Computational Projective Geometry, 2

Abstract Projective geometry in the form described in Chapter 2 does not involve any metrics (e.g., lengths and angles). It certainly suits the description of “parallelism” in 3-D scenes in terms of vanishing points and vanishing lines. However, “orthogonality” is also an important characteristic of 3-D scenes, since many man-made objects have rectangular shapes and orthogonal edges. In this chapter, we show that orthogonality in 3-D scenes can be described in terms of projective geometry if we introduce a special “conic” called the absolute conic. It is shown that the camera rotation transformation introduced in Section 2.3.1 is characterized as a collineation that preserves the absolute conic. It is also shown that the conjugacy with respect to the absolute conic can be interpreted as orthogonality in 3-D scenes. We then discuss two machine vision applications: camera calibration and 3-D road shape reconstruction, the latter being an important component of autonomously navigating land vehicles.