representative architecture, and secondarily, a method to evaluate the form properties is shown.

K. Fuchigami: Analysis of the Spiral Pattern*Karakusa*, 5 (2001) 035--044- I discuss the properties of a pattern known as "Karakusa". Karakusa is Japanese and means "foreign plant" or "winding plant". The pattern consists of various spirals, and these spirals take their shape from vines and other natural forms. I will examine and demonstrate how features of the pattern were abstracted from these natural forms. In addition, it will be argued that an algorithm employing a mathematical element could be involved in generating those features, and I will claim that we will be able to utilize this algorithm for generating new spiral forms. In this study I examine a Karakusa pattern familiar from its use on Japanese wrapping cloths. In this particular pattern only a geometric spiral is employed. It is a simple pattern constructed from many spirals, each of which extends freely in all directions. In this paper I examine such features in turn and describe my results.

K. Kojima, M. Hironaga, S. Nagae, Y. Kawamoto: A Human Motion Analysis Using the Rhythm -- A Reproducing Method of Human Motion, 5 (2001) 045--052- We estimate the postures of a human body in dance images (or repeated motions) recorded in an image database (especially recorded movies). We propose a new method by defining the "Rhythm Points", i.e., equally spaced points on the time axis located at the moment of the start or end of the motion. With these rhythm points we are able to separate each repetition of the motions and to measure coordinates only at the moment of rhythm points and of a few points between (key frames), while coordinates of other frames are interpolated automatically. We introduced new terms for these variables in the formula of the splines and we designed a new natural spline curve for interpolating the knee-angle. Thus we could improve our previous results. With our method the necessary human work could be remarkably reduced.

K. Mende: Light and Shadow in Painting -- Concerning the Expression of Shadows in Western Painting --, 5 (2001) 053--060- In realism paintings in Western art, as perspective drawing has developed,
so has the artists' ability to create the illusion of space as they see it.
One of the important techniques is the skill with which light and shadow are
used to create the illusion of a three dimensional space on a two
dimensional surface. In this representation of the space, if the light is
not depicted on a picture plane, the three dimensionality of the objects
will be weak. Then, my question is when the expression of shadow and shade
is correctly performed, does it assure recognition of the depicted space?

This study will take up Johannes Vermeer, who drew scenes of everyday life realistically, and consider his use of shadow and shade geometrically. I'd like to consider what kind of light and shadow he had in mind. As a conclusion, it can be said that his space was precise in terms of perspective; he depicted shade but not shadow precisely. That made his space lack unity, leaving his picture plane serene as if time had frozen in it. But if we suppose the world he wanted to express was not just a scene of daily life, but a story with a lesson to be learned, then without the precise depiction of shadow and shade, the viewer can tell the story is fiction.

T. Noguchi, Y. Ohno: A Deformation Algorithm of Railway Maps, 5 (2001) 061--070- Railway maps are often printed or posted in a deformed style for better understanding the topological connections of intersecting lines. In this paper we propose an algorithm for the automatic generation of deformed railway maps. The railway map is represented as an undirected graph; a vertex corresponds to a station and an edge corresponds to a railway between two stations. The data necessary for this algorithm are the position of each station, and a list of stations of each line. The algorithm proceeds as follows: (1) An angle value is assigned to each vertex based on the direction of edges attached to the vertex. (2) Adjacent vertices which are on a same railway line and which share similar angle values are gathered to make a group. (3) Assign a priority to each vertex based on the sizes of groups to which the vertex belongs. (4) Place each edge in the order of priority of its two end vertices. This algorithm is applied to some railway maps including the very complex one of Tokyo Metropolitan Area, and excellent results are obtained. The obtained deformed maps will be evaluated based on their accuracy and the understandability.

M. Ohnishi: A Photographic Method for Panoramic Sequence with a Regular Camera, Part 3: Application to Sky Photographs, 5 (2001) 071--080- In parts 1 and 2 the authors presented a method to produce a panorama by connecting photographs. In part 3 this method is applied to sky photographs. For this purpose a computer program is presented to calculate the relative position of the two photoprints. An example for this will be introduced, too. Next, a method for estimating the center of a sky photograph and the focal length of the lens of the camera will be presented. This method is based on the celestial coordinates of fixed stars being displayed on the sky photograph. The center and the focal length are important data but often there are gaps in the data when this method is used. The new method offers a more accurate placement of the two photoprints. This method can be applied to measuring the data needed to derive the positions of celestial bodies.

C. Leopold, R. A. Gorska, S. A. Sorby: International Experiences in Developing the Spatial Visualization Abilities of Engineering Students, 5 (2001) 081--092- Engineers communicate with one another largely by graphical means. For this reason, it is very important that the spatial visualization abilities of engineering students be well-developed. Unfortunately, there is little guarantee that our students come to the university with well-developed spatial abilities. In this paper we will compare the spatial visualization skill-levels for entering engineering students from our universities by means of several tests (Mental Rotations Test, Mental Cutting Test, and Differential Aptitude Test: Space Relations) and evaluate our experiences in improving the spatial abilities of both male and female engineering students. Through international comparisons of our experiences in teaching introductory courses, we will describe the teaching methods that seem to be especially helpful in the development of spatial visualization skills for engineering students.

E. Toledo Santos, J. I. Rojas Sola: A Proposal for an On-Line Library of Descriptive Geometry Problems, 5 (2001) 093--100- This paper describes an interactive system on the Internet, which implements an electronic library of Descriptive Geometry problems. All problems can be accessed as proposed exercises or solved problems (demonstrations). The library may register several alternative solutions for the same problem, allowing the user to choose the one he understands better. The system is implemented as a client-server architecture. The server side runs along with a web server and manages a database of problems and registered users. The client side is represented by a Java applet which provides graphical interaction with the users. Drawing tools are available for on-line problem solving. Demonstration of a solution is done through step-by-step animations and explanation texts. The system features some simple tutoring capabilities.

E. Tsutsumi, A. Ichikawa, N. Kadowaki: Evaluation of Mentally Perceived Differences Between the 3D Objects Used in Mental Cutting Tests, 5 (2001) 101--110- We examined the issue of recognizing 3D objects through the measurement of the mentally perceived differences between the 25 solid objects used in the MCT. Subjects were required to arrange actual solid models along a straight line based on their perception of similarities between the solid models. The distances between adjacent solids were transfered into a distance matrix which was analyzed using multidimensional scaling methods. The similarities between the solids were interpreted in a chart of constellation that was constructed using axes of dimension 1 and dimension 2, which were extracted by multidimensional scaling. The results show that subjects seem to pay intense attention to the following details of the solid objects: (1) impression of the solid shape as a whole which is then compared to some geometrically distinctive fundamental forms, and (2) characteristic local shapes which symbolize the solid. In judging mentally perceived differences between solids, high scoring subjects in the paper-pencil MCT were not influenced by the similarity of characteristic local shapes and they were able to classify the objects clearly considering the structural differences. Meanwhile, low scoring subjects in the paper-pencil MCT were influenced by the similarity of characteristic local shapes and had a tendency not to clearly separate the results of classification.

I. Juhasz, M. Hoffmann: The Effect of Knot Modifications on the Shape of B-spline Curves, 5 (2001) 111--120- This paper is devoted to the shape control of B-spline curves achieved by the modification of one of its knot values. At first those curves are described along which the points of a B-spline curve move under the modification of a knot value. Then we show that the one-parameter family of k-th order B-spline curves obtained by modifying a knot value has an envelope which is also a B-spline curve of order k-1.

J. Lang, S. Mick, O. Röschel: The Rigidity Rate of Positions of Stewart-Gough Platforms, 5 (2001) 121--132- We consider a 6-legged Stewart-Gough platform. The following investigations of such platforms will always be carried out at an arbitrary given position. If the leg lengths are kept constant, the platform in general will be rigid within the Euclidean displacement group, whereas viewed within the Euclidean similarity group it will yet be movable. There exists an infinitesimal transformation of this motion. Its deviation from the Euclidean displacement group is used to define the "rigidity rate" of the platform at this position. In order to obtain some geometric invariant measurement, Lie group methods are applied. An example eventually demonstrates the efficiency of the presented method.

E. Molnar, T. Schulz, J. Szirmai: Periodic and Aperiodic Figures on the Plane by Higher Dimensions, 5 (2001) 133--144- We extend de Bruijn's idea of constructing Penrose's non-periodic tilings of the plane to higher-dimensional analogons. On the base of d-dimensional space groups we can draw nice aperiodic coloured plane tilings with the aid of computers, especially interesting ones if d+1 is prime. Our proposed probabilistic method seems to produce attractive pictures, in particular.

Y. Yamaguchi: A Basic Evaluation Method of Subdivision Surfaces, 5 (2001) 145--156- A subdivision surface is a powerful tool to represent a smooth surface with arbitrary topology. It starts with a control polyhedron which roughly approximates the final surface. The polyhedron gradually approaches to the final surface by subdividing faces more closely to approximate the final surface. However, it is not so popular in engineering applications because of the procedural nature. Algorithms for parametric surfaces cannot be applicable to a procedural surface. This paper discusses a method for evaluating a subdivision surface with parametric values based on Stam's work. Our approach gained stability around an extraordinary point as well as the convergence of a normal vector at the point.

R. A. Wiggs: Geometry as Transformation, 5 (2001) 157--164- This paper utilizes geometry and graphics to demonstrate a process of three-dimensional spatial transformation. A transformation defines how structures behave in addition to how they are made. Here, the transformation process is applied to several different frames of reference. Material polyhedra transform into pairs of spatial lattices and the lattices transform into each other; the lattices also transform into twisted loops with exostructural and endostructural modes. The exo/endo interactions transform the hypercube into a pair of twisted loops and the twisted loops become a source for sculpture.

Z. B. Gradinscak: Constructional Graphics Application in Engineering Computer Graphics, 5 (2001) 165--180- In studying the drawing there is even today a tendency to concentrate upon what is marked on a piece of paper, and to forget that much of drawing that was marked in the past on the actual stone or wood. As the education and books on drawing were increasingly developed and used by academia, the drawing techniques became theoretically supported by developed geometrical principles extracted from basic empirical constructions. Geometrical concepts that provide developments of such empirical constructional methods, applications of which were utilised in the pre-Descriptive Geometry era, are denoted as constructional geometry. As the construction of 3D parametric solid models becomes recognised as the skill modern engineers need to posses, the ability to spatially construct and manipulate virtual geometrical elements will unavoidably become an elementary part in the engineering educational system. With the application of a vector space in computer graphics, the introduction of graphical techniques that conduct 3D problem solving by spatial construction instead of the planar projection becomes essential for engineering spatial graphics. Discussion of this need for introducing graphical concepts that deal with spatial relationships in a computer graphics vector space and subsequent application of such a constructional method in modern engineering computer graphics are presented

C. Leopold, A. Matievits: Studies of Geometry Integrated in Architectural Projects, 5 (2001) 181--192- Geometry plays the role of a basic science in engineering, especially in architecture. In the past years the importance of geometry was pushed back because of a wrong estimation of new technologies like CAD. Geometry researchers and teachers failed to convince the practical engineers of the importance of geometrical reasoning, even in the use of computerised methods. This paper will show the concept of an integrated teaching of geometry in architectural projects which is able to point out the importance of geometry in practising architecture as well as in the use of new technologies. Descriptive Geometry turns out not to be an antiquated science but a current one. In our integrated geometry concept we start with the architectural project that leads us to geometrical problems to be solved. The experiences with project-oriented studies for students of architecture in their major courses at our university in the past years are presented in this paper. In the examples like developing geometrical forms, projection methods for representations of architecture, photoreconstruction and photomontage, we reflect the way from the geometrical task, arising from the design project, to the geometrical solution.

K. Shiina, D. R. Short, C. L. Miller, K. Suzuki: Development of Software to Record Solving Process of a Mental Rotations Test, 5 (2001) 193--202- To collect data on the solving process for the Mental Rotations Test for large numbers of subjects, software was developed that restricts the appearance of alternatives. This software made it possible to record information corresponding to eye fixation data. This data was useful in analyzing the problem-solving process. A pilot study using the software was controlled in a computer network environment. The solving process data of each subject was collected via the network and analyzed. The time spent to judge each correct alternative was extracted from the time sequence data of each subject. Parameters were calculated from the extracted time. Its distribution corresponded to the variation of solving strategies observed in previous research in which eye fixation data and verbal protocol data were used. This approach presents the possibility of being able to use software to identify the source of the performance difficulties for subjects.