Jean Daniel Boissonnat

A practical exact motion planning algorithm for polygonal objects amidst polygonal obstacles

A general and simple algorithm is presented which computes the set FP of all free configurations for a polygonal object I (with m edges) which is free to translate and/or to rotate but not to intersect another polygonal object E. The worst-case time complexity of the algorithm is O(m/sup 3/n/sup 3/ log mn), which is close to optimal. FP is a three-dimensional curved object which can be used to find free motions within the same time bounds. Two types of motion have been studied in some detail. Motion in contact, where I remains in contact with E, is performed by moving along the faces of the boundary of FP. By partitioning FP into prisms, it is possible to compute motions when I never makes contact with E. In this case, the theoretical complexity does not exceed O(m/sup 6/n/sup 6/ alpha (mn)) but it is expected to be much smaller in practice. In both cases, pseudo-optimal motions can be obtained with a complexity increased by a factor log mn. >

Polygon Placement Under Translation and Rotation

We present a gênerai algorithm which computes an exact description of the set of ail placements for a polygon I (with m edges) which is free to translate and/or to rotate but not to intersect another polygon E (with n edges). The time complexity of our algorithm is O(mn log mn) which is close to optimal in the worst-case. Moreover, in some practical situations, the time complexity is only O (n log n). This algorithm is rather simple and has been implemented. It can be used as an efficient tool in several applications such as cutting stock, inspection and motion planning for a two dimensional robot admidst polygonal obstacles. Résumé. Cet article présente un algorithme général qui calcule une description analytique exacte de Vensemble des placements dun polygone I (ayant m arêtes) libre de se déplacer en translation et rotation dans le plan sans intersecter un polygone E (ayant n arêtes). La complexité de lalgorithme est O (m n log mn) ce qui est proche de loptimal dans le cas le pire. On montre de plus que la complexité réelle de Valgorithme, dans certaines situations pratiques est O (n log n). Valgorithme présenté est assez simple et a été implanté. Il recouvre un champ dapplications varié incluant les problèmes de découpe automatique, de conformité de pièces industrielles ou les problèmes de planifications de trajectoires pour un robot mobile plan évoluant au milieu dobstacles polygonaux.

3D volumetric modelling of Cadomian terranes (Northern Brittany, France): an automatic method using Voronoı̈ diagrams

Abstract An automatic method based on the use of Voronoi diagrams is proposed for the 3D volumetric reconstruction of geological objects. It enables volumes to be constructed starting from the combined use of geological maps and cross-sections defined in multiple directions, and can also take into account incomplete information on geological interfaces and faults. The method is suitable for global modelling of a set of geological objects; it gives a consistent 3D volumic partition of space according to geological information contained in the maps and the cross-sections. The method is described and applied to modelling the main domains of the Cadomian collisional orogenic belt of Panafrican age in Northern Brittany (France). Thanks to the contrasting densities of the main units involved in the collisional process, the gravimetric effect of the volumic model can be calculated. Its comparison with the observed anomaly allows us to discuss the validity of the geological model. Although some improvements appear to be necessary, it is concluded that the method can make geological modelling more efficient through providing the possibility of rapidly testing many hypotheses.

Steps Toward the Automatic Interpretation of 3D Images

We describe in this paper part of the research being performed at Inria on the automatic interpretation of three-dimensional images. We identify three common key problems which we call segmentation, representation, and matching of 3D regions. We describe our approach for solving these problems, our current results on 3D medical images, and give the trends of our future work.

A semidynamic construction of higher-order voronoi diagrams and its randomized analysis

Thek-Delaunay tree extends the Delaunay tree introduced in [1], and [2]. It is a hierarchical data structure that allows the semidynamic construction of the higher-order Voronoi diagrams of a finite set ofn points in any dimension. In this paper we prove that a randomized construction of thek-Delaunay tree, and thus of all the order≤k Voronoi diagrams, can be done inO(n logn+k3n) expected time and O(k2n) expected storage in the plane, which is asymptotically optimal for fixedk. Our algorithm extends tod-dimensional space with expected time complexityO(k⌈(d+1)/2⌉+1n⌊(d+1)/2⌋) and space complexityO(k⌈(d+1)/2⌉n⌊(d+1)/2⌋). The algorithm is simple and experimental results are given.

Efficient algorithms for line and curve segment intersection using restricted predicates

1 Introduction We consider whether restricted sets of geometric predicates support efficient algorithms to solve line and curve segment intersection problems in the plane. Our restrictions are based on the notion of algebraic degree, proposed by Preparata and others as a way to guide the search for efficient algorithms that can be implemented in more realistic computational models than the Real RAM. Suppose that n (pseudo-)segments have k intersections at which they cross. We show that intersection algorithms for monotone curves that use only comparisons and above/below tests for end-points, and intersection tests, must take at least Q(n&) time. There are optimal O(n log n + k) algorithms that use a higher-degree test comparing x coordinates of an endpoint and intersection point; for line segments we show that this test can be simulated using CCW() tests with a logarithmic loss of efficiency. We also give an optimal 0(n log n + k) algorithms for red/blue line and curve segment intersection , in which the segments are colored red and blue so that there are no red/red or blue/blue crossings. Permission to make digital or hard copies of all or part ofthis work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the tirst page. To copy otherwise, to republish, to post on servers or to redistribute to lists. requires prior specific permission and/or a fee. All too often, a proof that a geometric algorithm is correctness for the Real RAM computational model [21] does not imply that a correct implementation will run correctly on the limited precision arithmetic of a real computer. This fact has spurred three branches of research: First, researchers have studied how to correctly and efficiently evaluate predicates used by geometric algorithms. Much recent work has been devoted to combining floating point filters and exact evaluation of predicates; exact computation is performed when the floating point filter fails to provide a certified answer, which is usually rare. New methods have been designed for the exact evaluation of signs of determinants and arithmetic expressions [13, 2, 71, and various exact, adaptive arithmetics [9, 22, 23,251, and various floating point filters, both static and dynamic, have been experimentally tested [S, 15, 61. Second, researchers have investigated algorithms that give approximate results with prov-able properties and …

An optimal algorithm for the boundary of a cell in a union of rays—corrigendum

PROOF. Without loss of generality we assume that S = {el . . . . . e,} is a connected set of segments, otherwise the argument is applied separately to each connected component of S. In addition, we identify 1 with the x-axis, so that each line segment e i has an endpoint p+ with positive ordinate and the other P7 with negative ordinate. We denote by A a region of the planar subdivision induced by S. The condition that all segment endpoints lie on a given convex curve implies that no internal region of the subdivision induced by S may contain one such endpoint. Therefore, each internal region A has O(n) edges, because each segment of S may contribute at most one edge to it. Thus, let us assume that A is the unbounded region of the subdivision. We consider segment ei as the intersection of two rays r + with terminus p+ and r7 with terminus PT. Let R + = {r+: i = 1 . . . . , n} and R= {r/-: i = 1, . . . , n}. We consider four distinguished rays, specifically: r~and r + , the first and last elements of R + (in their ABY-order), and analogously defined, r{, and rR,. These four rays partition the boundary of A into four portions, naturally denoted upper (between p~ and p~), right (between pff and PL), lower (between PL and PR), and left (between PR and p~). The assumption on the endpoints lying on a convex curve permits us to replace segments with straight lines without altering the subdivision in the interior of this curve. It follows that the upper and lower portions are subsets of the external boundaries of R + and R, respectively, and each has O(n) edges by Theorem 1. Let

A geometric approach to inspection

An exact solution is presented to the following inspection problem in two dimensions. A manufactured part is desired to have a polygonal shape E/sub d/. Tolerances are given in terms of two polygons, one, called E/sup -/ lying within E/sub d/, and the other, called E/sup +/, lying outside E/sub d/. A vision system yields a description of each manufactured part as a polygon I. It is desired to determine whether I can be translated and rotated to lie between the tolerance polygons E/sup -/ and E/sup +/, in which case the part is considered to conform to the desired tolerances.<<ETX>>

Scene reconstruction from rays application to stereo data

The problem of reconstructing shapes of objects from sparse measurements such as points on the boundary of an object is considered. In most situations, the points are the endpoints of a curve or a ray which does not cross the objects. For example, if the sensor is an optical device, the ray is the straight line (the optical ray) joining the camera center to the point. It is shown that the information provided by rays is crucial when determining the shapes of objects, and nonheuristic reconstruction methods in 2-D and 3-D space are described. An efficient method is derived for the reconstruction of surfaces from 3-D segments provided by a stereo vision process.<<ETX>>