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.
- 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.
- We study piecewise linear approximation of quadratic functions defined
on Rn. 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.
- The aim of this paper is to give a survey on analytic representations of
central and orthographic projections from R4 to R3 or
R2. 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 R3 can be obtained by
applying particular projections on a point with given cartesian coordinates
in R4. We would also like to demonstrate that by projecting curves
or 2-surfaces of R4 interesting shapes in R3 and
R2 can be obtained.
- 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.
- 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.
- 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.
- 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.
- 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.
- This article presents generalizations of the theorems of Ceva and
Menelaus for n-dimensional Euclidean space.
- 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.
- 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.
- 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]).
- It is shown that the examples presented 1998 by A. Walz are special cases
of a more general class of flexible cross-polytopes in E4. 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.
- A point like light source in Rd induces a certain illumination intensity
at hypersurface elements of Rd. 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 C2-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.
- 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.
- 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.
- 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.
- 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.