An algorithm to compute the Minkowski sum outer-face of two simple polygons

Ramkumar, G.

doi:10.1145/237218.237374

Cited by 32 publications

(17 citation statements)

References 6 publications

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“…The main challenge of the convolution-based methods is in trimming edges and facets in the interior of the convolution to obtain the Minkowski sum boundary [15,19,30,3,4,27]. For example, Wein [30] computes the arrangement induced by the convolution and keeps the cells with non-zero winding numbers.…”

Section: Preliminariesmentioning

confidence: 99%

Dynamic Minkowski sums under scaling

Behar¹,

Lien²

2013

Computer-Aided Design

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In many common real-world and virtual environments, there are a significant number of repeated objects, primarily varying in size. Similarly, in many complex machines, there are a significant number of parts which also vary in size rather than shape. This repetition saves in both design and production costs. Recent research in robotics has also shown that exploiting workspace repetition can significantly increase efficiency. In this paper, we address the need to support computation reuse in fundamental operations. To this end, we propose an algorithm to reuse the computation of the Minkowski sum when an object is transformed by uniform scaling. The Minkowski sum is a fundamental operation in many areas of robotics, such as motion planning, computer vision, and mathematical morphology, and has been studied extensively over the last four decades. We present two methods for dynamically updating Minkowski sums under scaling, the first of which updates the sum under uniform scaling of arbitrary polygons and polyhedra, and the second of which updates the sum under non-uniform scaling of convex polyhedra. Ours are the first methods that study the M-sum under this type of transformation. Our results show speed gains between one and two orders of magnitude over recomputing the Minkowski sum from scratch for each repeated object, and we discuss applications for motion planning, CAD, rapid prototyping, and shape decomposition.

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Section: Preliminariesmentioning

confidence: 99%

Dynamic Minkowski sums under scaling

Behar¹,

Lien²

2013

Computer-Aided Design

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“…Ramkumar [16] uses a convolution-based method that detects self-intersections using a ray-shooting method and removes them, leaving only the M-sum boundary. Ramkumar's algorithm is asymptotically slower than existing algorithms of its time though its practical performance is acceptable.…”

Section: Related Workmentioning

confidence: 99%

Fast and robust 2D Minkowski sum using reduced convolution

Behar

Lien

2011

2011 IEEE/RSJ International Conference on Intelligent Robots and Systems

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Abstract-We propose a new method for computing the 2-d Minkowski sum of non-convex polygons. Our method is convolution based. The main idea is to use the reduced convolution and filter the boundary by using the topological properties of the Minkowski sum. The main benefit of this proposed approach is from the fact that, in most cases, the complexity of the complete convolution is much higher than the complexity of the final Minkowski sum boundary. Therefore, the traditional approach often wastes a large portion of the computation on computing the arrangement induced by the complete convolution that is later on thrown away. Our method is designed to specifically avoid this waste of computation. We experimentally demonstrate that the proposed method is more efficient than the existing methods.

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“…Further, the linking edges between sequences can also be removed since their inclusion is to define the correct starting position of the next sequence and they do not represent potential sliding between the polygons. Figure 6c In order to introduce this method, we recall briefly some terms introduced by Ramkumar (1996). A state is a pair consisting of a position s in the plane and a direction of a tracing as being traversed by a car which always faces in the direction of the state it is currently following.…”

Section: Computing the Boundary Of The Nfpmentioning

confidence: 99%

A comprehensive and robust procedure for obtaining the nofit polygon using Minkowski sums

Bennell

Song

2008

Computers & Operations Research

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The nofit polygon an important tool in the area of irregular shape stock cutting problems. It provides efficient handling of the geometric characteristics of the problem. The paper presents a new algorithmic procedure for deriving this tool. AbstractThe nofit polygon is a powerful and effective tool for handling the geometric requirements of solution approaches to irregular cutting and packing problems. Although the concept was first described in 1966, it was not until the early 90s that the general trend of research moved away from direct trigonometry to favour the nofit polygon. Since then, the ability to calculate the nofit polygon has practically become a pre-requisite for researching irregular packing problems. However, realisation of this concept in the form of a robust algorithm is a highly challenging task with few instructive approaches published. In this paper, a procedure using the mathematical concept of Minkowski sums for the calculation of the nofit polygon is presented. The described procedure is more robust than other approaches using Minkowski Sum knowledge and includes details of the removal of internal edges to find holes, slits and lock and key positions. The procedure is tested on benchmark data sets and gives examples of complicated cases. In addition the paper includes a description of how the procedure is modified in order to realise the inner-fit polygon.3

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An algorithm to compute the Minkowski sum outer-face of two simple polygons

Cited by 32 publications

References 6 publications

Dynamic Minkowski sums under scaling

Dynamic Minkowski sums under scaling

Fast and robust 2D Minkowski sum using reduced convolution

A comprehensive and robust procedure for obtaining the nofit polygon using Minkowski sums

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