Chapter 8 - Projective maps

Abstract

This chapter highlights the perspectives of projective maps. The idea of projective maps grew out of the perspective rendering techniques developed by renaissance artists. The perspective projection consists of extending a line from each point of the scene to the observer's eye, and marking the point where that line intersects the picture plane. However, even the flat objects appear greatly distorted when viewed in perspective. The projection does not preserve many common geometric properties—angles, distances, areas (or their ratios), parallelism, perpendicularity, congruence, and so forth. The perspective projections do preserve some attributes of the image, however, they always take straight lines to straight lines. Therefore, they also preserve the incidence relations between the points and lines, and all geometric properties that can be defined in terms of incidence. Intuitively, a projective map or collineation is a generalization of the perspective projection map; it is a function from one projective space into another that takes straight lines to straight lines, and therefore preserves the essential geometric structure of its domain.