Prof. Dr. Kurt Endl 

Software for Geometry

There have been two reasons for the development of this software:

Mathematical Education

The "Software for Geometry" enables the student on the programming level to transfer his geometric problems to programs in objectoriented Pascal. It can be described as a box of various programming tools for the treatment of geometrical problems and their realization on the computer. All necessary tools and methods are mathematically developed and immediately transformed in types, procedures, functions ect. In applying this software the student does not only learn to understand and handle the geometric background of a problem but also to transfer it into a program. This opens wide areas for realizing ideas of his own.

Solution of Geometric Problems in Research

In many fields of research in science and industry there are geometric problems which cannot be handled by any commercial software. The „Software for Geometry" allows the comfortable and fast construction of programs for solution of such problems. Applications have been made in a variety of fields: simulation and analysis of curves in medical problems, construction of form shoulders, approximation of curves by lines and circles for cutting plotters and  various problems in Descriptive Geometry.  Impressive applications have also been made in Differential Geometry and Ray Tracing.

The „Software for Geometry" has been developped over a period of eight years at the Ma-thematical 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"


„Best Teaching Software in Mathematics".

The programs of these volumes can be easily ported to Delphi. Examples of programs

ported from Computergraphics 2 to Delphi can be found on the CD-ROM

Wunderworld Interactive Computer Graphics

For further information see the homepage:


To demonstrate the possibilities of the „Software for Geometry" three CD-ROMs have been produced in the years 1994-1997:

Beauty of Geometry 

Science and Aesthetics

Faszination Interactive Computer Graphics

Wunderworld Interactive Computer Graphics


The intention of these CD-ROMs is not only to show the various possibilites and applications of the „Software for Geometry" but also to seduce the user to appreciate the beauty of geometry and to animate him creating geometric figures by 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 „Faszination of Interaktive Com-puter Graphics" were shown in the „Deutsches Museum" in Munich at the occasion of the exposition „Digital Worlds" . The „Faszination of Interaktive Computer Graphics" was awarded - at the bookfair 1996 in Frankfurt - the

„German Education Software Prize digita 96"

In addition the three CD-ROMs are also permanently shown in the „Digital Workshop" of the Heinz Nixdorf Museum in Paderborn.

Two summaries of Computergraphic 1,2

Computer Graphics 1

In this volume the authors present a "Software for Geometry" developped in Turbo-Pascal 4.0. In systematic form the geometrical tools for computer graphics are developped. The theoretical reasoning is immediatly transformed to types, functions and procedures which are collected in seven units and twelve include files. These tools allow the immediate realization of complex 2D- and 3D-Graphics. 

In the toolbox the user finds - besides the procedures for the 2D- and 3D-vector calculus - important procedures for establishing 2D- and 3D-coordinate systems, for rotation of points, lines and planes around around arbitrary axes, furthermore for cutting lines with planes and planes with planes ect. In addition a perspective in the most genereal form is contained. 

Applications of this software are presented in 82 programs, where the use of the diverse toolbox items is thoroughly explained. Starting with programs in 2D-graphics, the user will be led to complex programs for the representation of platonic bodies and the rotation surfaces sphere, cone and cylinder. At the same time arbitrarily many of these bodies resp. surfaces can be drawn including their intersectional lines. Here it is also possible to cut out the bodies resp. surfaces themselves. Methods for producing ornamentics are presented as well as for 2D- as for 3D-graphics. All programs are at the same time screen- and plot-programs where the HP-language is used. 

The complete source code - units, include files und programs - is on the disc. Immediately after installation the user can load and run a program.

Computer Graphics 2

With the software of Computergraphics 2 it is possible to create 3D-scenes with arbitrarily many planes, rotation surfaces (sphere, cone and cylinder) and platonic surfaces (tetrahedron, cube, octahedron, dodecahedron and icosahedron). Here, all surfaces can be cut out in arbitrarily complex form using families of star curves, polygons and leaves. At the same time also the background can be formed in manyfold ways. In addition all surfaces can be provided with different ornamentics inclusive text. 

In constrast to the software of Computergraphics 1 which was programmed with Turbo Pascal 4.0, now the software is conceived in objectoriented dynamic programming. It runs on Turbo Pascal 5.5 and up. 

Computergraphics 2 is divided  in two parts: 

The first part is the application part and contains 89 programs. Here - in very concise form - all object types together with the decisive methods are presented and immediatly applied in exemplary programs. This part does not suppose any knowledge of Computergraphics 1. The second part is the implementation part. Here the geometrical methods are used which have been developped in Computergraphics 1. The corresponding text is put in a single unit, whose declaration part is given in the annex. To understand this second part some reference to Computergraphics 1 cannot be avoided. 

Again all programs are at the same time screen- and plot-programs where the HP-language is used. The total text - 89 programs and 12 units with 78 object types - include files und programs - are on the 51/4 disc. Immediately after installation the user can load and run a program.

Application to Ray-Tracing

There have been various applications of the „Software for Geometry" to Ray Tracing. One of them is the a contribution to the problem of Ray Tracing Acceleration Techniques. Extensive Information on this topic are to be found on the CD-ROM „Beauty of Geometry".


Applications to Differential Geometry

Quite a number of applications of the „Software for Geometry" were made to Differential Geometry. In this direction two members of the Mathematical Institut at the University of Giessen – Eberhard Malkowsky and Wolfgang Nickel – extended the „Software for Geometry" to a „Software of Differential Geometry". This again is an objectoriented software in Pascal on the programming level. Here the whole arsenal of notions and methods of Differential Geometry has been added. Quite a number of applications of this software has been made since the appearance of this book. 

Another interesting application was made by Markus Failing, at the same Institute. He investigated with methods of Differential Geometry and Computer Graphics the famous theorie of G. Wulff on the growth of crystals and their connection with potential surfaces.

Information on these applications are found - together with demos - on the CD-ROM „Wunderworld Interactive Computer Graphics".

Downloads from the Homepage

On the homepage you will find full information on the books, the CD-ROMs, various prizes, literature, press reactions, expositions, information on the authors ect. Many programs can be downloaded. 

Prof. Dr. Kurt Endl 10.12.2001 

Hainstraße 27 Tel 06409-6516 
D-35444 Biebertal Fax 06409-6433