Journal for Geometry and Graphics

Volume 1 (1997)


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.

J. E. Baker: The Single Screw Reciprocal to the General Plane-Symmetric Six-Screw Linkage, 1 (1997) 005--012
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.

A. Elsonbaty, H. Stachel: Generating Solids by Sweeping Polyhedra, 1 (1997) 013--022
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.

M. Flasinski, R. Schaefer, W. Toporkiewicz: Supporting CAE Parallel Computations with IE-Graph Solid Representation, 1 (1997) 023--030
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.

K. Mende: The Representation of Pictorial Space in ``Ukie'', 1 (1997) 031--040
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.

O. Tebeleva, G. Elber, Y. Charit, M. Roffman: Geometric Problems in Computerized Preoperational Planning of a Robot Assisted Total Knee Replacement, 1 (1997) 041--050
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.

L. Zakowska: Dynamic Road View Research for Road Safety and Aesthetics Evaluation, 1 (1997) 051--058
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.

L. Gilewicz, W. Gilewicz: Perspective Drawing in the Architectural Design Process, 1 (1997) 059--066
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.

R. D. Jenison: New Directions for Introductory Graphics in Engineering Education, 1 (1997) 067--074
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.

A. Schmid-Kirsch: Teaching Descriptive Geometry at the Faculty of Architecture, 1 (1997) 075--082
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.

E. Tsutsumi: Descriptive Geometry Education at the Department of Clothing and Textiles, Otsuma Women's University, 1 (1997) 083--090
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.

G. Aumann: Subdivision of Linear Corner Cutting Curves, 1 (1997) 091--104
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.

H. Dirnboeck, H. Stachel: The Development of the Oloid, 1 (1997) 105--118
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.

O. Giering: Affine and Projective Generalization of Wallace Lines, 1 (1997) 119--134
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.

S. Gorjanc: The Pedal Surfaces of (1,2)-Congruences with a One-Parameter Set of Ellipses, 1 (1997) 135--150
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.

M. Hoffmann: On the Theorems of Central Axonometry, 1 (1997) 151--156
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.

G. Weiss: (n,2)-Axonometries and the Contour of Hyperspheres, 1 (1997) 157--168
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.

B. Chilla: Virtual Movement Through Planar Geometry: Fundamental Concepts in Visual Art, 1 (1997) 169--178
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.

L. D. Goss: Presentation of Visualization Problems Using an Expanded Coded Plan Technique, 1 (1997) 179--184
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.

K. Shiina, T. Saito, K. Suzuki: Analysis of Problem Solving Process of a Mental Rotations Test -- Performance in Shepard-Metzler Tasks, 1 (1997) 185--194
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.