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Advanced Programming Techniques Applied to Cgal’s Arrangement Package
, 2007
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Code flexibility and program efficiency by genericity: Improving cgal’s arrangements
 In Proc. 12th Annu. Euro. Sympos. Alg
, 2004
"... 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 impr ..."
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Cited by 17 (13 self)
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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 genericprogramming techniques, which make dexterous use of the compilation process. To the contrary, we expedited key operations as we demonstrate by experiments. 1
Generalized penetration depth computation
 In SPM ’06: Proceedings of the 2006 ACM symposium on Solid and physical modeling
, 2006
"... Penetration depth (PD) is a distance metric that is used to describe the extent of overlap between two intersecting objects. Most of the prior work in PD computation has been restricted to translational PD, which is defined as the minimal translational motion that one of the overlapping objects must ..."
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Cited by 11 (2 self)
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Penetration depth (PD) is a distance metric that is used to describe the extent of overlap between two intersecting objects. Most of the prior work in PD computation has been restricted to translational PD, which is defined as the minimal translational motion that one of the overlapping objects must undergo in order to make the two objects disjoint. In this paper, we extend the notion of PD to take into account both translational and rotational motion to separate the intersecting objects, namely generalized PD. When an object undergoes rigid transformation, some point on the object traces the longest trajectory. The generalized PD between two overlapping objects is defined as the minimum of the longest trajectories of one object under all possible rigid transformations to separate the overlapping objects. We present three new results to compute generalized PD between polyhedral models. First, we show that for two overlapping convex polytopes, the generalized PD is same as the translational PD. Second, when the complement of one of the objects is convex, we pose the generalized PD computation as a variant of the convex containment problem and compute an upper bound using optimization techniques. Finally, when both the objects are nonconvex, we treat them as a combination of the above two cases, and present an algorithm that computes a lower and an upper bound on generalized PD. We highlight the performance of our algorithms on different models that undergo rigid motion in the 6dimensional configuration space. Moreover, we utilize our algorithm for complete motion planning of polygonal robots undergoing translational and rotational motion in a plane. In particular, we use generalized PD computation for checking path nonexistence.
Finding Large Sticks and Potatoes in Polygons
, 2006
"... We study a class of optimization problems in polygons that seek to compute the "largest" subset of a prescribed type, e.g., a longest line segment ("stick") or a maximumarea triangle or convex body ("potato"). Exact polynomialtime algorithms are known for some of thes ..."
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Cited by 5 (0 self)
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We study a class of optimization problems in polygons that seek to compute the "largest" subset of a prescribed type, e.g., a longest line segment ("stick") or a maximumarea triangle or convex body ("potato"). Exact polynomialtime algorithms are known for some of these problems, but their time bounds are high (e.g., O(n^7) for the largest convex polygon in a simple ngon). We devise efficient approximation algorithms for these problems. In particular, we give nearlinear time algorithms for a (1&epsilon;)approximation of the biggest stick, an O(1)approximation of the maximumarea convex body, and a O(1&epsilon;)approximation of the maximumarea fat triangle or rectangle. In addition, we give efficient methods for computing large ellipses inside a polygon (whose vertices are a dense sampling of a closed smooth curve). Our algorithms include both deterministic and randomized methods, one of which has been implemented (for computing large area ellipses in a well sampled closed smooth curve).
Approximate Pyramidal Shape Decomposition
"... A shape is pyramidal if it has a flat base with the remaining boundary forming a height function over the base. Pyramidal shapes are optimal for molding, casting, and layered 3D printing. However, many common objects are not pyramidal. We introduce an algorithm for approximate pyramidal shape deco ..."
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Cited by 5 (3 self)
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A shape is pyramidal if it has a flat base with the remaining boundary forming a height function over the base. Pyramidal shapes are optimal for molding, casting, and layered 3D printing. However, many common objects are not pyramidal. We introduce an algorithm for approximate pyramidal shape decomposition. The general exact pyramidal decomposition problem is NPhard. We turn this problem into an NPcomplete problem which admits a practical solution. Specifically, we link pyramidal decomposition to the Exact Cover Problem (ECP). Given an input shape S, we develop clustering schemes to derive a set of building blocks for approximate pyramidal parts of S. The building blocks are then combined to yield a set of candidate pyramidal parts. Finally, we employ Knuth’s Algorithm X over the candidate parts to obtain solutions to ECP as pyramidal shape decompositions. Our solution is equally applicable to 2D or 3D shapes, and to shapes with polygonal or smooth boundaries, with or without holes. We demonstrate our algorithm on numerous shapes and evaluate its performance.
Peeling Meshed Potatoes
, 2009
"... We study variants of the potato peeling problem on meshed (triangulated) polygons. Given a polygon with holes, and a triangular mesh that covers its interior (possibly using additional vertices), we want to find a largestarea connected set of triangles of the mesh that is convex, or has some other ..."
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Cited by 2 (1 self)
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We study variants of the potato peeling problem on meshed (triangulated) polygons. Given a polygon with holes, and a triangular mesh that covers its interior (possibly using additional vertices), we want to find a largestarea connected set of triangles of the mesh that is convex, or has some other shaperelated property. In particular, we consider (i) convexity, (ii) monotonicity, (iii) bounded backturn, and (iv) bounded total turning angle. The first three problems are solved in polynomial time, whereas the fourth problem is shown to be NPhard.
Polygon Simplification for LocationBased Services Using Population Density
"... An important group of locationbased services (LBS), including 911 service, rely on the mapping between a user’s location and a service boundary in order to select the appropriate service provider. In such cases, mobile clients can cache the mapping information to reduce service latency and serve ..."
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Cited by 1 (1 self)
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An important group of locationbased services (LBS), including 911 service, rely on the mapping between a user’s location and a service boundary in order to select the appropriate service provider. In such cases, mobile clients can cache the mapping information to reduce service latency and server load. However, caching such a mapping can be burdensome on mobile devices because representing the polygon that defines a service boundary requires a large amount of data. We present GeoPSPD, a polygon simplification algorithm designed for LBS applications. Unlike existing algorithms, GeoPSPD never produces a false positive, is tunable at runtime for the desired balance between target polygon size and area coverage, and optionally takes into account the population density. We demonstrate the efficacy of GeoPSPD using the US state boundary data. For New York, GeoPSPD produces a simplified polygon which is only 3 % of the original size, yet covers 95 % of the original area, and makes the LBS queries 3.17 times faster.
Code Flexibility and Program Eciency by Genericity: Improving Cgal's Arrangements?
"... 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 impr ..."
Abstract
 Add to MetaCart
(Show Context)
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
exibility of the code does not come at the expense of eciency as we mainly use genericprogramming techniques, which make dexterous use of the compilation process. To the contrary, we expedited key operations as we demonstrate by experiments. 1