Exact Minkowski sums of polygons with holes

Baram, Alon; Fogel, Efi; Halperin, Dan; Hemmer, Michael; Morr, Sebastian

doi:10.1016/j.comgeo.2018.06.005

Cited by 11 publications

(8 citation statements)

References 11 publications

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“…In other words, there is no need to compile any library before an application that depends on Cgal and written in C++ is built. 3 On the other hand, when running an application written in Python that depends on Cgal the libraries that contain the bindings must be accessible. Typically, those bindings are generated ahead of time with little knowledge about the application itself.…”

Section: Bindings For Cgalmentioning

confidence: 99%

“…Arrangements are space subdivisions induced by curves and surfaces, which have been intensively studied in discrete and computational geometry [19], and have applications in various domains, from robotics [18] and assembly planning [17] through Geographic Information Systems (GIS) [30] to protein structure determination [24], to mention just a few uses. The arrangements packages of Cgal have been developed since the early days of Cgal, first for planar arrangements and maps [14,33], Boolean operations, and Minkowski sums [3]. Then, envelopes of surfaces in three-dimensions have been added [26].…”

Section: D Arrangement Bindingsmentioning

confidence: 99%

“…3. When developing code in pure C++, there is no Listing 25: Automatic generation of binding for CGAL : : minkowski_sum_2 exploiting SFINAE.…”

mentioning

confidence: 99%

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CGAL Made More Accessible

Goren¹,

Fogel²,

Halperin³

2022

Preprint

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We introduce bindings that enable the convenient, efficient, and reliable use of software modules of Cgal (Computational Geometry Algorithm 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.

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Section: Bindings For Cgalmentioning

confidence: 99%

Section: D Arrangement Bindingsmentioning

confidence: 99%

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CGAL Made More Accessible

Goren¹,

Fogel²,

Halperin³

2022

Preprint

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“…A simple approximated algorithm [26] is proposed that avoids computing 3D arrangement and winding numbers, and reduces the trimming issue that many convolution-based methods might face via collision detection. [27] proposes an exact Minkowski sums for polytopes that contain holes using convolution. In addition, point-based methods avoid convex decomposition which is an expensive step of the decomposition-based methods.…”

Section: B Computations Of Minkowski Sumsmentioning

confidence: 99%

Efficient Path Planning in Narrow Passages for Robots with Ellipsoidal Components

Ruan¹,

Poblete²,

Wu³

et al. 2021

Preprint

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Path planning has long been one of the major research areas in robotics, with PRM and RRT being two of the most effective classes of path planners. Though generally very efficient, these sampling-based planners can become computationally expensive in the important case of "narrow passages". This paper develops a path planning paradigm specifically formulated for narrow passage problems. The core is based on planning for rigid-body robots encapsulated by unions of ellipsoids. The environmental features are enclosed geometrically using convex differentiable surfaces (e.g., superquadrics). The main benefit of doing this is that configuration-space obstacles can be parameterized explicitly in closed form, thereby allowing prior knowledge to be used to avoid sampling infeasible configurations. Then, by characterizing a tight volume bound for multiple ellipsoids, robot transitions involving rotations are guaranteed to be collision-free without traditional collision detection. Furthermore, combining the stochastic sampling strategy, the proposed planning framework can be extended to solving higher dimensional problems in which the robot has a moving base and articulated appendages. Benchmark results show that, remarkably, the proposed framework outperforms the popular sampling-based planners in terms of computational time and success rate in finding a path through narrow corridors and in higher dimensional configuration spaces.

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“…But the local properties cannot be expressed by individual points themselves. Another type of methods is based on convolutions of two bodies, with the fact that Minkowski sum of two solid bodies is the support of the convolution of their indicator functions [19,20,11,21]. Convolution-based methods were also widely applied into Minkowski sum computations between surfaces or curves with algebraic expressions [22,23,24,25,26].…”

Section: Related Workmentioning

confidence: 99%

Closed-Form Minkowski Sums of Convex Bodies with Smooth Positively Curved Boundaries

Ruan,

Chirikjian

2020

Preprint

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This paper proposes a closed-form parametric formula of the Minkowski sum boundary for broad classes of convex bodies in d-dimensional Euclidean space. With positive sectional curvatures at every point, the boundary that encloses each body can be characterized by the surface gradient. The first theorem directly parameterizes the Minkowski sums using the unit normal vector at each body surface. Although simple to express mathematically, such a parameterization is not always practical to obtain computationally. Therefore, the second theorem derives a more useful parametric closed-form expression using the gradient that is not normalized. In the special case of two ellipsoids, the proposed expressions are identical to those derived previously using geometric interpretations. In order to further examine the results, numerical verifications and comparisons of the Minkowski sums between two superquadric bodies are conducted. The application for the generation of configuration space obstacles in motion planning problems is introduced and demonstrated.

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Exact Minkowski sums of polygons with holes

Cited by 11 publications

References 11 publications

CGAL Made More Accessible

CGAL Made More Accessible

Efficient Path Planning in Narrow Passages for Robots with Ellipsoidal Components

Closed-Form Minkowski Sums of Convex Bodies with Smooth Positively Curved Boundaries

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