Out of frame: expanding perspective into real space

It might sound strange to hear that our - locally - Euclidean environment can be thought as a special case inside the Projective space. According to linear perspective, indeed, we use to think at the opposite situation, or at projective images as special cases inside the isometric Euclidean space. But what happens when perspective breaks down the flatness of the picture plane (π) expanding its pattern beyond or across it? It becomes a three-dimensional space as well, distorted according to the projective pattern, namely a Relief-perspective, whose illusionary strength is empowered by its physical accessibility. Paraphrasing Rudolph Arnheim, it becomes a polarized pyramidal space whose ideal apex (A), working as a vanishing point, can stay in front as well as behind the sight point (S), along the viewing line (Fig. 1). In the first case an accelerated perspective illusion is generated, where represented space appears longer than it actually is (i.e. apex at Aa). In the second case, a decelerated perspectival illusion is achieved, where represented space appears shorter than it really is (i.e. apex at Ad). The two effects can also be combined in the same space. In both situations the illusion can be more or less emphasized by increasing or decreasing the distance between the perspective apex and viewpoint. From a geometrical point of view, the unlimited range of visually equivalent projective configurations descends from the non-bijective DNA of any single projection. Therefore two limit cases emerge: when the apex belongs to picture plane π (Ao) a pla-nar perspective is generated; when it moves at infinity (A∞) we are back to Euclidean space. In this sense Euclidean space can be seen as a special case of projective space. From an historical point of view, Leonardo Da Vinci already stated that perspective painting is based on two opposite pyramids, one having its apex at the sight point, the other at the horizon distance. Geometrical principles and graphic applications carried on by either traditional procedures or digital simulations will be dis-cussed, emphasizing the homological relationships among real space, relief-perspective spaces, perspective images. Last but not least, the educational power of this approach in the architectural design curricula will be highlighted, and some selected projects concerning three-dimensional thrompe l’œil, developed by master students from the course Geometrical Complements of Graphic Representation at the Polytechnic of Milan, will be shown.