Computational Projective Geometry, 1

Abstract This chapter presents a computational formalism that deals with collinearity of points and concurrency of lines on a 2-D plane from the standpoint of projective geometry. The image plane is regarded as a 2-D projective space, and points and lines are represented by unit vectors consisting of homogeneous coordinates, which we call N-vectors. Fundamental notions of projective geometry such as collineations, correlations, polarities, poles, polars, and conics are all reformulated as computational processes in terms of N-vectors. The camera rotation transformation, which will later play an important role, is defined as a special collineation resulting from rotation of the camera. The cross ratio is expressed in terms of N-vectors, and its invariance is proved with respect to general projective transformations, which include perspective projection. Then, one and two-dimensional projective coordinates based on cross ratios are introduced, and typical applications that take advantage of the perspective invariance of cross ratios are presented. Implications of harmonic ranges are also studied with regard to 3-D interpretation of scenes.