Showing papers on "Minkowski addition published in 2007"

Convolutions and multiplier transformations of convex bodies

Abstract: Rotation intertwining maps from the set of convex bodies in into itself that are continuous linear operators with respect to Minkowski and Blaschke addition are investigated. The main focus is on Blaschke-Minkowski homomorphisms. We show that such maps are represented by a spherical convolution operator. An application of this representation is a complete classification of all even Blaschke-Minkowski homomorphisms which shows that these maps behave in many respects similar to the well known projection body operator. Among further applications is the following result: If an even Blaschke-Minkowski homomorphism maps a convex body to a polytope, then it is a constant multiple of the projection body operator

Exact Minkowksi sums of polyhedra and exact and efficient decomposition of polyhedra in convex pieces

Abstract: We present the first exact and robust implementation of the 3D Minkowski sum of two non-convex polyhedra. Our implementation decomposes the two polyhedra into convex pieces, performs pairwise Minkowski sums on the convex pieces, and constructs their union. We achieve exactness and the handling of all degeneracies by building upon 3D Nef polyhedra as provided by CGAL. The implementation also supports open and closed polyhedra. This allows the handling of degenerate scenarios like the tight passage problem in robot motion planning. The bottleneck of our approach is the union step. We address efficiency by optimizing this step by two means: we implement an efficient decomposition that yields a small amount of convex pieces, and develop, test and optimize multiple strategies for uniting the partial sums by consecutive binary union operations. The decomposition that we implemented as part of the Minkowski sum is interesting in its own right. It is the first robust implementation of a decomposition of polyhedra into convex pieces that yields at most O(r2) pieces, where r is the number of edges whose adjacent facets comprise an angle of more than 180 degrees with respect to the interior of the polyhedron.

Tolerance Analysis and Synthesis by Means of Deviation Domains, Axi-Symmetric Cases

Abstract: The small displacement torsors are generally used for the represeolation of the geometrical deviations. The standardised tolerances can then be translated by a set of inequalities between the components of a deviation torsor. hi me case of cylindrical possible to reduce the space to three dimensions at the maximum instead of six in the general case. Topological operations like the Minkowski sum to carry OUT the domains presented application relates to metro-logic inspection for a specification with maximum material condition on both the toleranced surface and the datum. The second example makes it possible to determine the deviation between two surfaces belonging to two different parts after mating them by two contact features.

Point-Based Minkowski Sum Boundary

Abstract: Minkowski sum is a fundamental operation in many geometric applications, including robotics, penetration depth estimation, solid modeling, and virtual prototyping. However, due to its high computational complexity and several nontrivial implementation issues, computing the exact boundary of the Minkowski sum of two arbitrary polyhedra is generally a difficult task. In this work, we propose to represent the boundary of the Minkowski sum approximately using only points. Our results show that this point-based representation can be generated efficiently. An important feature of our method is its straightforward implementation and parallelization. We also demonstrate that the point-based representation of the Minkowski sum boundary can indeed provide similar functionality as mesh-based representations can. We show several applications in motion planning, penetration depth approximation and modeling.

Convolution of convex valuations

Abstract: We show that the natural “convolution” on the space of smooth, even, translation-invariant convex valuations on a euclidean space V, obtained by intertwining the product and the duality transform of S. Alesker J. Differential Geom. 63: 63–95, 2003; Geom.Funct. Anal. 14:1–26, 2004 may be expressed in terms of Minkowski sum. Furthermore the resulting product extends naturally to odd valuations as well. Based on this technical result we give an application to integral geometry, generalizing Hadwiger’s additive kinematic formula for SO(V) Convex Geometry, North Holland, 1993 to general compact groups \(G \subset O(V)\) acting transitively on the sphere: it turns out that these formulas are in a natural sense dual to the usual (intersection) kinematic formulas.

The distance function from the boundary in a minkowski space

Abstract: Let the space be endowed with a Minkowski structure (that is, is the gauge function of a compact convex set having the origin as an interior point, and with boundary of class ), and let be the (asymmetric) distance associated to . Given an open domain of class , let be the Minkowski distance of a point from the boundary of . We prove that a suitable extension of to (which plays the role of a signed Minkowski distance to ) is of class in a tubular neighborhood of , and that is of class outside the cut locus of (that is, the closure of the set of points of nondifferentiability of in ). In addition, we prove that the cut locus of has Lebesgue measure zero, and that can be decomposed, up to this set of vanishing measure, into geodesics starting from and going into along the normal direction (with respect to the Minkowski distance). We compute explicitly the Jacobian determinant of the change of variables that associates to every point outside the cut locus the pair , where denotes the (unique) projection of on , and we apply these techniques to the analysis of PDEs of Monge-Kantorovich type arising from problems in optimal transportation theory and shape optimization.

Minkowski sum boundary surfaces of 3D-objects

Abstract: Given two objects A and B with piecewise smooth boundary we discuss the computation of the boundary @C of the Minkowski sum A+B. This boundary surface @C is part of the envelope when B is moved by translations defined by vectors [email protected]?A, or vice versa. We present an efficient algorithm working for dense point clouds or for triangular meshes. Besides this the global self-intersections of the boundary @C are detected and resolved. Additionally we point to some relations between Minkowski sums and kinematics, and compute local quadratic approximations of the envelope.

f-Vectors of Minkowski Additions of Convex Polytopes

Abstract: The objective of this paper is to present two types of results on Minkowski sums of convex polytopes. The first is about a special class of polytopes we call perfectly centered and the combinatorial properties of the Minkowski sum with their own dual. In particular, we have a characterization of the face lattice of the sum in terms of the face lattice of a given perfectly centered polytope. Exact face counting formulas are then obtained for perfectly centered simplices and hypercubes. The second type of results concerns tight upper bounds for the f-vectors of Minkowski sums of several polytopes.

Regularity and Integration of Set-Valued Maps Represented by Generalized Steiner Points

Abstract: A family of probability measures on the unit ball in $\mathbb{R}^{n}$ generates a family of generalized Steiner (GS-)points for every convex compact set in $\mathbb{R}^{n}$ . Such a “rich” family of probability measures determines a representation of a convex compact set by GS-points. In this way, a representation of a set-valued map with convex compact images is constructed by GS-selections (which are defined by the GS-points of its images). The properties of the GS-points allow to represent Minkowski sum, Demyanov difference and Demyanov distance between sets in terms of their GS-points, as well as the Aumann integral of a set-valued map is represented by the integrals of its GS-selections. Regularity properties of set-valued maps (measurability, Lipschitz continuity, bounded variation) are reduced to the corresponding uniform properties of its GS-selections. This theory is applied to formulate regularity conditions for the first-order of convergence of iterated set-valued quadrature formulae approximating the Aumann integral.

Exact and approximate construction of offset polygons

Abstract: The Minkowski sum of two sets A,[email protected]?R^2, denoted [email protected]?B, is defined as {a+b|[email protected]?A,[email protected]?B}. We describe an efficient and robust implementation of the construction of the Minkowski sum of a polygon in R^2 with a disc, an operation known as offsetting the polygon. Our software package includes a procedure for computing the exact offset of a straight-edge polygon, based on the arrangement of conic arcs computed using exact algebraic number-types. We also present a conservative approximation algorithm for offset computation that uses only rational arithmetic and decreases the running times by an order of magnitude in some cases, while having a guarantee on the quality of the result. The package will be included in the next public release of the Computational Geometry Algorithms Library, Cgal Version 3.3. It also integrates well with other Cgal packages; in particular, it is possible to perform regularized Boolean set-operations on the polygons the offset procedures generate.

The expected convex hull trimmed regions of a sample

Abstract: Given a data set in the multivariate Euclidean space, we study regions of central points built by averaging all their subsets with a fixed number of elements. The averaging of these sets is performed by appropriately scaling the Minkowski or elementwise summation of their convex hulls. The volume of such central regions is proposed as a multivariate scatter estimate and a circular sequence algorithm to compute the central regions of a bivariate data set is described.

Minkowski sums of polytopes

Abstract: Minkowski sums are a very simple geometrical operation, with applications in many different fields. In particular, Minkowski sums of polytopes have shown to be of interest to both industry and the academic world. This thesis presents a study of these sums, both on combinatorial properties and on computational aspects. In particular, we give an unexpected linear relation between the f-vectors of a Minkowski sum and that of its summands, provided these are relatively in general position. We further establish some bounds on the maximum number of faces of Minkowski sums with relation to the summands, depending on the dimension and the number of summands. We then study a particular family of Minkowski sums, which consists in summing polytopes we call perfectly centered with their own duals. We show that the face lattice of the result can be completely deduced from that of the summands. Finally, we present an algorithm for efficiently computing the vertices of a Minkowski sum of polytopes. We show that the time complexity is linear in terms of the output for fixed size of the input, and that the required memory size is independent of the size of the output. We also review various algorithms computing different faces of the sum, comparing their strong and weak points.

A geometric framework for solving subsequence problems in computational biology efficiently

Abstract: In this paper, we introduce the notion of a constrained Minkowski sumwhich for two (finite) point-sets P,Q⊆ R2 and a set of k inequalities Ax≥ b is defined as the point-set (P ⊕ Q)Ax≥ b= x = p+q | ∈ P, q ∈ Q, , Ax ≥ b. We show that typical subsequenceproblems from computational biology can be solved by computing a setcontaining the vertices of the convex hull of an appropriatelyconstrained Minkowski sum. We provide an algorithm for computing such a setwith running time O(N log N), where N=|P|+|Q| if k is fixed. For the special case (P⊕ Q)x1≥ β, where P and Q consistof points with integer x1-coordinates whose absolute values arebounded by O(N), we even achieve a linear running time O(N). Wethereby obtain a linear running time for many subsequence problemsfrom the literature and improve upon the best known running times forsome of them.The main advantage of the presented approach is that it provides a generalframework within which a broad variety of subsequence problems canbe modeled and solved.This includes objective functions and constraintswhich are even more complexthan the ones considered before.

Lattice points in Minkowski sums

Abstract: Fakhruddin has proved that for two lattice polygons P and Q any lattice point in their Minkowski sum can be written as a sum of a lattice point in P and one in Q, provided P is smooth and the normal fan of P is a subdivision of the normal fan of Q. We give a shorter combinatorial proof of this fact that does not need the smoothness assumption on P.

Fuzzy Clustering with Minkowski Distance Functions

Abstract: Distances in the well known fuzzy c-means algorithm of Bezdek (1973) are measured by the squared Euclidean distance. Other distances have been used as well in fuzzy clustering. For example, Jajuga (1991) proposed to use the L1-distance and Bobrowski and Bezdek (1991) also used the L1-distance. For the more general case of Minkowski distance and the case of using a root of the squared Minkowski distance, Groenen and Jajuga (2001) introduced a majorization algorithm to minimize the error. One of the advantages of iterative majorization is that it is a guaranteed descent algorithm, so that every iteration reduces the error until convergence is reached. However, their algorithm was limited to the case of Minkowski parameter between 1 and 2, that is, between the L1-distance and the Euclidean distance. Here, we extend their majorization algorithm to any Minkowski distance with Minkowski parameter greater than (or equal to) 1. This extension also includes the case of the L1-distance. We also investigate how well this algorithm performs and present an empirical application.

Semicontinuous functions and convex sets in C(K) spaces

Abstract: The stability properties of the family ℳ of all intersections of closed balls are investigated in spaces C(K), where K is an arbitrary Hausdorff compact space. We prove that ℳ is stable under Minkowski addition if and only if K is extremally disconnected. In contrast to this, we show that ℳ is always ball stable in these spaces. Finally, we present a Banach space (indeed a subspace of C[0, 1]) which fails to be ball stable, answering an open question. Our results rest on the study of semicontinuous functions in Hausdorff compact spaces.

On the exact maximum complexity of Minkowski sums of convex polyhedra

Abstract: Reference ROSO-CONF-2007-002doi:10.1145/1247069.1247126 URL: http://doi.acm.org/10.1145/1247069.1247126 Record created on 2007-01-08, modified on 2016-08-08

Intersection properties of polyhedral norms

Abstract: We investigate the familyM of intersections of balls in a finite dimensional vector space with a polyhedral norm. The spaces for whichM is closed under Minkowski addition are completely determined. We characterize also the polyhedral norms for which M is closed under adding a ball. A subset P ofM consists of the Mazur sets K, defined by the property that for any hyperplane H not meeting K there is a ball containing K and not meeting H. We characterize the Mazur sets in terms of their normal cones and also as summands of closed balls. As a consequence, we characterize the polyhedral spaces with only trivial Mazur sets as those whose unit ball is indecomposable.

A monotonic convolution for minkowski sums

Abstract: We present a monotonic convolution for planar regions A and B bounded by line and circular arc segments. The Minkowski sum equals the union of the cells with positive crossing numbers in the arrangement of the convolution, as is the case for the kinetic convolution. The monotonic crossing number is bounded by the kinetic crossing number, and also by the maximum number of intersecting pairs of monotone boundary chains, which is typically much smaller. We give a Minkowski sum algorithm based on the monotonic convolution. The running time is O(s + nα(n)log(n) + m2), versus O(s + n2) for the kinetic algorithm, with s the input size and with n and m the number of segments in the kinetic and monotonic convolutions. For inputs with a bounded number of turning points and inflection points, the running time is O(sα(s) log s), versus Ω(s2) for the kinetic algorithm. The monotonic convolution is 37% smaller than the kinetic convolution and its arrangement is 62% smaller based on 21 test pairs.

Rotation invariant Minkowski classes of convex bodies

Abstract: A Minkowski class is a closed subset of the space of convex bodies in Euclidean space ℝ n which is closed under Minkowski addition and non-negative dilatations. A convex body in ℝ n is universal if the expansion of its support function in spherical harmonics contains non-zero harmonics of all orders. If K is universal, then a dense class of convex bodies M has the following property. There exist convex bodies T 1 , T 2 such that M + T 1 = T 2 , and T 1 , T 2 belong to the rotation invariant Minkowski class generated by K . It is shown that every convex body K which is not centrally symmetric has a linear image, arbitrarily close to K , which is universal. A modified version of the result holds for centrally symmetric convex bodies. In this way, a result of S. Alesker is strengthened, and at the same time given a more elementary proof.

On the hardness of minkowski addition and related operations

Abstract: For polytopes P,Q ⊂ Rd we consider the intersection P ∪ Q; the convex hull of the union CH(P ∪ Q); and the Minkowski sum P+Q. We prove that given rational H-polytopes P1,P2,Q it is impossible to verify in polynomial time whether Q=P1+P2, unless P=NP. In particular, this shows that there is no output sensitive polynomial algorithm to compute the facets of the Minkowski sum of two arbitrary H-polytopes even if we consider only rational polytopes. Since the convex hull of the union and the intersection of two polytopes relate naturally to the Minkowski sum via the Cayley trick and polarity, similar hardness results follow for these operations as well.

Some indecomposable polyhedra

Abstract: We complete the classification, in terms of decomposability, of all combinatorial types of polytopes with 14 or fewer edges. Recall that a polytope P is said to be decomposable if it is equal to a Minkowski sum of two polytopes Q and R which are not similar to P. Our main contribution here is to consider the 42 types of polyhedra with 8 faces and 8 vertices. It turns out that 34 of these are always indecomposable, and 5 are always decomposable. The remaining 3 are ambiguous, i.e. each of them has both decomposable and indecomposable geometric realizations. †Dedicated to the memory of Prof. Dr Alexander Moiseevich Rubinov.

Chebyshev sets in hyperspaces over a Minkowski space

Abstract: In this paper we extend our previous results on Ceby sev sets in hyperspaces over a Euclidean n-space to hyperspaces over a Minkowski space.

Unit disk of the smallest self-perimeter in a Minkowski plane

Abstract: On a Minkowski plane with nonsymmetric metric, we find all unit disks with self-perimeter 6, the smallest possible value.

On Summands Properties and Minkowski Subtraction

Abstract: In this paper we generalise the Sallee theorem from [J. Geom. 29 (1987)(1), 1–11, Theorem 4.3] into non-symmetric sets and give its proof in the terms of Minkowski subtraction.

Strong Minkowski Separation and Co-Drop Property

Abstract: In the framework of topological vector spaces, we give a characterization of strong Minkowski separation, introduced by Cheng, et al., in terms of convex body separation. From this, several results on strong Minkowski separation are deduced. Using the results, we prove a drop theorem involving weakly countably compact sets in locally convex spaces. Moreover, we introduce the notion of the co-drop property and show that every weakly countably compact set has the co-drop property. If the underlying locally convex space is quasi-complete, then a bounded weakly closed set has the co-drop property if and only if it is weakly countably compact.