1,721,014 research outputs found
Counting projections of rational curves
Get PDFGiven two general rational curves of the same degree in two projective spaces, one can ask whether there exists a third rational curve of the same degree that projects to both of them. We show that, under suitable assumptions on the degree of the curves and the dimensions of the two given ambient projective spaces, the number of curves and projections fulfilling the requirements is finite. Using standard techniques in intersection theory and the Bott residue formula, we compute this number
The diffeomorphism type of small hyperplane arrangements is combinatorially determined
Get PDFIt is known that there exist hyperplane arrangements with the same underlying matroid that admit
non-homotopy equivalent complement manifolds. Here we show that, in any rank, complex central hyperplane arrangements with up to 7 hyperplanes and the same underlying matroid are isotopic. In particular, the
diffeomorphism type of the complement manifold and the Milnor fiber and fibration of these arrangements
are combinatorially determined, that is, they depend only on the underlying matroid. To prove this, we associate to every such matroid a topological space, that we call the reduced realization space; its connectedness,
shown by means of symbolic computation, implies the desired result
Ambiguities in a Problem in Planar Geodesy
No full textThis is a study of a problem in geodesy with methods from complex algebraic geometry: for a fixed number of measure points and target points at unknown position in the Euclidean plane, we study the problem of determining their relative position when the viewing angles between target points seen from measure points are known. In particular, we determine all situations in which there is more than one solution
Moebius photogrammetry
No full textMotivated by results on the mobility of mechanical devices called pentapods, this paper deals with a mathematically freestanding problem, which we call Möbius Photogrammetry. Unlike traditional photogrammetry, which tries to recover a set of points in three-dimensional space from a finite set of central projection, we consider the problem of reconstructing a vector of points in TeX starting from its orthogonal parallel projections. Moreover, we assume that we have partial information about these projections, namely that we know them only up to Möbius transformations. The goal in this case is to understand to what extent we can reconstruct the starting set of points, and to prove that the result can be achieved if we allow some uncertainties in the answer. Eventually, the techniques developed in the paper allow us to show that for a pentapod with mobility at least two, either some anchor points are collinear, or platform and base are similar, or they are planar and affine equivalent
Counting Realizations of Laman Graphs on the Sphere
Get PDFWe present an algorithm that computes the number of realizations of a Laman graph on a sphere for a general choice of the angles between the vertices. The algorithm is based on the interpretation of such a realization as a point in the moduli space of stable curves of genus zero with marked points, and on the explicit description, due to Keel, of the Chow ring of this space
Bond theory for pentapods and hexapods
No full textThis paper deals with the old and classical problem of determining necessary conditions for the overconstrained mobility of some mechanical device. In particular, we show that the mobility of pentapods/hexapods implies either a collinearity condition on the anchor points, or a geometric condition on the normal projections of base and platform points. The method is based on a specific compactification of the group of direct isometries of R^3
Probabilities of incidence between lines and a plane curve over finite fields
Get PDFWe study the probability for a random line to intersect a given plane curve, over a finite field, in a given number of points over the same field. In particular, we focus on the limits of these probabilities under successive finite field extensions. Supposing absolute irreducibility for the curve, we show how a variant of the Chebotarev density theorem for function fields can be used to prove the existence of these limits, and to compute them under a mildly stronger condition, known as simple tangency. Partial results have already appeared in the literature, and we propose this work as an introduction to the use of the Chebotarev theorem in the context of incidence geometry. Finally, Veronese maps allow us to compute similar probabilities of intersection between a given curve and random curves of given degree
A NEW LINE-SYMMETRIC MOBILE INFINITY-POD
Get PDFWe construct parallel manipulators with one degree of freedom and admitting infinitely many legs lying on a curve of degree ten and genus six. Our technique relies upon a duality between the spaces parametrizing all the possible legs and all the possible configurations of a manipulator. Before describing our construction, we show how this duality helps explaining several known phenomena regarding mobility of parallel manipulators
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