Enumeration problems in geometry, robotics and vision

Abstract

The objective of this work is to give intersection formulas in algebraic problems appearing in mechanics and vision. We begin with problems in ℙ2 where distances are involved. The second section describes the case of surfaces in ℙ3 which have a conic at infinity (called the umbilic) in common. In other words, we are interested here in intersection problems on the space of spheres. A formula is given for the number of common points outside this umbilic. The third section deals with the degree of varieties, corresponding to segments whose extremities are on two curves. Following this progression, the next section is devoted to intersection in the variety of displacements. In this section, we analyze precisely the ring of functions on the variety of displacements and give its multiplicity, which allows us to bound the number of solutions in the direct kinematic problem of a parallel robot and in the problem of reconstruction from points in vision.