The 360° Curvilinear Perspective: A Hybrid Hypercubic Angular Space Grid Based on the 1968 Barre and Flocon proposal

Abstract
We analyze in detail a specific image of curvilinear perspective developed by Andre Barre  and  Albert  Flocon  in  1968  which  depicts  a  square  room  from  its  interior  by  a 360° total perspective. Since its publication, their work has become an important point of reference, but they did not say how it was done. We discuss how this specific case  of  a  pseudo-spherical  projection  for  a  hypercube  was  achieved  by  hybridized  using  ellipse  curves  in  one  of  the  spatial  directions  and  a  different  spherical  projection  in  the  other  direction.  For  this  analysis  we  develop  the  mathematical  expressions   and   study   how   the   ellipsoidal   and   spherical   transformations   are   hybridized into one solution. Barre and Flocon attempted to use both projections in an angular curvilinear grid as a practical drawing method. One of the main interests of having access to a 360° curvilinear grid arises its ability to act as a drawing guide. We provide an example of its use to demonstrate its practicality.