Francis Avnaim and Jean Daniel Boissonnat. Polygon placement under translation and rotation. Technical Report 889, INRIA Sophia-Antipolis, 1988. 145  Home/Search   Document Not in Database   Summary   Related Articles   Check

....ffl Determining if two polygons intersect. ffl Moving a polygon for a given distance along a specified direction. 3 Translating and rotating the polygons is made easy as we define the parts in a coordinate free manner. We have used an algorithm similar to that proposed by Avnaim and Boissonnat [11], which computes all the placements for a polygonal object I (with m edges) which is free to rotate translate but not to intersect another polygonal object E. In order to achieve motion of a polygon in a given direction under the constraint that it does not intersect with any other polygons, we ....

Avnaim, F. and Boissonnat, Jean-Daniel, Polygon placement under translation and rotation, Theoretical Informatics and Applications, vol. 23, pp 5-28, 1989. 7

....Chazelle [6] studied several variants of this problem. One of his results is that, given two convex polygons P and Q, one can decide in linear time whether can be translated such that it is contained in P. Other papers compute the largest copy of a polygon that can be placed inside another one [5, 10, 11, 20]. 2 The number of distinct placements Let P be a simple polygon with n vertices in the plane and let Q be a simple polygon with m vertices. The position and orientation of P are fixed, but Q is free to translate. In this section we bound the number of distinct placements of Q with respect to P. ....

F. Avnaim and J.-D. Boissonnat. Polygon placement under translation and rotation. In Proc. 5th Sympos. Theoret. Aspects Cornput. Sci., volume 294 of Lecture Notes in Computer Science, pages 322-333. Springer-Verlag, 1988.

....Chazelle [6] studied several variants of this problem. One of his results is that, given two convex polygons P and Q, one can decide in linear time whether Q can be translated such that it is contained in P . Other papers compute the largest copy of a polygon that can be placed inside another one [5, 10, 11, 20]. 2 The number of distinct placements Let P be a simple polygon with n vertices in the plane and let Q be a simple polygon with m vertices. The position and orientation of P are xed, but Q is free to translate. In this section we bound the number of distinct placements of Q with respect to P . We ....

Francis Avnaim and Jean-Daniel Boissonnat. Polygon placement under translation and rotation. In Proc. 5th Sympos. Theoret. Aspects Comput. Sci., volume 294 of Lecture Notes Comput. Sci., pages 322333. Springer-Verlag, 1988.

....and an algorithm that computes largest disjoint placements of two polygons in a third. Some papers study the fixed size polygon containment problem, in which (the convex) P is only allowed to translate and rotate and we wish to determine whether there is any placement of a copy of P inside Q [Ch, AB1]. Chazelle [Ch] studies the problem for the case where P and Q are arbitrary simple polygons and presents a naive algorithm that takes time O(k 3 n 3 (k n) log(k n) A more restricted # A preliminary version of the results in this paper apeared in [To] Work on this paper by the first ....

.... University 1 case of the problem, in which both P and Q are convex, is also studied by Chazelle [Ch] who solves this case in time O(kn 2 ) Chazelle gives a simple solution to an even more restricted version in which P is a triangle; this version runs in time O(n 2 ) Avnaim and Boissonnat [AB1] present an algorithm for the case where both P and Q are non convex, possibly non connected polygons, which runs in time O(k 3 n 3 log(kn) In another paper Avnaim and Boissonnat [AB] investigate the problem of simultaneous placement of two or three not necessarily convex polygons in a ....

F. Avnaim and J. D. Boissonnat, Polygon placement under translation and rotation, 5th Annual Symp. on Theoretical Aspects of Computer Science, Lectures Notes in Comp. Science 294, Springer-Verlag, New-York, 1988, 322--333.

....and an algorithm that computes largest disjoint placements of two polygons in a third. Some papers study the fixed size polygon containment problem, in which (the convex) P is only allowed to translate and rotate and we wish to determine whether there is any placement of a copy of P inside Q [Ch, AB1]. Chazelle [Ch] studies the problem for the case where P and Q are arbitrary simple polygons and presents a naive algorithm that takes time O(k 3 n 3 (k n) log(k n) A more restricted case of the problem, in which both P and Q are convex is also studied by Chazelle [Ch] who solves this ....

.... more restricted case of the problem, in which both P and Q are convex is also studied by Chazelle [Ch] who solves this case in time O(kn 2 ) Chazelle gives a simple solution to an even more restricted version in which P is a triangle; this version runs in time O(n 2 ) Avnaim and Boissonnat [AB1] present an algorithm for the case where both P and Q are non convex, possibly non connected polygons, which runs in 32 time O(k 3 n 3 log(kn) In another paper Avnaim and Boissonnat [AB] investigate the problem of simultaneous placement of two or three not necessarily convex polygons in a ....

F. Avnaim and J. D. Boissonnat, Polygon placement under translation and rotation, 5th Annual Symp. on Theoretical Aspects of Computer Science, Lectures Notes in Comp. Science 294, Springer-Verlag, NewYork, 1988, 322--333.

....and an algorithm that computes largest disjoint placements of two polygons in a third. Some papers study the fixed size polygon containment problem, in which (the convex) P is only allowed to translate and rotate and we wish to determine whether there is any placement of a copy of P inside Q [Ch, AB1]. Chazelle [Ch] studies the problem for the case where P and Q are arbitrary simple polygons and presents a naive algorithm that takes time O(k 3 n 3 (k n) log(k n) A more restricted case of the problem, in which both P and Q are convex is also studied by Chazelle [Ch] who solves this ....

.... more restricted case of the problem, in which both P and Q are convex is also studied by Chazelle [Ch] who solves this case in time O(kn 2 ) Chazelle gives a simple solution to an even more restricted version in which P is a triangle; this version runs in time O(n 2 ) Avnaim and Boissonnat [AB1] present an algorithm for the case where both P and Q are nonconvex, possibly non connected polygons, which runs in time O(k 3 n 3 log(kn) In another paper Avnaim and Boissonnat [AB] investigate the problem of simultaneous placement of two or three not necessarily convex polygons in a closed ....

F. Avnaim and J. D. Boissonnat, Polygon placement under translation and rotation, 5th Annual Symp. on Theoretical Aspects of Computer Science, Lectures Notes in Comp. Science 294, Springer-Verlag, New-York, 1988, 322--333.

....sheet metal [35] and leather [20] both of which permit rotations. Chazelle [6] introduced the single polygon containment problem: place m gon P into n gon container Q. ForconvexQ, the running time bound is O(mn 2 ) and for non convex P and Q,O(m 3 n 3 (m n) log(m n) Avnaim et al. [3,4] improve this to O(m 3 n 3 log(m n) Most recent work deals with finding the largest copy of a convex P that can be placed, which is equivalent to finding the minimum (scaled) enclosure. For convex Q the best running time is O(mn 2 log n) 1] and for non convex Q, O(m 2 n 2 ) 2] ....

....This repetition (cascading) can lead to numerical difficulties which we deal with using nearest pair rounding (Section 4) Note: for k = 2, geometric restriction is an exact algorithm. In this special case, it becomes essentially the same as Avnaim and Boissonnat s exact formula for this case [3,4]. If either geometric restriction or linear programming restriction generates the empty set, then there is no solution in the current hypothesis. 2.4. Evaluation We have previously developed a practical algorithm for rotational overlap minimization [32] find translations and rotations that ....

F. Avnaim, J. Boissonnat, Polygon placement under translation and rotation, in: Proceedings of the 5th Annual Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science, Vol. 294, Springer, 1988, pp. 322--333.

....the robot manipulates. People typically do not think of all of these problems as similar, as they correspond to entirely different fields of research. However, some work in robot motion planning and computational geometry has noted the correspondence between placement and motion planning problems [Cha83, CK89, AB88, Avn89]. For example, Avnaim [Avn89] solves placement and displacement (motion planning) problems, noting that the former type is useful in layout applications and the latter in robotics. The relationship between cutting, packing, 1 One possible maximality criterion is the number of objects. Another is ....

....removed. He gives an O(mn 2 ) time algorithm for solving the problem when C is convex, and an O(m n) time algorithm if C is convex and only translations are allowed. He gives a naive algorithm for the general case requiring O(m 3 n 3 (m n) log(m n) time. Avnaim [Avn89] and Avnaim et al. [AB88] give an O(m 3 n 3 log mn) time algorithm for a single nonconvex polygon in a nonconvex container. Their algorithm yields all the solutions. Grinde and Cavalier [GC93] use mathematical programming to solve the rotational containment problem for a convex polygon in a convex container. Their ....

F. Avnaim and J. Boissonnat. Polygon placement under translation and rotation. In Proceedings of the 5th Annual Symposium on Theoretical Aspects of Computer Science; Lecture Notes in Computer Science (294), pages 322--333. Springer-Verlag, 1988.

....[Cha83] studied several variants of this problem. One of his results is that, given two convex polygons P and Q, one can decide in linear time whether Q can be translated such that it is contained in P . Other papers compute the largest copy of a polygon that can be placed inside another one [AB88, CK93, For85, SCK 86] 2 The number of distinct placements Let P be a simple polygon with n vertices in the plane and let Q be a simple polygon with m vertices. The position and orientation of P are xed, but Q is free to translate. In this section we bound the number of distinct placements ....

F. Avnaim and J.-D. Boissonnat. Polygon placement under translation and rota tion. In Proc. 5th Sympos. Theoret. Aspects Comput. Sci., volume 294 of Lecture Notes in Computer Science, pages 322333. Springer-Verlag, 1988.

....ffl Determining if two polygons intersect. ffl Moving a polygon for a given distance along a specified direction. 4 Translating and rotating the polygons is made easy as we define the parts in a coordinate free manner. We have used an algorithm similar to that proposed by Avnaim and Boissonnat [11], which computes all the placements for a polygonal object I (with m edges) which is free to rotate translate but not to intersect another polygonal object E. In order to achieve motion of a polygon in a given direction under the constraint that it does not intersect with any other polygons, we ....

Avnaim, F. and Boissonnat, Jean-Daniel, Polygon placement under translation and rotation, Theoretical Informatics and Applications, vol. 23, pp 5-28, 1989.

....and N00014 89 J1988. Figure 1: An example pattern and background. In this case there is match of the pattern to the background under Euclidean motion for = 0:1 in. 1. Introduction The problem of determining whether a given planar geometric set can be found in another planar geometric set [2, 3, 4, 5, 6, 7, 8, 11, 19, 20, 21, 22] is central to computer vision and also appears in such diverse fields as biology, astronomy, and robotics. For example, a typical computational problem in astronomy is to locate a certain configuration of stars in the night sky (see Figure 1) This problem can be modeled as a pattern matching ....

....at endpoints) line segments. Our approach will be similar to that of the previous section. Recall that our goal is to determine if there is a Euclidean motion g such that g(A) lies entirely within B . The well versed reader may suspect that we could use a result of Avnaim and Boissonnat [8] to solve this problem, but this does not seem possible. They solve the problem of placing a general polygonal shape P with m sides and corners, in another general polygonal shape Q, with n sides and corners, when P is allowed to translate and rotate. Let us call Q the environment. Avnaim and ....

[Article contains additional citation context not shown here]

F. Avnaim and J.-D. Boissonnat, "Polygon placement under translation and rotation," Proc. 5th Symp. Theoret. Aspects Comput. Sci., Lecture Notes in Computer Science, 294, Springer-Verlag, 1988, 322--333.

....where a composite probe comprises in parallel several supporting line probes or finger probes. The polygon containment problem, that is, deciding whether an n gon P can fit into an m gon Q under translations and or rotations, has been studied by various researchers in computational geometry. See [5], 6] 12] 23] In the case where Q is convex, the best known algorithm runs in time O(m 2 n) when both translations and rotations are allowed [12] Here we will deal with a special case of containment in which each edge of Q must touch P ; this constraint causes a reduction of the running ....

Francis Avnaim and Jean Daniel Boissonnat. Polygon placement under translation and rotation. Technical Report 889, INRIA Sophia-Antipolis, 1988.

....3D solids. We expect the same to be true when rotation is added. Chazelle [5] introduced the single polygon containment problem: place m gon P into n gon container Q. For convex Q, the running time bound is O(mn 2 ) and for nonconvex P and Q, O(m 3 n 3 (m n) log(m n) Avnaim et al. [4, 3] improve this to O(m 3 n 3 log(m n) Most recent work deals with finding the largest copy of a convex P that can be placed, which is equivalent to finding the minimum (scaled) enclosure. For convex Q the best running time is O(mn 2 log n) 1] and for non convex Q, O(m 2 n 2 ) 2] ....

F. Avnaim and J. Boissonnat. Polygon placement under translation and rotation. In Proceedings of the 5th Annual Symposium on Theoretical Aspects of Computer Science; Lecture Notes in Computer Science (294), pages 322--333. Springer-Verlag, 1988.

....Chazelle [6] studied several variants of this problem. One of his results is that, given two convex polygons P and Q, one can decide in linear time whether Q can be translated such that it is contained in P . Other papers compute the largest copy of a polygon that can be placed inside another one [5, 10, 11, 20]. 2 The number of distinct placements Let P be a simple polygon with n vertices in the plane and let Q be a simple polygon with m vertices. The position and orientation of P are fixed, but Q is free to translate. In this section we bound the number of distinct placements of Q with respect to P . ....

F. Avnaim and J.-D. Boissonnat. Polygon placement under translation and rotation. In Proc. 5th Sympos. Theoret. Aspects Comput. Sci., volume 294 of Lecture Notes in Computer Science, pages 322--333. Springer-Verlag, 1988.

....The derived contact equations can be used to determine the curvature of an unknown object at a point of contact. The effects of friction on two objects in contact has been analyzed extensively in [Goy89] There are other significant contributions to planning for manipulation, of which we note [PS88, AB89, Fea86]. In the domain of grasp planning, grasp analysis with respect to such properties as equilibrium, force closure, form closure, and positivity has been presented in [MNP90, Ngu86, KMY92, MSS87, Pon93] Grasp planning algorithms that treat uncertainty include [Bro88, Gol90] A survey of the grasping ....

F. Avnaim and J. D. Boissonnat, Polygon placement under translation and rotation, in Informatique Theorique et Applications, pp 5-28, 1989.

....on the Minkowski sum, we represent the space of contact free configurations for A in B as the complementary set of Sym(A) Phi B where Sym(A) is the symmetric of A with respect to the origin. Early works have treated the case of linear polygons whose boundaries are only made of line segments [AB] or arcs of circles [JPL] and the case of curved polygons restricted to translations [JJR] We propose a generalization of both cases allowing rotation translation to convex curved polygons. More precisely, we present a theorical study as a fondation for an efficient algorithm to compute the ....

F. Avnaim et J.D. Boissonnat, 1989 Polygon placement under translation and rotation, Informatique Th'eorique et Applications (vol.23, n o 1, 1989, p.5 `a 28).

....execute the task autonomously by robot manipulators. One of the problems is the motion planning of objects in contact, which plays an important role in the mechanical assembly. Avnaim and Boissonnat studied the problem in which a polygon translates and rotates among polygonal obstacles in 2 space [1], and developed the algorithm to give the explicit representation of the obstacles in the configuration space of the moving polygon, where the configuration space is defined by the space whose elements are all possible configurations. The obstacle in the configuration space is called the ....

F.Avnaim and J.D.Boissonnat, "Polygon Placement under Translation and Rotation", INRIA TR-No.889, 1988.

....connected workspace obstacle and then take the union of all constructed boundaries to find the boundary of all C obstacles. Path planners operating in discretized C spaces feed the results produced by these methods to routines that build the C 1 This method is an adaptation of the algorithm in [1] which constructs an exact representation of a C obstacle in the case where A can translate and rotate in the plane. space bitmap. For example, Latombe [11] and Lengyel et al. 14] used the O(nA n B ) algorithm of Lozano P erez and Guibas to compute the vertices of the C obstacles and then ....

F. Avnaim, J.D. Boissonnat, Polygon Placement Under Translation and Rotation, Technical Report No. 890, INRIA, 1988.

....sheet metal [35] and leather [20] both of which permit rotations. Chazelle [6] introduced the single polygon containment problem: place m gon P into n gon container Q. For convex Q, the running time bound is O(mn 2 ) and for nonconvex P and Q, O(m 3 n 3 (m n) log(m n) Avnaim et al. [4, 3] improve this to O(m 3 n 3 log(m n) Most recent work deals with finding the largest copy of a convex P that can be placed, which is equivalent to finding the minimum (scaled) enclosure. For convex Q the best running time is O(mn 2 log n) 1] and for non convex Q, O(m 2 n 2 ) 2] ....

....This repetition (cascading) can lead to numerical difficulties which we deal with using nearest pair rounding (Section 4) Note: for k = 2, geometric restriction is an exact algorithm. In this special case, it becomes essentially the same as Avnaim and Boissonnat s exact formula for this case [4, 3]. If either geometric restriction or linear programming restriction generates the empty set, then there is no solution in the current hypothesis. 2.4 Evaluation. We have previously developed a practical algorithm for rotational overlap minimization [32] find translations and rotations that ....

F. Avnaim and J. Boissonnat. Polygon placement under translation and rotation. In Proceedings of the 5th Annual Symposium on Theoretical Aspects of Computer Science; Lecture Notes in Computer Science (294), pages 322--333. Springer-Verlag, 1988.

....and N00014 89 J 1988. Figure 1: An example pattern and background. In this case there is match of the pattern to the background under Euclidean motion for = 0:1 in. 1. Introduction The problem of determining whether a given planar geometric set can be found in another planar geometric set [2, 3, 4, 5, 6, 7, 8, 11, 19, 20, 21, 22] is central to computer vision and also appears in such diverse fields as biology, astronomy, and robotics. For example, a typical computational problem in astronomy is to locate a certain configuration of stars in the night sky (see Figure 1) This problem can be modeled as a pattern matching ....

....at endpoints) line segments. Our approach will be similar to that of the previous section. Recall that our goal is to determine if there is a Euclidean motion g such that g(A) lies entirely within B . The well versed reader may suspect that we could use a result of Avnaim and Boissonnat [8] to solve this problem, but this does not seem possible. They solve the problem of placing a general polygonal shape P with m sides and corners, in another general polygonal shape Q, with n sides and corners, when P is allowed to translate and rotate. Let us call Q the environment. Avnaim and ....

[Article contains additional citation context not shown here]

F. Avnaim and J.-D. Boissonnat, "Polygon placement under translation and rotation," Proc. 5th Symp. Theoret. Aspects Comput. Sci., Lecture Notes in Computer Science, 294, Springer-Verlag, 1988, 322--333.

....= z in the admissible space. So, we are actually looking for the intersection of A(O;P) with Z . 3. 2 Gaussian Diagrams of Convex Polyhedra We shall dene a mapping of a convex polyhedron P onto the unit sphere of IR 3 denoted S 2 that is called the normal or Gaussian diagram (see [LP83, GS87, AB89, AB87] which can be used to calculate a Minkowski sum and to bound its complexity. On S 2 , the vertical direction is determined by the vector normal to the wall. Any plane containing this vector is said to be vertical, and any plane orthogonal to this vertical direction is called ....

....of signs of determinants which is more costly than AEoating point arithmetic, it is a lot faster. We also implemented another method for computing the Minkowski sum. It consists in computing all vertices vA vB for all mn pairs of vertices v A 2 A; vB 2 B and then calculating their convex hull [AB89] with the Quick Hull algorithm [BDH93] This algorithm also uses AEoating point arithmetic. The performance were even poorer. Instrument N p N v Nm K N s K s WALL 54 837 ESRI 1 3 32 162 30.8 44 11.0 ESRI 2 3 32 171 30.3 46 10.8 HORN 7 HST 7 188 420 51.3 146 11.9 SVS ANTENNA 10 727 ....

F. Avnaim and J.-D. Boissonnat. Polygon placement under translation and rotation. RAIRO Inform. Theor., 23:528, 1989.

....Z = z in the admissible space so we are actually looking for the intersection of A(O;P) with Z. 3. 2 Gaussian diagrams of Convex Polyhedra We shall dene a mapping of a convex polyhedron P onto the unit sphere of IR 3 denoted S 2 that is called the normal or Gaussian diagram (see [LP83, GS87, AB89, Avn89, CEGS93] and which can be used to calculate a Minkowski sum and bound its complexity. For each facet f 2 P , let n f denote its normal vector. The Gaussian diagram P 0 on the unit sphere S 2 consists of the following elements (we intentionally use another terminology to avoid ....

....and is therefore much more robust than the convolution method, it is a lot faster. In fact we also implemented the method that consists, for computing the Minkowski sum, in computing all vertices v A v B for all mn pairs of vertices v A 2 A; v B 2 B and then calculating their convex hull [AB89, Avn89] with the Quick Hull algorithm [BDH93] The performances were even poorer. Equipment N p N v Nm K N s K s INIT 54 837 ESRI 1 3 32 162 30.8 44 11.0 ESRI 2 3 32 171 30.3 46 10.8 HORN 7 HST 7 188 420 51.3 146 11.9 SVS ANTENNA 10 727 670 144.5 257 22.3 VOGO 10 296 770 91.5 161 ....

F. Avnaim and J.-D. Boissonnat. Polygon placement under translation and rotation. RAIRO Inform. Theor., 23:528, 1989.

....z in the admissible space so we are actually looking for the intersection of A(O;P) with Z . 3. 2 Gaussian diagrams of Convex Polyhedra We shall de ne a mapping of a convex polyhedron P onto the unit sphere of IR 3 denoted S 2 that is called the normal or Gaussian diagram (see [LP83, GS87, AB89, AB87] and which can be used to calculate a Minkowski sum and bound its complexity. For each facet f 2 P , let n f denote its normal vector. The Gaussian diagram P 0 on the unit sphere S 2 consists of the following elements (we intentionally use another terminology to avoid confusion ....

....sections with Z . much more robust than the convolution method, it is a lot faster. In fact we also implemented the method that consists, for computing the Minkowski sum, in computing all vertices vA v B for all mn pairs of vertices vA 2 A; v B 2 B and then calculating their convex hull [AB89] with the Quick Hull algorithm [BDH93] The performances were even poorer. 6 Conclusions We have proposed an optimal output sensitive algorithm to compute a section of a three dimensional Minkowski sum by a plane without computing the whole sum. Degeneracies and numerical problems were solved. ....

F. Avnaim and J.-D. Boissonnat. Polygon placement under translation and rotation. RAIRO Inform. Theor., 23:528, 1989.

No context found.

Francis Avnaim and Jean Daniel Boissonnat. Polygon placement under translation and rotation. Technical Report 889, INRIA Sophia-Antipolis, 1988. 145

No context found.

F. Avnaim and J. D. Boissonnat, "Polygon placement under translation and rotation," Institut National de Recherche en Informatique et en Automatique, Le Chesnay Cedex, France, INRIA Report #899, August 1988.

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