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Visualisation - interpretation understanding: Traveling on the royal road to mathematical abstraction in old boots: Descriptive geometry

Weiss Gunter (Institute of Geometry, Dresden University of Technology, Germany)

Sketching and computer visualization are standard communication media in Technology and Natural Science as well as in Mathematics. Of any visualization we demand easy interpretability by the 'educated' viewer not only can 'read' the meaning of the figure but also gains some understanding of the visualized problem. Such a 'visual communication' needs schooling and training. Descriptive Geometry provides some simple but effective rules and techniques for such a visual communication. Using properties classical geometric mappings, e.g. normal projections or cyclography, can give insight to problems, which sometimes are rather hard to tackle purely by mathematical calculation. Sometimes we receive even an easy proof of the problem, a proof 'by looking at the figure', such that one is encouraged to speak of a 'geometric royal road' to the problem. Some special examples of such problems shall illustrate this statement. Most of the shown examples are not new, but they are not at all very well known! What should be shown is that Descriptive Geometry is much more than just an engineering graphics tool for visualizing 3D-objects. It strongly supports mathematicians, too. To emphasize that Descriptive Geometry is an intellectual tool besides for visualization the figures in this work are freehand drawn sketches instead of perhaps more beautiful computer generated drawings.

Keywords: descriptive geometry, geometric mappings, normal projection, cyclography, generalized stereographic mapping, quadrics of revolution, pipe connections