Arrangements of planar curves are fundamental structures in computational geometry. Recently, the arrangement package of Cgal, the Computational Geometry Algorithms Library, has been redesigned and re-implemented exploiting several advanced programming techniques. The resulting software package, which constructs and maintains planar arrangements, is easier to use, to extend, and to adapt to a variety of applications. It is more efficient space- and time-wise, and more robust. The implementation is complete in the sense that it handles degenerate input, and it produces exact results. In this paper we describe how various programming techniques were used to accomplish specific tasks within the context of computational geometry in general and Arrangements in particular. These tasks are exemplified by several applications, whose robust implementation is based on the arrangement package. Together with a set of benchmarks they assured the successful application of the adverted programming techniques. 1

We show how to compute and maintain the two-dimensional arrangement on a quadric that is induced by intersection curves with other quadrics. The key idea is to parameterize the quadric by two variables, which then allows to implicitly compute the arrangement in a modified parameter space. We give details of a possible parameterization and explain how to implement the needed geometric and topological predicates.

Probabilistic and deterministic planners are two major approximate-based frameworks for solving motion planning problems. Both approaches have their own advantages and disadvantages. In this work, we provide an investigation to the following question: Is there a planner that can take the advantages from both probabilistic and deterministic planners? Our strategy to achieve this goal is to use the point-based Minkowski sum of the robot and the obstacles in workspace. Our experimental results show that our new method, called M-sum planner, which uses the geometric properties of Minkowski sum to solve motion planning problems, provides advantages over the existing probabilistic or deterministic planners. In particular, M-sum planner is significantly more efficient than the Probabilistic Roadmap Methods (PRMs) and its variants for problems that can be solved by reusing configurations.

The offset surfaces to non-developable quadratic triangular Bézier patches are rational surfaces. In this paper we give a direct proof of this result and formulate an algorithm for computing the parameterization of the offsets. Based on the observation that quadratic triangular patches are capable of producing C 1 smooth surfaces, we use this algorithm to generate rational approximations to offset surfaces of general free–form surfaces.

We describe the algorithms and implementation details involved in the concretizations of a generic framework that enables exact construction, maintenance, and manipulation of arrangements embedded on certain two-dimensional orientable parametric surfaces in three-dimensional space. The fundamentals of the framework are described in a companion paper. Our work covers arrangements embedded on elliptic quadrics and cyclides induced by intersections with other algebraic surfaces, and a specialized case of arrangements induced by arcs of great circles embedded on the sphere. We also demonstrate how such arrangements can be used to accomplish various geometric tasks efficiently, such as computing the Minkowski sums of polytopes, the envelope of surfaces, and Voronoi diagrams embedded on parametric surfaces. We do not assume general position. Namely, we handle degenerate input, and produce exact results in all cases. Our implementation is realized using Cgal and, in particular, the package that provides the underlying framework. We have conducted experiments on various data sets, and documented the practical efficiency of our approach.

We present a tight bound on the exact maximum complexity of Minkowski sums of convex polyhedra in R³. In particular, we prove that the maximum number of facets of the Minkowski sum of two convex polyhedra with m and n facets respectively is bounded from above by f(m, n) = 4mn − 9m − 9n + 26. Given two positive integers m and n, we describe how to construct two convex polyhedra with m and n facets respectively, such that the number of facets of their Minkowski sum is exactly f(m, n). We generalize the construction to yield a lower bound on the maximum complexity of Minkowski sums of many convex polyhedra in R3. That is, given k positive integers m1, m2,..., mk, we describe how to construct k convex polyhedra with corresponding number of facets, such that the number of facets of their Minkowski sum is � 1≤i<j≤k (2mi − 5)(2mj − 5) + � � k � 2 + 1≤i≤k mi. We also provide a conservative upper bound for the general case. Snapshots of several preconstructed convex polyhedra, the Minkowski sum of which is maximal, are available at http://www.cs.tau.ac.il / ~ efif/Mink. The polyhedra models and an interactive program that computes their Minkowski sums and visualizes them can be downloaded as well.

We present an exact implementation of an efficient algorithm that computes Minkowski sums of convex polyhedra in R 3. Our implementation is complete in the sense that it does not assume general position, namely, it can handle degenerate input, and produces exact results. Our software also includes applications of the Minkowski-sum computation to answer collision and proximity queries about the relative placement of two convex polyhedra in R 3. The algorithms use a dual representation of convex polyhedra, and their implementation is mainly based on the Arrangement package of Cgal, the Computational Geometry Algorithm Library. We compare our Minkowski-sum construction with a naïve approach that computes the convex hull of the pairwise sums of vertices of two convex polyhedra. Our method is significantly faster. The video demonstrates the techniques used on simple cases as well as on degenerate cases. The relevant programs, source code, data sets, and documentation are available at

This paper presents an easy-to-implement and efficient method to calculate the Minkowski Sums of simple convex objects. The method is based on direct geometrical manipulation of planes in 3D space. The paper also explains the translational and topological invariance properties of Minkowski sums, their use in distance calculations and presents some performance results.

Abstract: Computing the Minkowski sum of two polyhedra exactly has been shown difficult. Despite its fundamental role in many geometric problems in robotics, to the best of our knowledge, no 3-d Minkowski sum software for general polyhedra is available to the public. One of the main reasons is the difficulty of implementing the existing methods. There are two main approaches for computing Minkowski sums: divide-and-conquer and convolution. The first approach decomposes the input polyhedra into convex pieces, computes the Minkowski sums between a pair of convex pieces, and unites all the pairwise Minkowski sums. Although conceptually simple, the major problems of this approach include: (1) The size of the decomposition and the pairwise Minkowski sums can be extremely large and (2) robustly computing the union of a large number of components can be very tricky. On the other hand, convolving two polyhedra can be done more efficiently. The resulting convolution is a superset of the Minkowski sum boundary. For non-convex inputs, filtering or trimming is needed. This usually involves computing (1) the arrangement of the convolution and its substructures and (2) the winding numbers for the arrangement

Project co-funded by the European Commission within FP6 (2002–2006) We present a tight bound on the exact maximum complexity of Minkowski sums of convex polyhedra in R3. In particular, we prove that the maximum number of facets of the Minkowski sum of two convex polyhedra with m and n facets respectively is bounded from above by f(m, n) = 4mn − 9m − 9n + 26. Given two positive integers m and n, we describe how to construct two convex polyhedra with m and n facets respectively, such that the number of facets of their Minkowski sum is exactly f(m, n). We generalize the construction to yield a lower bound on the maximum complexity of Minkowski sums of many convex polyhedra in R3. That is, given k positive integers m1, m2,..., mk, we describe how to construct k convex polyhedra with corresponding number of facets, such that the number of facets of their Minkowski sum is ∑