Imagining projective transformations: aligned orientations in spatial organization

Four experiments were conducted to investigate whether variations in orientation that profoundly affect the ability to imagine rotations also affect the ability to imagine projective transformations. For a basic rectilinear object and the three simpler Platonic Solids, imagining projective transformations (e.g., the casting of a shadow) was quite successful when the objects were aligned with the direction of projection. For the solids, this alignment occurred when the objects were generalized cylinders about axes aligned with the projection. As the objects were made more oblique to the projection, performance deteriorated markedly. When the objects were moderately aligned with the projection, performance depended on the orientation of the object and the orientation of the projection to the environment. We suggest that the imagination of projection and of rotation is a type of problem solving in which spatial structures are organized in relation to initially given properties of the objects and transformations. When there is alignment among the various structural components, this process of imagination works efficiently. Without such alignment, nonexperts often fail. We suggest that aligned (i.e., parallel and perpendicular) orientations are effective in spatial imagination because they are categorically distinct and singular, and they provide a critical form of redundancy.