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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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
- 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.
- 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.