THE RELIEF PERSPECTIVE CAMERA. A JOURNEY INTO DEEP SPACE

This paper describes the functioning of the Relief Perspective Camera (RPC) and its application to some projective transformations. This Camera has been designed through mathematical representation in a generative modelling system. It allows managing polygonal surfaces and curves in real time. Essentially this perspective Camera, as the ones used by the Renaissance painters, creates perspectives automatically: these perspectives can be two-dimensional, i.e. classical perspective views as a photography, or relief perspectives that expand in three-dimensional space. The Camera allows understanding and seeing how classical linear perspective can be conceived as a particular case of relief perspective: a bijective relation exists between projective anisotropic space and the affine isotropic space. This geometrical correlation can be described by simple equations that connect the points of the projective space with the points of the affine space and vice versa. The PRC allows us to appreciate in real time the relation between the two spaces. We can observe and experiment the compression and dilatation of the projective space. This PRC camera can transform architectures or achieve theatrical stage effects. Equally fascinating is to observe how this machine can be applied to study geometry and its projective transformations: a round ruled hyperboloid, crossing the second limit plane, can be transformed into another elliptical hyperboloid and, touching the second limit plane, can be transformed into a hyperbolic paraboloid. We can consider the two limit planes as two gates for travelling. This paper describes what happens when an object passes through the limit gates and takes a journey into the deep space.