**Journal for Geometry and Graphics**

**Volume 4 (2000)
Abstracts**

G. Glaeser, H.-P. Schröcker: Reflections on Refractions, 4 (2000) 001--018- In computer graphics, it is often an advantage to calculate refractions
directly, especially when the application is time-critical or when line
graphics have to be displayed. We specify efficient formulas and parametric
equations for the refraction on straight lines and planes. Furthermore, we
develop a general theory of refractions, with reflections as a special case.

In the plane case, all refracted rays are normal to a characteristic conic section. We investigate the relation of this conic section and the diacaustic curve. Using this, we can deduce properties of reciprocal refraction and a virtual object transformation that makes it possible to produce 2D-refraction images with additional depth information.

In the three-dimensional case, we investigate the counter image of a straight line. It is a very special ruled surface of order four. This yields results on the order of the refrax of algebraic curves and on the shading of refracted polygons. Finally, we provide a formula for the diacaustic of a circle.

H. Abdelmoez, Y. Aly Abas: On the Bisectors of Weakly Separable Sets, 4 (2000) 019--030- This article plagiarizes the original article of Lee R. Nackman and V. Srinivasan,
Bisectors of Linearly Separable Sets, Discrete and Computational Geometry 6(1) (1991) 263--275.

The editors regret the publication of the copied article.

H. Pottmann, R. Krasauskas, B. Hamann, K. Joy, W. Seibold: On Piecewise Linear Approximation of Quadratic Functions, 4 (2000) 031--053- We study piecewise linear approximation of quadratic functions defined
on R
^{n}. Invariance properties and canonical Cayley/Klein metrics that help in understanding this problem can be handled in arbitrary dimensions. However, the problem of optimal approximants in the sense that their linear pieces are of maximal size by keeping a given error tolerance, is a difficult one. We present a detailled discussion of the case n = 2, where we can partially use results from convex geometry and discrete geometry. The case n = 3 is considerably harder, and thus just a few results can be formulated so far.

S. Zacharias, D. Velichova: Projection from 4D to 3D, 4 (2000) 055--069- The aim of this paper is to give a survey on analytic representations of
central and orthographic projections from R
^{4}to R^{3}or R^{2}. There are discussed various aspects of these projections, whereby some special relations were revealed, e.g., the fact that homogeneous coordinates or barycentric coordinates in R^{3}can be obtained by applying particular projections on a point with given cartesian coordinates in R^{4}. We would also like to demonstrate that by projecting curves or 2-surfaces of R^{4}interesting shapes in R^{3}and R^{2}can be obtained.

C. Bovill: Fractal Geometry as Design Aid, 4 (2000) 071--078- From Charles Jencks in England to Itsuko Hasegawa in Japan, there is discussion in the architectural press of chaos, fractals, complexity theory, and self-organization. Architecture and design should be informed by and express the emerging scientific view that the world around us is more chaotic and complex than previously thought. However, the architectural response has a tendency to be fairly shallow. Twists and folds and waves, jumps in organizing grids, and superposition of different ordering systems are used to express in architectural form the new scientific ideas about complexity. These are moves in the right direction toward connecting architecture with contemporary cosmic concepts. However, knowledge of the mathematics of fractal geometry can provide a path to an even deeper expression.

A. M. Farag, G. Weiss: Reconstruction of the Satellite Orbit via Orientation Angles, 4 (2000) 079--088- This paper presents an efficient geometric method to find the mathematical model for the normal orbit of a moving satellite observed from a given station on the earth. The method relies on getting a sufficient number of observations oriented from the earth station to the satellite which moves on its predictable orbit on the celestial space. The concurrence of the revolution of the earth and the motion of the satellite is utilized to orient the calculated normal orbit in its fixed plane. Rather than deriving the geometric model for the case of a known orbital plane, we reformulate the method of solution to study the case of an unknown orbital plane. Since the earth station rotates with the earth and the satellite moves, the lines of observation are generatrices of a ruled surface with the elliptic orbit as one directrix. In this paper we assume that the satellite obeys the Keplerian laws and that the true anomaly of the orbit is the only time-dependent Kepler element.

A. C. Clark, B. Matthews: Scientific and Technical Visualization: A New Course Offering that Integrates Mathematics, Science and Technology, 4 (2000) 089--098- This paper is an explanation of the Scientific and Technical Visualization project that North Carolina State University, the North Carolina State Department of Public Instruction, and Wake Technical Community College created as a joint effort funded by a Tech-Prep Innovation Grant. The purpose of this effort was to develop a model program to improve science and graphics instruction in North Carolina. This improvement consists of the use and integration of three specific components: the Scientific and Technical Visualization curriculum, Scientific and Technical Visualization tools, and technology.

T. Yonemura, S. Nagae: Design Procedure on a Newly Developed Paper Craft, 4 (2000) 099--107- 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.

H. S. M. Coxeter: Five Spheres in Mutual Contact, 4 (2000) 109--114- Consider five mutually tangent spheres having (5 over 2) = 10 distinct points of contact. If O is one of these ten points, we obtain by inversion two parallel planes with three ordinary spheres sandwiched between them. Since these three are congruent and mutually tangent, their centres are the vertices of an equilateral triangle. Analogously, if four congruent spheres are mutually tangent, their centres are the vertices of a regular tetrahedron. A fifth sphere, tangent to all these four, may be either a larger sphere enveloping them or a small one in the middle of the tetrahedral cluster. In this article it will be shown that here are fifteen spheres, each passing through six of the ten points of contact of the five mutually tangent spheres.

M. Buba-Brzozowa: Ceva's and Menelaus' Theorems for the n-Dimensional Space, 4 (2000) 115--118- This article presents generalizations of the theorems of Ceva and Menelaus for n-dimensional Euclidean space.

E. Kozniewski, R. A. Gorska: Gergonne and Nagel Points for Simplices in the n-Dimensional Space, 4 (2000) 119--128- Properties of triangles related to so called Gergonne and Nagel points are known in elementary geometry. We present a discussion on some extensions of these theorems. First, we refer to a relation between a tetrahedron and a sphere inscribed into this tetrahedron in the 3-dimensional space. Next, we generalize the obtained results to simplices in n-dimensional geometry. The problem concerning tetrahedra occurs to be no longer as easy to solve as it is for triangles. It has been shown that there are both tetrahedra, which have Gergonne and Nagel points, and tetrahedra with no such a point. We give conditions necessary and sufficient for a simplex to satisfy the Gergonne and Nagel property.

H. Abdelmoez: Generation and Recovery of Highway Lanes, 4 (2000) 129--146- Highway lanes of planar shapes can be defined by specifying an arc or a straight line called the axis and a geometrical figure such as a disk or a line segment called the generator that wipes the internal boundary of the lane by moving along the axis, possibly changing size as it moves. Medial axis transformations of this type have been considered by Blum, Schwarts, Sharir and others. This research work considers such transformations for both the generation and the recovery processes. For a given highway lane generated in this way, we determine the medial axis and the generation rule that gave rise to it.

O. Mermoud, M. Steiner: Visualisation of Configuration Spaces of Polygonal Linkages, 4 (2000) 147--158- Using Morse theory, the configuration space of a 4-gonal (respectively 5-gonal) linkage in the plane or on the unit-sphere can be visualised as a curve in [0, 2p] ´ [0, 2p] (respectively as a surface in [0, 2p] ´ [0, 2p] ´ [0, 2p]).

H. Stachel: Flexible Cross-Polytopes in the Euclidean 4-Space, 4 (2000) 159--168- It is shown that the examples presented 1998 by A. Walz are special cases
of a more general class of flexible cross-polytopes in E
^{4}. The proof is given by means of 4D descriptive geometry. Further, a parameterization of the one-parameter self-motions of Walz's polytopes is presented.

G. Weiss, H. Martini: On Curves and Surfaces in Illumination Geometry, 4 (2000) 169--180- A point like light source in R
^{d}induces a certain illumination intensity at hypersurface elements of R^{d}. Manifolds of such elements with the same intensity of illumination are called isophotic. A uniformly radiating light source causes isophotic strips along sinusoidal spirals. In the present paper this investigation is extended in two directions. First all isophotic C^{2}-hypersurfaces are found, and also manifolds of hypersurface elements which are isophotic with respect to two and more central illuminations are discussed. It suggests itself to treat such illumination problems also in non-Euclidean spaces. The second part of the paper deals with the generating curves of isophotic strips. They belong to the well-known families of Clairaut curves and sinusoidal spirals. Their known relations to each other and to other curve families (such as Ribaucour curves and roses) are extended by some perhaps new aspects.

M. Amrani, F. Jaillet, B. Shariat: Deformable Objects Modeling and Animation: Application to Organs' Interactions Simulation, 4 (2000) 181--188- we describe a methodology for the calculation and animation of volumetric deformable objects. The goal of this work is to obtain realistic models of internal organs in order to simulate their motion and their form alteration during a radiotherapy process. Thus, these models should be able to represent the internal movements due to rhythmic motions, respiration, filling/emptying processes and organs' interactions. So we show how this can be done using particle systems and implicit surfaces and how to mix both models in an hybrid scene making organ's interaction simulation easier.

A. Blach: Determination of Thickness of Rotary Building Shells, 4 (2000) 189--196- This paper presents a non-invasive method for determining the thickness of rotary building shells. The method is based on the geometric locus of centres of circles which pass through a given point and intersect a given circle at angles of given measure.

P. Rubinowicz: Chaos and Geometric Order in Architecture and Design, 4 (2000) 197--208- Since the beginning of human history, the geometric order and chaos exists in the architectural and urban structures together. In context of future dissertation, this paper presents an opinion, that for a good quality of architectural space the balance between order and chaos is necessary. The architectonic space is created by design and other self-organising processes as well. In the long term it is unforeseeable and unstable. The development of the chaos theory creates a new perspective for better understanding of chaos and complex processes in architecture. Some aspects of this theory can by applied in design.

C. Pütz: Descriptive Geometry Courses for Students of Architecture -- On the Selection of Topics, 4 (2000) 209--222- Descriptive Geometry is an applied mathematical discipline dealing with the practical performance of the process of representation as well as with the analysis and generation of objects in three-dimensional space by methods of drawing. Due to the decreasing share of Descriptive Geometry in the curriculum of architectural studies it is no longer possible to teach Descriptive Geometry even roughly to its full extent. As contribution to the development of a curriculum "Descriptive Geometry for architects" the geometrical topics actually used by professional architects as well as those which assist the student directly or indirectly in developing skills fundamental in the daily work of professional architects are explored.