Enumeration Problems in Geometry, Robotics and Vision

نویسنده

  • B Mourrain
چکیده

This paper presents diierent examples of enumerative geometry in robotics and vision. The aim is to obtain intersection formulas giving the number of solutions (on C) of some classes of problems which appear in such applied elds. The rst family of examples involves curves and surfaces dealing with distances. In such problems, the varieties have a common part at innnity, called the umbilic, that we must not take into account in the enumeration. A new formula is obtained by means of simple algebraic manipulations and the approach is compared with the usual techniques of blowing up. In a second part, we consider the variety of displacements from an algebraic point of view. A structure, similar to what is called an algebra with straightening laws, is exhibited and allows us to compute the degree of the algebra representing the functions on the displacements. This approach has an immediate application in the direct kinematic problem of a parallel robot and in the problem of reconstruction from points in vision. The objective of this work is to give intersection formulas in algebraic problems appearing in Mechanics and Vision. We begin with problems in P 2 where distances are involved. The second section describes the case of surfaces in P 3 which have a conic at innnity (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.

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تاریخ انتشار 1994