Analysis of Conics

Abstract This chapter studies various aspects of computations concerning conics. We first describe the representation of conics in terms of N-vectors and discuss fundamental properties of projective geometry such as poles, polars, and conjugate pairs in terms of conics. It is shown that the 3-D geometry of three orthogonal space Jines can be computed by operations involving conics. The procedure for fitting a conic to image points by least squares is also discussed. Then, we describe analytical procedures for interpreting the 3-D geometry of conics in the scene from their projections. Finally, we study how much information is available if a motion of a single conic is observed on the image plane. It is shown that an image motion of a conic is a composition of a motion that maps one conic to the other (a particular motion) and a motion that does not cause a visible change of the conic (an invisible motion). We discuss the isomorphism between groups of invisible motions and a special conic called the standard circle, for which the group of invisible motions is the (three-dimensional) Lorentz group. Similar results are obtained for invisible optical flows, and we discuss the adjoint transformation of optical flows and the quotient space of the linear space of optical flows modulo the linear space of invisible flows. Their 3-D interpretation is also discussed.