# Exact Minkowski Sums of Polygons With Holes

@inproceedings{Baram2015ExactMS, title={Exact Minkowski Sums of Polygons With Holes}, author={Alon Baram and Efi Fogel and Dan Halperin and Michael Hemmer and Sebastian Morr}, booktitle={ESA}, year={2015} }

We present an efficient algorithm that computes the Minkowski sum of two polygons, which may have holes. The new algorithm is based on the convolution approach. Its efficiency stems in part from a property for Minkowski sums of polygons with holes, which in fact holds in any dimension: Given two polygons with holes, for each input polygon we can fill up the holes that are relatively small compared to the other polygon. Specifically, we can always fill up all the holes of at least one polygon…

## 2 Citations

The State Interval Estimation Based on Ellipsoid Minkowski Sum and Intersection

- MathematicsRCAE 2019
- 2019

This paper proposes an approach that utilizes an ellipsoid as the envelope of the intersection of multi-ellipsoids, which is applied to the system's interval estimation. To precisely obtain the…

Efficient Path Planning in Narrow Passages via Closed-Form Minkowski Operations

- Computer ScienceArXiv
- 2021

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.

## References

SHOWING 1-10 OF 34 REFERENCES

Exact and Efficient Construction of Planar Minkowski Sums Using the Convolution Method

- Mathematics, Computer ScienceESA
- 2006

An efficient and robust implementation for the construction of Minkowski sums of polygons in R 2 using the convolution of the polygon boundaries, which allows for faster computation of the sum of non-convex polygons.

Polygon decomposition for efficient construction of Minkowski sums

- Mathematics, Computer ScienceComput. Geom.
- 2002

A procedure for simultaneously decomposing the two polygons such that a "mixed" objective function is minimized and there are optimal decomposition algorithms that significantly expedite the Minkowski-sum computation, but the decomposition itself is expensive to compute.

A Simple Method for Computing Minkowski Sum Boundary in 3D Using Collision Detection

- Mathematics, Computer ScienceWAFR
- 2008

The premise of the method is to reduce the trimming problem to the problems of computing 2-d arrangements and collision detection, which are much better understood in the literature, and to maintain the simplicity, intentionally sacrifice the exactness.

Fast and robust 2D Minkowski sum using reduced convolution

- Mathematics, Computer Science2011 IEEE/RSJ International Conference on Intelligent Robots and Systems
- 2011

The main idea is to use the reduced convolution and filter the boundary by using the topological properties of the Minkowski sum to avoid the waste of computation.

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

- Mathematics, Computer ScienceALENEX
- 2006

An exact implementation of an efficient algorithm that computes Minkowski sums of convex polyhedra in R3 that can handle degenerate input, and it produces exact results is presented.

Accurate Minkowski sum approximation of polyhedral models

- Mathematics, Computer Science12th Pacific Conference on Computer Graphics and Applications, 2004. PG 2004. Proceedings.
- 2004

The algorithm decomposes the polyhedral objects into convex pieces, generates pairwise convex Minkowski sums and computes their union by generating a voxel grid, computing signed distance on the grid points and performing isosurface extraction from the distance field.

Exact Minkowksi Sums of Polyhedra and Exact and Efficient Decomposition of Polyhedra into Convex Pieces

- Mathematics, Computer ScienceAlgorithmica
- 2008

This work presents the first exact and robust implementation of the 3D Minkowski sum of two non-convex polyhedra, and implements an efficient decomposition that yields a small number of convex pieces.

A unified computational framework for Minkowski operations

- Computer ScienceComput. Graph.
- 1993

The two Minkowski operations are unify as a single operation by introducing the notion of negative shape, which makes possible further unification since exactly the same computational technique works for convex as well as nonconvex objects.

A Monotonic Convolution for Minkowski Sums

- Mathematics, Computer ScienceInt. J. Comput. Geom. Appl.
- 2007

We present a monotonic convolution for planar regions A and B bounded by line and circular arc segments. The Minkowski sum equals the union of the cells with positive crossing numbers in the…

A GPU-based voxelization approach to 3D Minkowski sum computation

- Mathematics, Computer ScienceSPM '10
- 2010

This work presents a new approach for computing the voxelized Minkowski sum of two polyhedral objects using programmable Graphics Processing Units (GPUs), which is at least one order of magnitude faster than existing boundary representation (B-rep) based algorithms for computing Minkingowski sums of objects with curved surfaces at similar accuracy.