THE CONNECTION BETWEEN PROJECTIVE SPACE AND AFFINE SPACE AND THE SOLID PERSPECTIVE MACHINE

This paper describes how to construct and use a perspective camera: the solid perspective machine, similar to those used in the past by painters, but able to work in a three-dimensional perspective space. I’m refering, for example, to the machine used by Albrecht Dürer in the sixteenth century. Perspective cameras were used to avoid having to construct a drawing through geometry and thus obtain the desired perspective mechanically. These machines were used to construct views or theat-rical stage effects. Today these are not used anymore except for educational purposes. The reasons are different and the most important is that painters and architects have different tools to get perfect perspectives, i.e. cameras and digital representation methods to automatically get any kind of repre-sentation. However, the perspective machine described in this study allows you to obtain a more general perspective than the usual linear perspective: a solid perspective (or a relief perspective), a perspective that is contained in a three dimensional space and not two-dimensional anymore. The solid perspective machine also allows you to observe in real time the relationship between the pro-jective space, generated by the perspective machine, and the affine space, that of the regular solid geometry. We'll see that linear perspective can be classified as a special case of the solid perspective. This digital camera can be used of course to obtain transformed architectures or to achieve theatrical stage effects. An alternative use, equally interesting, is to observe how this machine can be used to study geometry and, in particular, projective transformations: a round ruled hyperboloid can be transformed into another elliptic hyperboloid and, as a particular case, can be transformed into a hyperbolic paraboloid.