3d on Paper and Screen Displaying Three‐dimensional Figures Axonometric and Perspective Representation

The task of classical descriptive geometry is to produce precise, reconstructible drawings on three-dimensional figures; nowadays, in the case of drawings made by means of computer, this is often completed with movable, interactive parts. This work is a brief introduction to the axonometric and perspective representation, the tools for preparing descriptive images, keeping in view some principal aspects. We discuss both the ruler-and-compass construction methods and those in which computers are applied. The emphasize is not on the tutorial aspects of a software made for such purpose, but rather, on its mathematical backgroumd. Our aim is that to develop a certain awareness of our readers, in the sense that they not only look at either a hand-made or a computer-produced drawing with more appreciative eyes, but they themselves be able to prepare such drawings, or, with suitable programming knowledge, their own softwares as well. There are files attached to almost all the drawings presented here, which were made by the dynamic geometry softwares, either Euklides or Euler3D. These softwares can be downloaded, and basics for their use are also given here. We suggest our readers to study this writing together with interactive use of these softwares. 1. The tools of descriptive geometry Nowadays the computers supply beautifully designed, moving, furthermore, interactively movable drawings of real, or just virtual, spatial figures. In most cases they unburden us of the construction of these drawings; at the same time, they also deprive us of the pleasure of drawing. One cannot be content with passive reception of the view; even if one cannot make such beautiful drawings, one should know what happens while a spatial figure is produced on the paper sheet or on the screen. We would like to develop such awareness of our readers, in the sense that they not only look at either a hand-made or a computer-produced drawing with more appreciative eyes, but they themselves be able to prepare such drawings. To this end, we will look over the basics of descriptive geometry. The task of descriptive geometry is to produce a clear, uniquely reconstructible image of spatial figures, such that one should be able to read out from it both the structural and metric data of the given geometric figure. Its method is the projection: one project the geometric figure to be represented onto one or more (image) plane (the plane of the paper sheet or of the screen) by straight lines, which are parallel, or start from a common point. If the primary aim of the representation is simpler constructibility, easy reconstructibility, or that the metric data should be faithfully represented, then the most suitable method is the Monge projection. If one applies, as usual, only two image planes, than in order to uniquely represent e.g. a polyhedron (a solid body with planar side faces), one has to label the vertices, and also the two distinct images of each of them. Without such labeling, one Gaspard Monge (1746–1818) French mathematician, after the French Revolution Minister of the Navy, military organizer.