Journal for Geometry and Graphics
Volume 1 (1997)
Abstracts
M. Palej: A Simple Proof for the Theorems of Pascal and Pappus, 1 (1997) 001--004
- The theorem of Pascal concerning a hexagon inscribed in a conic
is very useful in many geometrical constructions and ought to be included
in a normal course on descriptive geometry. Though the citation of this
theorem is possible in a short lecture, its proof is very often omitted due
to a lack of time and because students at technical universities have no
basic knowledge in projective geometry. As far as we know, this discipline
is not contained in the curricula of technical universities. However, a
lecture without proofs is incomplete and satisfies neither lecturers nor
students. Therefore the author presents a proof of Pascal's theorem which
does not require any knowledge of projective geometry. The conic is seen as
the contour of a quadric $\Phi$, and some pairs of lines define conical
surfaces $\Gamma_1, \Gamma_2$. Then the intersections between these three
quadrics $\Phi, \Gamma_1, \Gamma_2$ lead to three collinear Pascal's
points. When the quadric $\Phi$ is replaced by a conical surface $\Gamma_3$
the analysis of intersections between the three surfaces leads to an
immediate proof of Pappus theorem.
- The degree of mobility of a given linkage depends upon the order
of its screw system, whereby a loop with six joint freedoms may be related
to its reciprocal screw system, a single screw axis with an associated pitch.
It has been postulated that this reciprocal screw can provide an alternative
means of identifying its parent linkage, and an investigation in this respect has
been recently carried out for the line-symmetric six-screw linkage.
Here we undertake a similar enquiry for the plane-symmetric six-screw loop
and so determine the essential characteristics of its reciprocal screw.
- Let a one-parametric motion $\beta$ and the boundary representation
of a polyhedron $P$ be given. Our goal is to determine the solid $S$ swept
by $P$ under $\beta$: The complete boundary $\partial S$ of $S$ contains a
subset of the enveloping surface $\Phi$ of the moving polyhedron's boundary
$\partial P$ together with portions of the boundaries of the initial and the
final positions of $P$. For each intermediate position of $P$ the curve of
contact $c_{\partial P}$ between $\partial P$ and $\Phi$ is called the
characteristic curve $c_{\partial P}$ of the surface $\partial P$.
However, in general only a subset of $c_{\partial P}$ gives the
characteristic curve $c_P$ of the solid $P$ which is defined as the curve of
contact between $\partial P$ and $\partial S$.
After a short introduction into instantaneous spatial kinematics, these two
characteristic curves $c_{\partial P}$ and $c_P$ are characterized locally.
Then some global problems are discussed that arise when the boundary
representation of a polyhedral approximation of $S$ is derived automatically.
The crucial point here is the determination of self-intersections at the
envelope $\Phi$. For the global point of view the motion $\beta$ is
restricted to the case of a helical motion with fixed axis and parameter.
- An algorithm providing a decomposition of the lumped structure
body into arbitrary parts having minimal interface has been introduced.
The presented approach may be applied in parallel domain decomposition
methods for mechanical analysis. Optimal partitioning is provided before the
computational mesh is generated on the basis of its prescribed node density
distribution. A new unambiguous solid representation is utilized.
- I analysed the methods of drawing used in the ukie picture titled,
`Interior of a Kabuki Theater', which shows a playhouse, including both the
audience space and the stage. The results of the analysis indicated that two
different methods of drawing were used within the same picture, one-point
perspective for the audience space and combined oblique projection for the
stage. This is not a mere result of incorrect technique but closely related
to what the artist of the picture wanted to express through the drawing.
The artist was also able to create the feeling of actually being inside the
theater through the use of one-point perspective and to show the content of
the play through the use of combined oblique projection, thus placing
emphasis on the main actors. The use of two different methods of drawing
also gives a sense of movement, a shakiness, which adds tension to the
scene. Space that has this tension is related to the Japanese concept of ma.
- A computer graphics system is developed enabling the surgeon to
pre-plan a total knee replacement. A series of geometric problems had to be
solved in order to introduce a device independent set of coordinates serving
the pre-planning as well as the robot that will assist the surgeon. A
special efficient algorithm is used for interactive planning and the
improvement of the performance of bone cuts.
- The dynamic road view perception study was designed and conducted
in order to get a better insight into the safety and speed effects of
alternative road design. The laboratory experiment was designed, primarily,
to test the effect of road category, geometric road design and road
environment elements, as well as driving speed, on subjective assessment of
road characteristics and drivers speed choice. The dynamic road
visualisation method applied has been tested for practical use in computer
aided 3D road design evaluation.
- The paper deals with the problems of the significance of a
perspective drawing in the architectural design. The role of a perspective
as a final presentation drawing in the project is not doubtful. Far more
complex is its importance throughout the design process. At the beginning of
this process perspective sketches become a graphic medium for design
associations, a quick notation of ideas and a field of research for
connections between the designed object and its surrounding. During such a
process the perspective sketch checks the correctness of the spatial
relations and helps modifying architecture. The recalling of pictorial notes
from a project which was carried out for an architectural competition in
Madrid serves as an illustration of the discussed problem. The several
sketches show how the perspective drawing acts as a design tool as well as a
mean of pictorial communication.
- Changes in engineering education, fueled by the rapid growth of
electronic information technology, have had a major impact on the content,
delivery, and role of engineering graphics. The past 30 years have seen a
sharp reduction in the use of graphical methods for solutions of geometry
problems. The engineering drawing, once the means of control of the design
process, has been replaced by the electronic database. Students in
introductory graphics courses are now able to observe the total design
process and create prototype devices from their models. The laboratories in
graphics courses are evolving from intensive drafting and graphical problem
solving activity, to computer modeling and prototype design. This paper
briefly traces the evolution of an introductory design/graphics course over
the past three decades and describes a number of innovative and new
activities that have been incorporated. Software for enhancing
visualization, solid modeling/rapid prototyping, and design-build projects
are among the new directions. The impetus behind these changes is the
engineering education reform movement spearheaded by the National Science
Foundation (NSF), industrial factions, and professional engineering
societies. Engineering education coalitions, sponsored by NSF and
encompassing over 70 universities, are the leaders in developing these new
directions. The author's perceptions of the impact of these new directions
on introductory design/graphics courses in the future are described.
- This paper presents a glimpse into how Descriptive Geometry is
taught in Hannover. After 15 years of teaching experience one goal of this
course is to bridge the gap between everyday experience and scientific
methods. Descriptive Geometry for architects should provide basic
information on the geometry of shapes as well as on projection methods.
Lectures should not only address the brain but all senses.
- The department of clothing and textiles, Otsuma women's
university, conducted an educational program as an initial step in getting
students to recognize the importance of accurate description in the proper
analysis of 3-dimensional objects. The program, which consisted of an
application of descriptive geometry in the teaching of clothing pattern
planning, met with success. This paper summarizes the results of the program
and its unsolved questions which it raises: One of the goals of the course
was to give students a theoretical line of thinking about clothing pattern
planning. With a clear concept of modeling, the student is able to visualize
and understand the relationship between the 3-dimensional shape of the human
body and a clothing pattern. And the student is able to recognize that
clothing pattern construction is closely related to the morphological aspect
of the human body. On the other hand, there still remain some unsolved
problems concerning the contents of the curriculum and policy.
- The most important properties of Bezier and B-spline curves are
the convex hull property, the affine invariance, the possibility to
subdivide and the variation diminishing property. Therefore it would be of
great interest to have a larger class of point controlled curves with the
same properties. It is known that all corner cutting curves have the first
two properties. In this paper we deal with the subdivision of corner cutting
curves, especially of linear corner cutting curves. For uniformly tangent
corner cutting curves (a subclass which contains B-spline curves) we present
a simple method for computing the control points of the new curves.
- Let two unit circles $k_A, k_B$ in perpendicular planes be given
such that each circle contains the center of the other. Then the convex hull
of these circles is called Oloid. In the following some geometric properties
of the Oloid are treated analytically. It is proved that the development of
the bounding torse $\Psi$ leads to elementary functions only. Therefore it
is possible to express the rolling of the Oloid on a fixed tangent plane
$\tau$ explicitly. Under this staggering motion, which is related to the
well-known spatial Turbula-motion, also an ellipsoid $\Phi$ of revolution
inscribed in the Oloid is rolling on $\tau$. We give parameter equations of
the curve of contact in $\tau$ as well as of its counterpart on $\Phi$. The
surface area of the Oloid is proved to equal the area of the unit sphere.
Also the volume of the Oloid is computed.
- If one draws in a plane from a point $X$ the perpendiculars onto
the sides $AB,BC,C A$ of a triangle $ABC$ and if the feet of these
perpendiculars $P\in AB$, $Q\in BC$, $R\in C A$ lie on a line -- the
Wallace line of $X$ -- then $X$ lies on the circumcircle of the triangle
$ABC$. We introduce two generalizations: If the affine feet $P, Q, R$ lie
on the affine Wallace line of $X$ with respect to a center $Z$ or if the
projective feet $P, Q, R$ lie on the projective Wallace line of $X$ with
respect to a center $Z$ and an axis $f$ then $X$ lies on a conic.
- In the 3-dimensional Euclidean space (the model of the
3-dimensional projective space) among the surfaces of 4th order with a nodal
line those through the absolute conic are extracted as a special class which
contains the pedal surfaces of (1,2)-congruences. One class of these
surfaces is classified with regard to the number and types of their real
singular points. These surfaces have been visualized using the program
Mathematica 3.0.
- One of the important questions of central axonometry is to give a
condition under which a central axonometric mapping is a central projection.
The aim of this paper is to prove that the well-known Stiefel's condition
can be considered as a limiting case of a recent theorem proved by Szabo,
Stachel and Vogel.
- The paper deals with special axonometric mappings of an
n-dimensional Euclidean space onto a plane $\pi'$. Such an
(n,2)-axonometry is given by the image of a cartesian n-frame in $\pi'$
and it is especially an isocline or orthographic axonometry, if the contour
of a hypershere is a circle in $\pi'$.
The paper discusses conditions under which the image of the cartesian
n-frame defines an orthographic axonometry. Also a recursive construction
of the hypersphere-contour in case of an arbitrary given oblique axonometry
is presented.
- I present the developments in my work using math and
geometry as major elements to construct small maquette-sized pieces to large
wall installations. I like to engage the viewer into my pieces for a certain
amount of time in order to experience another dimension. The eye of the
viewer is tricked with an impression of perpetual change and motion. And so
the viewer sees more than what is actually received by the eye; he will find
three-dimensional spaces within the two-dimensional surfaces. In this sense
I offer the viewer a chance for virtual movement through planar geometry.
- A teaching technique for presenting visualization problems is
discussed which requires students to generate both multiview (orthographic)
and pictorial views of simple to complex objects from symbolic information.
The technique makes use of an expanded coded plan or base design method for
describing objects in space, and transcends the traditional
isometric/orthographic, orthographic/isometric translation exercises. The
technique can be used in traditional or CAD-based engineering design
graphics courses and can be implemented using either sketching or instrument
construction methods.
- In order to clarify the ability reflected in scores in a Mental
Rotations Test (MRT), the performance in Shepard-Metzler tasks (S-M tasks)
was compared between experts and novices. The analysis indicated that the
variety of the strategy preference in S-M tasks was very similar to that
observed in the MRT. It can be said that the differences in strategies of
the MRT were evoked by individual differences in performing mental rotations.
The speed of mental rotation is one of the factors which have an effect on
the score in the MRT. The variety of strategies for novices may be evoked by
the low speed of mental rotation and the difficulty in unifying strategies
to mental rotation. It is summarized that the score in the MRT evaluates the
performance in mental rotations.