Efi Fogel

arXiv (Cornell University), Feb 28, 2022

We introduce Python bindings that enable the convenient, efficient, and reliable use of software ... more We introduce Python bindings that enable the convenient, efficient, and reliable use of software modules of Cgal (The Computational Geometry Algorithms Library), which are written in C++, from within code written in Python. There are different tools that facilitate the creation of such bindings. We present a short study that compares three main tools, which leads to the tool of choice. The implementation of algorithms and data structures in computational geometry presents tremendous difficulties, such as obtaining robust software despite the use of (inexact) floating point arithmetic, found in standard hardware, and meticulous handling of all degenerate cases, which typically are in abundance. The code of Cgal extensively uses function and class templates in order to handle these difficulties, which implies that the programmer has to make many choices that are resolved during compile time (of the C++ modules). While bindings take effect at run time (of the Python code), the type of the C++ objects that are bound must be known when the bindings are generated, that is, when they are compiled. The types of the bound objects are instances (instantiated types) of C++ function and class templates. The number of object types that can potentially be bound, in implementation of generic computational-geometry algorithms, is enormous; thus, the generation of the bindings for all these types in advance is practically impossible. For example, the programmer needs to choose among a dozen types of curves (e.g., line segments, circular arcs, geodesic arcs on a sphere, or polycurves of any curve type) to yield a desired arrangement type; often there are several choices to make, resulting in a prohibitively large number of combinations. We present a system that rapidly generates bindings for desired object types according to user prescriptions, which enables the convenient use of any subset of bound object types concurrently. After many years, in which the usage of these packages was restricted to C++ experts, the introduction of the bindings made them easily accessible to newcomers and practitioners in non-computing fields, as we report in the paper. Additional information can be found at http://acg.cs.tau.ac.il/cgal-python-bindings. The bindings software can be found at https://bitbucket.org/taucgl/cgal-python-bindings.

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Optimized Synthesis of Snapping Fixtures

arXiv (Cornell University), Sep 12, 2019

Fixtures for constraining the movement of parts have been extensively investigated in robotics, s... more Fixtures for constraining the movement of parts have been extensively investigated in robotics, since they are essential for using robots in automated manufacturing. This paper deals with the design and optimized synthesis of a special type of fixtures, which we call snapping fixtures. Given a polyhedral workpiece P with n vertices and of constant genus, which we need to hold, a snapping fixture is a semi-rigid polyhedron G, made of a palm and several fingers, such that when P and G are well separated, we can push P toward G, slightly bending the fingers of G on the way (exploiting its mild flexibility), and obtain a configuration, where G is back in its original shape and P and G are inseparable as rigid bodies. We prove the minimal closure conditions under which such fixtures can hold parts, using Helly's theorem. We then introduce an algorithm running in O(n 3) time that produces a snapping fixture, minimizing the number of fingers and optimizing additional objectives, if a snapping fixture exists. We also provide an efficient and robust implementation of a simpler version of the algorithm, which produces the fixture model to be 3D printed and runs in O(n 4) time. We describe two applications with different optimization criteria: Fixtures to hold add-ons for drones, where we aim to make the fixture as lightweight as possible, and small-scale fixtures to hold precious stones in jewelry, where we aim to maximize the exposure of the stones, namely minimize the obscuring of the workpiece by the fixture.

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Exact and Efficient Construction of Minkowski Sums of Convex Polyhedra with Applications

We present an exact implementation of an efficient algorithm that computes Minkowski sums of conv... more 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 it produces exact results. We also present 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 the only three other methods that produce exact results we are aware of. One is a simple approach that computes the convex hull of the pairwise sums of vertices of two convex polyhedra. The second is based on Nef polyhedra embedded on the sphere, and the third is an output sensitive approach based on linear programming. Our method is significantly faster. The results of experimentation with a broad family of convex polyhedra are reported. The relevant programs, source code, data sets, and documentation are available at http://www.cs.tau.ac.il/~efif/CD, and a short movie [16] that describes some of the concepts portrayed in this paper can be downloaded from http://www.cs.tau.ac.il/~efif/CD/Mink3d.avi.

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Proceedings of the twenty-fifth annual symposium on Computational geometry

Symposium on Computational Geometry, Jun 8, 2009

This volume collects the research papers and the short descriptions of the video presentations pr... more This volume collects the research papers and the short descriptions of the video presentations presented during the 25th Annual Symposium on Computational Geometry (SoCG'09), held June 8--10, 2009, at Aarhus University, Denmark. These proceedings also contain a summary of the symposium's invited talk by Robert Lang. The symposium is presented in cooperation with ACM SIGACT and SIGGRAPH, with support from Aarhus University and the Center for Massive Data Algorithmics (MADALGO). A total of 170 papers were submitted in response to the call for papers. The program committee reviewed the submissions carefully, with a distributed effort that included expert input from both the program committee and a host of external reviewers. The committee is very grateful to the external reviewers for the substantial help they provided. The program committee met electronically from January 23 to February 11 for discussions, and finally selected 44 papers for presentation at the symposium. The on-line paper submission process, the review process, and the electronic discussion and decision process were all performed using the free conference management system EasyChair. The papers presented here generally represent preliminary results of ongoing research. We expect that most of these papers will eventually appear in a more fully developed form in refereed journals. In addition, nine videos were submitted in response to the call for videos and multimedia; all nine were accepted for presentation at the symposium by the video and multimedia committee. The web proceedings showcase the revised videos at http://www.computational-geometry.org/.

Web-site with Benchmark Instances for Planar Curve Arrangements

Area Optimal Polygonization Using Simulated Annealing

ACM Journal of Experimental Algorithms, Mar 4, 2022

We describe a practical method to find near-optimal solutions for the area-optimal simple polygon... more We describe a practical method to find near-optimal solutions for the area-optimal simple polygonization problem: Given a set of points S in the plane, the objective is to find a simple polygon of minimum or maximum area defined by S . Our approach is based on the celebrated metaheuristic Simulated Annealing. The method consists of a modular pipeline of steps, where each step can be implemented in various ways and with several parameters controlling it. We have implemented several different algorithms and created an application that computes a polygon with minimal (or maximal) area. We experimented with the various algorithmic options and with the controlling parameters of each algorithm to tune up the pipeline. Then, we executed the application on each of the benchmark instances, exploiting a grid of servers, to obtain near optimal results.

CGAL Arrangements and Their Applications: A Step-by-Step Guide

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Session details: SIGGRAPH Core: CGAL - the computational geometry algorithms library

Sweeping and maintaining two-dimensional arrangements on quadrics

European Workshop on Computational Geometry, 2007

We introduce a general framework for processing a set of curves defined on a continuous two-dimen... more We introduce a general framework for processing a set of curves defined on a continuous two-dimensional parametric surface, while sweeping the parameter space. A major goal of our work is to maximize code reuse in implementing algorithms that employ the prevalent sweep-line paradigm, and consequently to minimize the effort needed to extend the implementation of the paradigm to various surfaces and families of curves embedded on them. We show how the sweep-line paradigm is used to construct an arrangement of curves embedded on an orientable parametric surface, and explain how the arrangement package of Cgal, which previously handled only arrangements of bounded planar curves, is extended to handle curves embedded on a general surface. To the best of our knowledge, this is the first software implementation of generic algorithms that can handle arrangements on general parametric surfaces.

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Optimized Synthesis of Snapping Fixtures

Algorithmic Foundations of Robotics XIV, 2021

Fixtures for constraining the movement of parts have been extensively investigated in robotics, s... more Fixtures for constraining the movement of parts have been extensively investigated in robotics, since they are essential for using robots in automated manufacturing. This paper deals with the design and optimized synthesis of a special type of fixtures, which we call snapping fixtures. Given a polyhedral workpiece P with n vertices and of constant genus, which we need to hold, a snapping fixture is a semi-rigid polyhedron G, made of a palm and several fingers, such that when P and G are well separated, we can push P toward G, slightly bending the fingers of G on the way (exploiting its mild flexibility), and obtain a configuration, where G is back in its original shape and P and G are inseparable as rigid bodies. We prove the minimal closure conditions under which such fixtures can hold parts, using Helly's theorem. We then introduce an algorithm running in O(n 3) time that produces a snapping fixture, minimizing the number of fingers and optimizing additional objectives, if a snapping fixture exists. We also provide an efficient and robust implementation of a simpler version of the algorithm, which produces the fixture model to be 3D printed and runs in O(n 4) time. We describe two applications with different optimization criteria: Fixtures to hold add-ons for drones, where we aim to make the fixture as lightweight as possible, and small-scale fixtures to hold precious stones in jewelry, where we aim to maximize the exposure of the stones, namely minimize the obscuring of the workpiece by the fixture.

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Minkowski Sum Construction and Other Applications of Arrangements

... 2D and 3D Minkowski sums of both convex and non-convex polyhedra based on a ... family of cur... more

2 D Arrangements Efi

Constructs, maintains, modifies, traverses, queries, and presents arrangements on two-dimensional... more Constructs, maintains, modifies, traverses, queries, and presents arrangements on two-dimensional parametric surfaces. Complete and Robust All inputs are handled correctly (including degenerate input). Exact number types are used to achieve robustness. Generic – easy to interface, extend, and adapt Modular – geometric and topological aspects are separated Supports among the others: various point location strategies zone-construction paradigm sweep-line paradigm vertical decomposition overlay computation

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Specification of the traits classes for CGAL arrangements of curves

This document describes the specification of the new and improved traits classes for CGAL arrange... more This document describes the specification of the new and improved traits classes for CGAL arrangements of curves. It is the product of collaboration between TAU in Israel, INRIA in France, and MPI in Germany. Together we were able to tighten the set of requirements of the arrangement traits, that is the interface between arrangements and the primitives they use. The dialog also emphasized the need for a redesign and a new implementation of the sweep-line module. This has been successfully completed recently.

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Annual Scientific Report Board of Governors Meeting the Deutsch Institute 1.1 History and Mission

The Deutsch Institute of Computer Science was founded through the generosity of the Deutsch famil... more The Deutsch Institute of Computer Science was founded through the generosity of the Deutsch family of Buenos Aires, Argentina. The Deutsch Institute of Computer Science supports state-of-the-art research in computer science, performed at Tel-Aviv University and presented at the most prestigious international computer science conferences and workshops. The Deutsch Institute believes that enabling graduate students to attend international conferences, major workshops and summer schools is perhaps the most significant contribution that one can make to advance the goals of both research and education. The Institute therefore encourages and offers financial support to graduate students for attending important international symposia and presenting their results in these forums. In the same spirit of promoting graduate student research and encouraging collaboration, the Deutsch Institute supports graduate students seminars at Tel-Aviv University. In addition, as of the year 2000, an annual...

Placement of map labels

Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere

This research project would not have been possible without the support of many people. I wish to ... more This research project would not have been possible without the support of many people. I wish to express my gratitude to my advisor, Prof. Dan Halperin who was abundantly helpful and offered invaluable assistance, support, insights, and guidance. This sentense is far short of describing the magnitude of positive influence Danny had on this research and this researcher. I would like to thank Ron Wein and Ophir Setter from Tel-Aviv University and Eric Berberich from Max-Planck-Insitut für Informatik, for ongoing and fruitful collaboration. I would also like to thank all other members of the applied computational geometry group at the computer science school in Tel-Aviv University who made the study duration festive. I would like to thanks the members of the Cgal Editorial Board in particular and all members of the Cgal developers' community in general for sharing their wisdom through many useful advises and rich discussions. During the years I participated in the EU-funded projects Ecg (Effective Computational Geometry for Curves and Surfaces; contract No. IST-2000-26473), Movie (Motion Planning in Virtual Environments, contract No. IST-2001-39250) and Acs (Algorithms for Complex Shapes; contract No. IST-006413) projects. I wish to thank all the other participants in these projects, with whom I enjoyed working. Finally, I would like to express my love and gratitude to my beloved families; for their understanding and endless love, through the duration of my studies.

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On the exact maximum complexity of Minkowski sums of convex polyhedra

Proceedings of the twenty-third …, 2007

We present a tight bound on the exact maximum complexity of Minkowski sums of convex polyhedra in... more We present a tight bound on the exact maximum complexity of Minkowski sums of convex polyhedra in R 3. 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 R 3. That is, given k positive integers m1, m2,. .. , m k , we describe how to construct k convex polyhedra with corresponding number of facets, such that the number of facets of their Minkowski sum is P 1≤i<j≤k (2mi − 5)(2mj − 5) +`k 2´+ P 1≤i≤k mi. We also provide a conservative upper bound for the general case. Snapshots of several pre-constructed 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.

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Video: Exact Minkowski sums of convex polyhedra

Proceedings of the 21st ACM Symposium on Computational Geometry (SoCG), 2005

We present an exact implementation of an efficient algorithm that computes Minkowski sums of conv... more 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

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Code flexibility and program efficiency by genericity: Improving cgal’s arrangements

Abstract. Arrangements of planar curves are fundamental structures in computational geometry. We ... more Abstract. Arrangements of planar curves are fundamental structures in computational geometry. We describe the recent developments in the arrangement package of Cgal, the Computational Geometry Algorithms Library, making it easier to use, to extend and to adapt to a variety of applications. This improved flexibility of the code does not come at the expense of efficiency as we mainly use generic-programming techniques, which make dexterous use of the compilation process. To the contrary, we expedited key operations as we demonstrate by experiments. 1

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