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Short Information on 
the three CD-ROMs


It is the intention of these CD-ROMs to show to the user the beauty of geometry and to animate him into creating geometrical figures on his own.

All three CD-ROMs were widely reviewed in computer journals, weekly journals and daily newspapers. In 1996-97 the Beauty of Geometry and the Fascination of Interactive Computer Graphics were shown in the Deutsches Museum in Munich on the occasion of the exposition Digital Worlds. In addition the three CD-ROMs are also on permanent show in the Digital Workshop of the Heinz Nixdorf Museum in Paderborn.

Beauty of Geometry, Science and Aesthetics

This CD-ROM represents a multimedia collection of 620 pages with innumerable hyperlinks. It is divided into two chapters:

Software for Geometry

This chapter contains vector graphics as a direct application of the Software for Geometry. In 2D-graphics a variety of classical curves such as astroids, rosettes, cycloids, etc. are discussed. Furthermore families of those curves are formed. In 3D-graphics planes, rotational surfaces (sphere, cone and cylinder) and Platonic and Archimedic bodies are introduced, leading to many surprising motifs. All surfaces can be cut out in complex forms with the curves introduced in 2D-graphics. At the same time it is possible to paint these surfaces with families of these curves, also with very sophisticated ornamentation.

There are 400 different motifs which can be seen. In addition all these motifs are saved as vector graphics and can be loaded and run. This is especially interesting because it shows the genesis of a motif giving insight also into the geometrical construction. During the drawing of such a motif one can change colors, the width of the pen and other parameters. There are also 10 demos showing the history of the Software for Geometry and various application in Differential Geometry.

Ray Tracing

This chapter gives an application of the Software for Geometry to ray tracing. Although the technique of ray tracing is completely different to that of vector graphics, the methods of the Software for Geometry are used to create very complex photo realistic images. There are sections on illumination models, acceleration techniques and 2D- and 3D-textures. In each of these sections a theoretical introduction is also given. All this leads to 220 motifs, including fractal motifs, Sierpinsky-Tetrahedron, scenes with chessmen and glasses, still lives, etc.

One of the chessmen scenes won the Computer Graphics Forum cover photograph prize competition in 1995.


Fascination of Interactive Computer Graphics

In 1995 the appearance of Borland’s Delphi made it possible to create interactive programs with the Software for Geometry very easily. The CD-ROM Fascination of Interactive Computer Graphics includes 30 interactive programs for 2D- and 3D-graphics. Here the user is able to realize his own artistic ideas. He can change the geometrical scene, the type of ornamentation on the surfaces or the background, the kind of curves used on the surfaces, the colors, the width of the pen and many more parameters. With simple movements of the mouse he can arbitrarily change the perspective. Furthermore he can save the results as bitmaps and HPGL-Files.

The graphical user interface of the CD-ROM starts with an introduction and a user’s guide. In a short and not always completely serious manner a brief introduction is given to the history of Euclidean Geometry and Analytic Geometry. Among the demos there is a demo on the Beauty of Geometry and 7 demos of applications to Differential Geometry. Furthermore there are various articles on the „Software for Geometry„ and references to the literature. In 2D-graphics various interactive programs on families of curves are presented. In 3D-graphics there is a brief theoretical introduction to the representation of planes, rotational surfaces and Platonic bodies. Many interactive motifs using these elements are presented. The background can be formed with 12 different kinds of ornamentations. 

The Fascination of Interactive Computer Graphic" was awarded the German Education Software Prize digita 96 at the 1996 book fair in Frankfurt.

Wunderworld of Interactive Computer Graphics

With this CD-ROM the authors have tried - even more than in the preceding CD-ROM - to lead the user in an intuitive way through handling the interactive programs. There are 53 interactive programs with 44 variations. The idea of the variations is to give an impression of how different the results can be obtained from the same program by simply altering a few parameters .

Now there are 20 kinds of ornamentations that can be used for the background as well as for the surfaces. Each change of a parameter can be tested immediately! This is very important because most of the programs are very complex. Again the perspective can be changed by simple movements of the mouse and the results can be saved as bitmaps and HPGL files.

As a new feature, a guided tour has been created as an insight in the workshop revealing some of the programming secrets. Among the demos one finds a demo on the CD-ROMs Beauty of Geometry and Fascination of Interactive Computer Graphics as well as 7 demos of applications to Differential Geometry.

In the chapter on ornamentations a new type of fill-ornamentation is introduced serving as a prototype for tiling surfaces. As applications of Platonic and Archimedic bodies there are interesting applications such as Euro-football, Adeno virus and the C60 molecule. There are also five Impossible Figures à la Escher and a gallery of 11 even more complex scenes.

Downloads from the Homepage

Recently there have been three initiatives of the Software for Geometry to the Internet. The purpose is to establish a public computer graphics forum for the three CD-ROMs and the Software for Geometry.


Picture of the month

Under this title a new interactive program for generating complex pictures is presented each month. The interactive executable can be downloaded and permits almost an infinite number of variations. The style of the programs follows those of the interactive CD-ROMs. Hence the newly created motifs can be saved as bitmaps or HPGL files. 


Under this title one can enter a gallery of complex motifs created using the programs of the month and the programs on the three CD-ROMs. The pictures are commented, they can be enlarged and of course downloaded

Program of the Month


Porting Turbo-Pascal-Programs to Delphi-Projects

All three CD-ROMs have been created with the Software for Geometry, which has been developed over a period of eight years at the Mathematical Institute of the Justus-Liebig-University in Giessen (Germany). It has been published in two volumes. These books are in German, however the source code is entirely in English.

The Software for Geometry was presented at the Cebit ´93 and the Cebit ´95 at the booth of the University of Giessen. In 1993 - at the occasion of the University-Computer-Forum in Berlin - it was awarded the German-Austrian University-Software Prize as Best Teaching Software in Mathematics.

Meanwhile Turbo-Pascal with all its glorious merits has been followed by Borland’s Delphi, an interactive development environment for creation of graphical user interfaces for MS-Windows. This software contains many more features and is logically much more complicated than even the most advanced versions of Turbo-Pascal. Therefore the transition has to be considered a big and complicated step. Now two master projects can be downloaded – for 2D-Graphics and 3D-Graphics – which allow the fast port of all object-oriented Turbo-Pascal programs of the Software for Geometry to Delphi projects. In addition every month there can be downloaded a Turbo-Pascal program for porting to a Delphi project.